Step 1: Identify a point, {eq}(x,y) {/eq} on the graph that is not the origin.

$$m = \frac{y_2 – y_1}{x_2 – x_1} $$. Where {eq}(x_1, y_1) {/eq} and {eq}(x_2, y_2) {/eq} are the origin and the chosen point, respectively.

Step 3: Write the equation of the line in the form {eq}y = mx {/eq}.

The equation for the slope of a line is {eq}m = \dfrac{y_1 – y_2}{x_1 – x_2} {/eq}.

We find ordered pairs along the line by evaluating the equation for different values of {eq}x {/eq}.

Let’s practice writing equations in the form y=mx for lines that pass through the origin by working through 2 examples.

Step 1: From the graph, we identify the point {eq}(1,5) {/eq}.

$$\begin{align*} m &= \dfrac{y-0}{x-0} \\ &= \dfrac{5-0}{1-0} \\ &= \dfrac{5}{1} \\ &= 5 \end{align*} $$. Step 3: Using the slope found in the previous step, the equation of the line is {eq}y = 5x {/eq}.

Write the equation of the line in the graph below.

Step 1: From the graph, we identify the point {eq}(6,2) {/eq}.

$$\begin{align*} m &= \dfrac{y-0}{x-0} \\ &= \dfrac{y}{x} \\ &= \dfrac{2}{6} \\ &= \dfrac{1}{3} \end{align*} $$. Step 3: Now that we have the slope of the line, the equation of the line is {eq}y = \dfrac{1}{3}x {/eq}.

Step 1: Rearrange the given equation into slope intercept form, {eq}y = mx + b {/eq}.

Look at the given graphs, and omit any that do not have this as their y-intercept.

If the slope is positive, the function increases from left to right. If the slope is negative, the function decreases from left to right.

Step 4: If there is still more than one graph as an answer option, check points on each of the remaining graphs to see if they satisfy the given linear equation. If a graph contains a point that does not satisfy the given equation, omit it.

Linear Equation: A linear equation is a mathematical function with a degree of 1, meaning that the highest exponent of the independent variable of the function is 1. We can also think of a linear equation as any equation that can be manipulated to be put in the form {eq}y=mx+b {/eq}, where m and b are constants.

Slope-Intercept Form: The slope-intercept form of a linear equation is {eq}y = mx + b, {/eq} where {eq}m {/eq} is the slope of the equation’s line, and {eq}b {/eq} is the y-intercept of the equation’s line.

The slope can be thought of as the rise (the change in y) over the run (the change in x) from one point to the next on a line.

Select the graph of the equation {eq}y – 2x = 3 {/eq}.

Step 1: Rearrange the given equation into slope intercept form, {eq}y = mx + b {/eq}.

The slope-intercept form of the equation is {eq}y = 2x + 3 {/eq}.

Look at the given graphs, and omit any that do not have this as their y-intercept.

Thus, the y-intercept of our equation is {eq}(0,b) = (0,3) {/eq}. Graphs C and D do not have a y-intercept of {eq}(0,3), {/eq} so we omit them.

Step 3: Note the slope of the function, {eq}m {/eq}, from the equation from step 1. If the slope is positive, the function increases from left to right.

Omit any graphs that do not follow the slope pattern of the given equation.

In our equation, the slope of the function is {eq}m = 2 {/eq}. Since our slope is positive, the function increases from left to right.

Therefore, we omit A.

If a graph contains a point that does not satisfy the given equation, omit it. The single graph that remains as an answer option after this step is the graph of the given linear equation.

We have successfully omitted 3 of the 4 given options, so we conclude that the correct graph of our equation is Graph B.

Select the correct graph of the equation {eq}3y + 2x = 3 {/eq}.

Step 1: Rearrange the given equation into slope intercept form, {eq}y = mx + b {/eq}.

The slope-intercept form of the function is {eq}y = -\dfrac{2}{3}x+1 {/eq}.

Look at the given graphs, and omit any that do not have this as their y-intercept.

In this problem, {eq}b = 1, {/eq} so the y-intercept is {eq}(0,1) {/eq}. Graph B does not have the correct y-intercept, so we omit it.

Step 3: Note the slope of the function, {eq}m {/eq}, from the equation from step 1. If the slope is positive, the function increases from left to right.

Omit any graphs that do not follow the slope pattern of the given equation.

Since the slope is negative, the function decreases from left to right. Graph A rises from left to right, so we omit it.

Step 4: If there is still more than one graph as an answer option, check points on each of the remaining graphs to see if they satisfy the given linear equation. If a graph contains a point that does not satisfy the given equation, omit it.

We have narrowed our selection to C and D. To determine which is the correct graph, we evaluate the function at a value of x, and compare the input and output to the points on the line.

{eq}y = – \dfrac{2}{3}(2)+1 = – \dfrac{4}{3}+1 = -\dfrac{1}{3} {/eq}. We see that the graph of our function should contain the point {eq}\left(2, -\dfrac{1}{3}\right) {/eq} on the line.

Therefore, we omit D, and the correct graph is Graph C.

The slope intercept form of linear equations is an algebraic representation of straight lines: y = mx + b. Use this formula to graph a line for two variables using the X and Y axes on the coordinate plane.

The prominence of these two properties gives this form of linear equation its name. It is the most common way to represent straight lines.

Using this linear equation requires understanding the x and y axes on the coordinate plane. To learn more, read X and Y Axes.

y = mx + b. Where:

For example, 3x – 4. This form allows you to pull important information from the equation quickly.

To find points on the line, enter x-values into the linear equation to calculate the corresponding y-values. Together, the (x, y) coordinates are points on the line.

Hence, (-1, -7) and (2, 2) are two points on the line. Frequently, you’ll want to graph the slope intercept linear equations to see the lines.

Here’s how all the components of the slope intercept form equation come together to create a line on a graph. Consider the previous linear equation: y = 3x – 4.

You can see the points (-1, -7) and (2, 2) that we calculated earlier. I also show the y-intercept point (0, -4).

You’ll see slope values using fractions or decimals. For instance, the following linear equations are equivalent:

The equations y = mx + b and y = b + mx are equivalent because the order does not matter for addition (i.e., it is commutative). Hence, the following equations have the same solutions:

Or you can start with the linear equations and find the slope and intercept values, as shown below: Let’s dig into the coefficients for the slope intercept form a bit more: m and b.

The slope defines the angle of the line on the x and y coordinate plane. Because you multiply m*x, the slope coefficient describes the change in y for every one-unit increase of x.

You can think of one-unit increases in x as moving right along the x-axis, and the slope intercept form equation tells you how y changes with each shift. Hence, positive slope coefficients cause y-values to increase as you move right.

Consequently, the sign of the coefficient determines whether you have an upward or downward slope as you move right. The absolute size of the coefficient determines how much y changes for each one-unit shift.

In other words, larger absolute slopes produce steeper lines. The y-intercept is where the line crosses the y-axis.

The y-intercept point on the line always has an x-value of zero because that falls right on the y-axis. Consequently, when you have an intercept of b, the slope intercept form produces a point on the line at (0, b).

y = m*0 + b = b. For example:

Using the slope intercept form of a linear equation to graph lines is easy. Simply calculate two (x, y) points by entering two x-values into the formula and finding the corresponding y-values.

In fact, it’s even easier than that because you can use m and b to find two points using minimal calculations: (0, b) and (1, b + m). Here are the step-by-step instructions for graphing a line using the slope intercept form:

These lines illustrate various properties, including positive and negative slopes and intercepts, and steeper and shallower angles. To create this graph, I found the two points for each line using the steps above.

Suppose you are given two points on a line and need to find the slope intercept form of the linear equation that fits them. How do you do that.

You just need a bit of algebra.

First, we need to calculate the slope. Slope is the rise over run.

For this calculation, it doesn’t matter which point is 1 vs. 2.

I’ll enter the values for our two points: (1, 6) and (3,10) into the slope formula.

Second, we calculate the intercept. We need to use the m we found before and some algebra.

It doesn’t matter which point you use. Pick the easier one.

So, b = 4. Now, enter the values we found for m and b into y = mx + b.

y = 2x + 4. This article looks at the slope intercept form of a linear equation from an algebraic viewpoint.

To learn more about that, click the following links:.

In mathematics, a linear equation is an equation that may be put in the form a 1 x 1 + … + a n x n + b = 0 , {\displaystyle a_{1}x_{1}+\ldots +a_{n}x_{n}+b=0,} where x 1 , … , x n {\displaystyle x_{1},\ldots ,x_{n}} are the variables (or unknowns), and b , a 1 , … , a n {\displaystyle b,a_{1},\ldots ,a_{n}} are the coefficients, which are often real numbers.

To yield a meaningful equation, the coefficients a 1 , … , a n {\displaystyle a_{1},\ldots ,a_{n}} are required to not all be zero.

The solutions of such an equation are the values that, when substituted for the unknowns, make the equality true.

Often, the term linear equation refers implicitly to this particular case, in which the variable is sensibly called the unknown.

The solutions of a linear equation form a line in the Euclidean plane, and, conversely, every line can be viewed as the set of all solutions of a linear equation in two variables. This is the origin of the term linear for describing this type of equation.

Linear equations occur frequently in all mathematics and their applications in physics and engineering, partly because non-linear systems are often well approximated by linear equations.

All of its content applies to complex solutions and, more generally, to linear equations with coefficients and solutions in any field. For the case of several simultaneous linear equations, see system of linear equations.

A linear equation in one variable x can be written as a x + b = 0 , {\displaystyle ax+b=0,} with a ≠ 0 {\displaystyle a\neq 0}.

A linear equation in two variables x and y can be written as a x + b y + c = 0 , {\displaystyle ax+by+c=0,} with a and b not both 0.

If b ≠ 0, the equation. is a linear equation in the single variable y for every value of x.

This defines a function. The graph of this function is a line with slope − a b {\displaystyle -{\frac {a}{b}}} and y-intercept − c b.

However, in linear algebra, a linear function is a function that maps a sum to the sum of the images of the summands. So, for this definition, the above function is linear only when c = 0, that is when the line passes through the origin.

Each solution (x, y) of a linear equation. may be viewed as the Cartesian coordinates of a point in the Euclidean plane.

Conversely, every line is the set of all solutions of a linear equation.

If b ≠ 0, the line is the graph of the function of x that has been defined in the preceding section. If b = 0, the line is a vertical line (that is a line parallel to the y-axis) of equation x = − c a , {\displaystyle x=-{\frac {c}{a}},} which is not the graph of a function of x.

Similarly, if a ≠ 0, the line is the graph of a function of y, and, if a = 0, one has a horizontal line of equation y = − c b. {\displaystyle y=-{\frac {c}{b}}.}.

In the following subsections, a linear equation of the line is given in each case.

In this case, its linear equation can be written. If, moreover, the line is not horizontal, it can be defined by its slope and its x-intercept x0.

or, equivalently,. These forms rely on the habit of considering a nonvertical line as the graph of a function.

these forms can be easily deduced from the relations. A non-vertical line can be defined by its slope m, and the coordinates x 1 , y 1 {\displaystyle x_{1},y_{1}} of any point of the line.

This equation can also be written. for emphasizing that the slope of a line can be computed from the coordinates of any two points.

A line that is not parallel to an axis and does not pass through the origin cuts the axes into two different points. The intercept values x0 and y0 of these two points are nonzero, and an equation of the line is.

Given two different points (x1, y1) and (x2, y2), there is exactly one line that passes through them. There are several ways to write a linear equation of this line.

If x1 ≠ x2, the slope of the line is y 2 − y 1 x 2 − x 1. {\displaystyle {\frac {y_{2}-y_{1}}{x_{2}-x_{1}}}.} Thus, a point-slope form is.

which is valid also when x1 = x2 (for verifying this, it suffices to verify that the two given points satisfy the equation).

(exchanging the two points changes the sign of the left-hand side of the equation).

There are two common ways for that.

The equation ( y 1 − y 2 ) x + ( x 2 − x 1 ) y + ( x 1 y 2 − x 2 y 1 ) = 0 {\displaystyle (y_{1}-y_{2})x+(x_{2}-x_{1})y+(x_{1}y_{2}-x_{2}y_{1})=0} can be obtained by expanding with respect to its first row the determinant in the equation. Besides being very simple and mnemonic, this form has the advantage of being a special case of the more general equation of a hyperplane passing through n points in a space of dimension n – 1.

A linear equation with more than two variables may always be assumed to have the form. The coefficient b, often denoted a0 is called the constant term (sometimes the absolute term in old books ).

When dealing with n = 3 {\displaystyle n=3} variables, it is common to use x , y {\displaystyle x,\. y} and z {\displaystyle z} instead of indexed variables.

A solution of such an equation is a n-tuple such that substituting each element of the tuple for the corresponding variable transforms the equation into a true equality.

If every variable has a zero coefficient, then, as mentioned for one variable, the equation is either inconsistent (for b ≠ 0) as having no solution, or all n-tuples are solutions.

In the case of three variables, this hyperplane is a plane.

If the coefficients are real numbers, this defines a real-valued function of n real variables.

This is my solution. First, we recognize that the quadratic has roots at $(0,0)$ and $(1,0)$.

$\int_0^1 x-x^2.dx=\left[\frac{x}{2}-\frac{x^3}{3}\right]_0^1$. $=\frac{1}{6}$.

Let $x=a$ be the abscissa of the intersection point of line $y=mx$ and curve $y=x(1-x)$ different from $0$. Then:

$m=(1-a)$. So, the equation of the straight line is $y=(1-a)x$.

$\int_0^a-x^2+ax.dx=\frac{1}{12}$. $\left[-\frac{x^3}{3}+\frac{ax^2}{2}\right]_0^a=\frac{1}{12}$.

$\frac{a^3}{6}=\frac{1}{12}$. $a^3=\frac{1}{2}$.

As earlier determined, the equation of the straight line is $y=(1-a)x$, so $y=(1-\sqrt {\frac{1}{2}})x$. You can simplify further from your last step,.

1 1 What you will learn today 1. Review of slope 2.

How to graph a linear equation in y = mx + b form 4. Slopes of parallel and perpendicular lines 5.

2 Objective: 2.2 Slope and 2.3 Quick Graphs 2 Fun with Slope Slope is the ratio of vertical change to horizontal change – rise over run.

Look at the graph and “count” the vertical change over the horizontal change. 2.

4 Objective: 2.2 Slope and 2.3 Quick Graphs 4 Finding Slope Using the Formula  Example: Find the slope of a line passing through (-3, 5) and (2, 1). 5 Objective: 2.2 Slope and 2.3 Quick Graphs 5 You Try  Find the slope of the line passing through (-2, -4) and (3, -1).

7 Objective: 2.2 Slope and 2.3 Quick Graphs 7 Classifying Lines Using Slope  Example: Without graphing tell whether the line through the given points rises, falls, is horizontal, or is vertical. a.

(2, -1), (2, 5). 8 Objective: 2.2 Slope and 2.3 Quick Graphs 8 Comparing Steepness of Lines Example: Tell which line is steeper: Line 1: through (2,3) and (4,7) or Line 2: through (-1,2) and (4,5).

Perpendicular lines have slopes that are the negative reciprocal of one another (e.g. 2 and – 1/2).

Line 1 through (-3, 3) and (3, 1) Line 2 through (-2,-3) and (2,3) You Try: Line 1 through (1,-2) and (3,-2) Line 2 through (-5,4) and (0,4). 11 Objective: 2.2 Slope and 2.3 Quick Graphs 11 Graphing Using the Slope-Intercept Form y = mx + b is the slope intercept form of a linear equation.

12 Objective: 2.2 Slope and 2.3 Quick Graphs 12 Steps for Graphing in Slope Intercept Form 1. Put the equation in slope-intercept form.

Find the y-intercept and use it to plot the point where the graph crosses the y-axis. 3.

Draw a line through the two points. 13 Objective: 2.2 Slope and 2.3 Quick Graphs 13 Example  Graph y = 3/4x – 2 1.

y-intercept is -2 3. the slope is ¾ 4.

14 Objective: 2.2 Slope and 2.3 Quick Graphs 14 You Try  Graph y = 1/2x + 1. 15 Objective: 2.2 Slope and 2.3 Quick Graphs 15 A Real World Example  You are buying an $1100 computer on layaway.

Step 1: rewrite as a = -50t + 850 Step 2: y-intercept is 850 Step 3: slope is -50 Step 4: connect the dots. 16 Objective: 2.2 Slope and 2.3 Quick Graphs 16 Using the Standard Form to Graph an Equation The standard form of a linear equation is Ax + By = C.

17 Objective: 2.2 Slope and 2.3 Quick Graphs 17 The Steps Step 1: Put the equation in standard form Step 2: Set y equal to zero and solve for x to get the x-intercept. Step 3: Set x equal to zero and solve for y to get the y-intercept.

18 Objective: 2.2 Slope and 2.3 Quick Graphs 18 Example  Graph 2x + 3y = 12. 19 Objective: 2.2 Slope and 2.3 Quick Graphs 19 Horizontal and Vertical Lines  The graph of y = some number is a horizontal line through (0, the number).

20 Objective: 2.2 Slope and 2.3 Quick Graphs 20 Example  Graph y = 3  Graph x = -2. 21 Objective: 2.2 Slope and 2.3 Quick Graphs 21 Homework  Page 79, 18, 22, 24, 26, 27, 32-35 all, 41, 42, 44, 46  Page 86, 16-18 all,20, 26, 34, 37-39 all, 44, 52.

In geometry, the equation of a line can be written in different forms and each of these representations is useful in different ways. The equation of a straight line is written in either of the following methods:

Learn what is the intercept of a line here. Let’s have a look at the slope-intercept form definition.

This form of the linear equation is called the slope-intercept form, and the values of m and c are real numbers. The slope, m, represents the steepness of a line.

The y-intercept, b, of a line, represents the y-coordinate of the point where the graph of the line intersects the y-axis. In this section, you will learn the derivation of the equation of a line in the slope-intercept form.

Here, the distance c is called the y-intercept of the given line L. So, the coordinate of a point where the line L meets the y-axis will be (0, c).

We know that, the equation of a line in point slope form, where (x1, y1) is the point and slope m is: (y – y1) = m(x – x1).

Substituting these values, we get.

y – c = mx. y = mx + c.

Note: The value of c can be positive or negative based on the intercept is made on the positive or negative side of the y-axis, respectively. As derived above, the equation of the line in slope-intercept form is given by:

Here,. (x, y) = Every point on the line.

c = y-intercept of the line. Usually, x and y have to be kept as the variables while using the above formula.

y = m(x – d). Here,.

d = x-intercept of the line. Sometimes, the slope of a line may be expressed in terms of tangent angle such as:

Also, try: Slope Intercept Form Calculator. We can derive the slope-intercept form of the line equation from the equation of a straight line in the standard form as given below:

Ax + By + C = 0. Rearranging the terms as:

⇒y = (-A/B)x + (-C/B). This is of the form y = mx + c.

When we plot the graph for slope-intercept form equation we get a straight line. Slope-intercept is the best form.

We just have to put the x-values and the equation is solved for y. The best part of the slope-intercept form is that we can get the value of slope and the intercept directly from the equation.

Find the equation of the straight line that has slope m = 3 and passes through the point (–2, –5). Solution:

y = mx+c. Given,.

As per the given point, we have.

Hence, putting the values in the above equation, we get.

-5 = -6+c. c = -5 + 6 = 1.

y = 3x+1. Example 2:.

Solution: By the slope-intercept form we know.

y = mx+c. Given,.

As per the given point, we have.

Hence, putting the values in the above equation, we get.

-3 = -2 + c. c = -3+2 = -1.

y = -x-1. Example 3:

(i) y-intercept is -5. (ii) x-intercept is 7/3.

Given, tan θ = 1/2. So, slope = m = tan θ = 1/2.

Equation of the line using slope intercept form is: y = mx + c.

2y = x – 10. x – 2y – 10 = 0.

Equation of slope intercept form with x-intercept is: y = m(x – d).

2y = (3x – 7)/3. 6y = 3x – 7.

In mathematics, the slope or gradient of a line is a number that describes both the direction and the steepness of the line. Slope is often denoted by the letter m.

Slope is calculated by finding the ratio of the “vertical change” to the “horizontal change” between (any) two distinct points on a line. Sometimes the ratio is expressed as a quotient (“rise over run”), giving the same number for every two distinct points on the same line.

The line may be practical – as set by a road surveyor, or in a diagram that models a road or a roof either as a description or as a plan.

A slope with a greater absolute value indicates a steeper line. The direction of a line is either increasing, decreasing, horizontal or vertical.

The rise of a road between two points is the difference between the altitude of the road at those two points, say y1 and y2, or in other words, the rise is (y2 − y1) = Δy. For relatively short distances, where the Earth’s curvature may be neglected, the run is the difference in distance from a fixed point measured along a level, horizontal line, or in other words, the run is (x2 − x1) = Δx.

In mathematical language, the slope m of the line is. The concept of slope applies directly to grades or gradients in geography and civil engineering.

Thus, a 45° rising line has a slope of +1 and a 45° falling line has a slope of −1.

When the curve is given by a series of points in a diagram or in a list of the coordinates of points, the slope may be calculated not at a point but between any two given points. When the curve is given as a continuous function, perhaps as an algebraic expression, then the differential calculus provides rules giving a formula for the slope of the curve at any point in the middle of the curve.

This generalization of the concept of slope allows very complex constructions to be planned and built that go well beyond static structures that are either horizontals or verticals, but can change in time, move in curves, and change depending on the rate of change of other factors. Thereby, the simple idea of slope becomes one of the main basis of the modern world in terms of both technology and the built environment.

The slope of a line in the plane containing the x and y axes is generally represented by the letter m, and is defined as the change in the y coordinate divided by the corresponding change in the x coordinate, between two distinct points on the line. This is described by the following equation:.

Given two points ( x 1 , y 1 ) {\displaystyle (x_{1},y_{1})} and ( x 2 , y 2 ) {\displaystyle (x_{2},y_{2})} , the change in x {\displaystyle x} from one to the other is x 2 − x 1 {\displaystyle x_{2}-x_{1}} (run), while the change in y {\displaystyle y} is y 2 − y 1 {\displaystyle y_{2}-y_{1}} (rise).

The formula fails for a vertical line, parallel to the y {\displaystyle y} axis (see Division by zero), where the slope can be taken as infinite, so the slope of a vertical line is considered undefined.

By dividing the difference in y {\displaystyle y} -coordinates by the difference in x {\displaystyle x} -coordinates, one can obtain the slope of the line:. As another example, consider a line which runs through the points (4, 15) and (3, 21).

For example, consider a line running through points (2,8) and (3,20). This line has a slope, m, of.

The angle θ between −90° and 90° that this line makes with the x-axis is. Consider the two lines: y = −3x + 1 and y = −3x − 2.

They are not the same line. So they are parallel lines.

Consider the two lines y = −3x + 1 and y = x/3 − 2. The slope of the first line is m1 = −3.

The product of these two slopes is −1. So these two lines are perpendicular.

In statistics, the gradient of the least-squares regression best-fitting line for a given sample of data may be written as:. This quantity m is called as the regression slope for the line y = m x + c {\displaystyle y=mx+c}.

This may also be written as a ratio of covariances:. There are two common ways to describe the steepness of a road or railroad.

See also steep grade railway and rack railway.

where angle is in degrees and the trigonometric functions operate in degrees. For example, a slope of 100% or 1000‰ is an angle of 45°.

A third way is to give one unit of rise in say 10, 20, 50 or 100 horizontal units, e.g. 1:10.

For example, steepness of 20% means 1:5 or an incline with angle 11.3°.

The concept of a slope is central to differential calculus. For non-linear functions, the rate of change varies along the curve.

If we let Δx and Δy be the distances (along the x and y axes, respectively) between two points on a curve, then the slope given by the above definition,. is the slope of a secant line to the curve.

For example, the slope of the secant intersecting y = x2 at (0,0) and (3,9) is 3. (The slope of the tangent at x = 3⁄2 is also 3 − a consequence of the mean value theorem.).

Using differential calculus, we can determine the limit, or the value that Δy/Δx approaches as Δy and Δx get closer to zero. it follows that this limit is the exact slope of the tangent.

Therefore, the slope of the tangent is the limit of Δy/Δx as Δx approaches zero, or dy/dx. We call this limit the derivative.

The value of the derivative at a specific point on the function provides us with the slope of the tangent at that precise location. For example, let y = x2.

The derivative of this function is dy⁄dx = 2x. So the slope of the line tangent to y at (−2,4) is 2 ⋅ (−2) = −4.

An extension of the idea of angle follows from the difference of slopes. Consider the shear mapping.

The slope of ( 1 , 0 ) {\displaystyle (1,0)} is zero and the slope of ( 1 , v ) {\displaystyle (1,v)} is v {\displaystyle v}. The shear mapping added a slope of v {\displaystyle v}.

has slope increased by v {\displaystyle v} , but the difference n − m {\displaystyle n-m} of slopes is the same before and after the shear. This invariance of slope differences makes slope an angular invariant measure, on a par with circular angle (invariant under rotation) and hyperbolic angle, with invariance group of squeeze mappings.

The concept of a slope or gradient is also used as a basis for developing other applications in mathematics:.

Step 1: Graph the first equation by plotting the {eq}y {/eq}-intercept and plotting another point using the slope. Connect the two points.

Step 2: Repeat step 1 with the second equation.

There can be one solution, infinitely many solutions (if the two lines are identical – which in slope-intercept form means the lines have the same equation), or no solution (if the two lines are parallel).

A solution to a system of equations is any point that, when substituted into all of the equations, results in a true statement.

The point {eq}(0,b) {/eq} is the {eq}y {/eq}-intercept and {eq}m {/eq} is the slope.

The slope can be thought of as {eq}m = \dfrac{\text{ rise}}{\text{ run}} {/eq} or {eq}m = \dfrac{\text{ change in } y}{\text{ change in } x} {/eq}.

Choose the graph that shows the correct solution to the system of equations:. {eq}y = 4x – 3\\ y = -2x + 3 {/eq}.

Step 1: Graph the first equation by plotting the {eq}y {/eq}-intercept and plotting another point using the slope. Connect the two points.

The {eq}y {/eq}-intercept of the first equation is {eq}(0,-3) {/eq}. The slope is {eq}4 {/eq}, and so we can find another point by moving up {eq}4 {/eq} and to the right {eq}1 {/eq} to get the point {eq}(1,1) {/eq}.

Graph of y = 4x – 3. Looking at the answer choices, our line only appears in choices A and C, so those are the only two possible answers.

Step 2: Repeat step 1 with the second equation.

The slope is {eq}-2 {/eq}, and so we can find another point by moving down {eq}2 {/eq} and to the right {eq}1 {/eq} to get the point {eq}(1,1) {/eq}. Connecting, we have:.

Graph of y = -2x + 3. Looking at the answer choices, this line only appears in choices C and D.

Step 3: The solution to the system of equations, if it exists, is the intersection of the two lines. There can be one solution, infinitely many solutions (if the two lines are identical – which in slope-intercept form means the lines have the same equation), or no solution (if the two lines are parallel).

Based on steps 1 and 2, the solution is shown in graph C. The solution is the point {eq}(1,1) {/eq}.

Choose the graph that shows the correct solution to the system of equations:. {eq}y = 3x – 2\\ y = 3x + 2 {/eq}.

Step 1: Graph the first equation by plotting the {eq}y {/eq}-intercept and plotting another point using the slope. Connect the two points.

Graph of y = 3x – 2. Looking at the answer choices, our line only appears in choices C and D, so those are the only two possible answers.

Step 2: Repeat step 1 with the second equation.

Graph of y = 3x + 2. Looking at the answer choices, this line only appears in choices A and D.

Step 3: The solution to the system of equations, if it exists, is the intersection of the two lines. There can be one solution, infinitely many solutions (if the two lines are identical – which in slope-intercept form means the lines have the same equation), or no solution (if the two lines are parallel).

Based on steps 1 and 2, the solution is shown in graph D. The graphs never intersect and so there is no solution.

The standard form, also known as the general form of linear equations, is Ax + By = C, where A, B, and C are constants. Using the standard form to represent a straight line makes it easier to obtain the y and x-intercepts using substitution.

The slope-intercept form is the most commonly used form and is of the form y = mx + b, where m and b are constants. The benefit of using the slope-intercept is that the slope and y-intercept of the straight line can be obtained directly by observing the equation, as m is the slope, and b is the y-intercept.

As the name suggests, to construct an equation in the point-slope form we require a point on the straight line and its slope. Definition: The point-slope form of a line is expressed using the slope of the line and and point that the line passes through.

The general structure of an equation in the point-slope form is: y – y1 = m*(x – x1). Here (x1,y1) is any arbitrary point on the straight line, and m is the slope/gradient of the straight line.

We previously wrote the equation for a linear function from a graph. Now we can extend what we know about graphing linear functions to analyze graphs a little more closely.

We can see right away that the graph crosses the y-axis at the point (0, 4), so this is the y-intercept. Then we can calculate the slope by finding the rise and run.

To get from this point to the y-intercept, we must move up 4 units (rise) and to the right 2 units (run). So the slope must be:

Substituting the slope and y-intercept into slope-intercept form of a line gives: [latex]y=2x+4[/latex].

Analyze the information for each function. Now we can re-label the lines.

So far we have been finding the y-intercepts of functions: the point at which the graph of a function crosses the y-axis. A function may also have an x-intercept, which is the x-coordinate of the point where the graph of a function crosses the x-axis.

To find the x-intercept, set the function f(x) equal to zero and solve for the value of x. For example, consider the function shown:

Set the function equal to 0 and solve for x. [latex]\begin{array}{l}0=3x – 6\hfill \\ 6=3x\hfill \\ 2=x\hfill \\ x=2\hfill \end{array}[/latex].

Do all linear functions have x-intercepts.

However, linear functions of the form y = c, where c is a nonzero real number are the only examples of linear functions with no x-intercept. For example, y = 5 is a horizontal line 5 units above the x-axis.

The x-intercept of a function is the value of x where f(x) = 0. It can be found by solving the equation 0 = mx + b.

Set the function equal to zero to solve for x. [latex]\begin{array}{l}0=\frac{1}{2}x – 3\\ 3=\frac{1}{2}x\\ 6=x\\ x=6\end{array}[/latex].

A graph of the function is shown below. We can see that the x-intercept is (6, 0) as expected.

Find the x-intercept of [latex]f\left(x\right)=\frac{1}{4}x – 4[/latex]. [latex]\left(16,\text{ 0}\right)[/latex].

There are two special cases of lines on a graph—horizontal and vertical lines. A horizontal line indicates a constant output or y-value.

The change in outputs between any two points is 0. In the slope formula, the numerator is 0, so the slope is 0.

In other words, the value of the function is a constant. This graph represents the function [latex]f\left(x\right)=2[/latex].

A vertical line indicates a constant input or x-value. We can see that the input value for every point on the line is 2, but the output value varies.

Notice that between any two points, the change in the input values is zero. In the slope formula, the denominator will be zero, so the slope of a vertical line is undefined.

This graph represents the line x = 2. The vertical line [latex]x=2[/latex] which does not represent a function.

A horizontal line is a line defined by an equation of the form [latex]f\left(x\right)=b[/latex] where [latex]b[/latex] is a constant. A vertical line is a line defined by an equation of the form [latex]x=a[/latex] where [latex]a[/latex] is a constant.

For any x-value, the y-value is [latex]–4[/latex], so the equation is [latex]y=–4[/latex].

Write the equation of the line graphed below. The constant x-value is 7, so the equation is [latex]x=7[/latex].

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1 TBF 121 – General Mathematics – I Lecture – 4 : Polynomials and Rational Functions Prof. Dr.

2 Polynomial Functions. Let a 0, a 1,.., an an be real numbers.

The numbers a 0, a 1,.., an an are called the coefficients of the polynomial. a 0 : constant term, ai ai : i-th coefficient Zero polynomial : the polynomial with all coefficients equal to zero.

The domain of every polynomial function is ℝ.ℝ. if a n ≠ 0, then is called the leading coefficient of f and n is called the degree of f.

Given a polynomial The degree of f the zero polynomial is defined to be ..

Constant function: f(x) = b, b b ℝ If c c ℝ and f(c) = 0, c is called a root of f. If f and g are polynomials and c is a real number such that f(x)=(x–c)g(x), then c is a root of f.f.

If c  ℝ is a root of f, then there is a polynomial g of degree n–1 such that f(x)=(x–c)g(x). It follows that Linear function: f(x) = mx+b.

Quadratic Function: f(x) = ax 2 +bx+c. a, a, b, b, c  ℝ, a ≠ 0.

Note that roots of a polynomial give its x-intercepts.

x–1 3x23x2 3x 3 – 3x 2 5x 2 –7x–7x +5x 5x 2 – 5x –2x+2 -2 -2x+2 0. 5 Let c  ℝ and k  ℕ.ℕ.

6 For the polynomial f (x) = 3×5 3×5 +2x 2 –7x+2. 7 Constant Functions.

x y (0,0) (–1,b) (0,b) (2,b) f (x) = b. 8 Linear Functions.

f(x) f(x) = mx + b. m, b  ℝ, m ≠ 0.

The identity function f(x) f(x) = x is a linear function: m=1 and b=0. x y (0,0) y = x x y (0,0) y = mx x y (0,0) y = mx+ b (0,b) The graph of any linear function is a line.

9 The graph of the linear function f(x) f(x) = 2x + 4 : x-intercept : f(x) f(x) = 0  x = – 2  (-2, 0). y-intercept : (0, f(0)) = (0, 4) x y (0,0) (-2, 0) (0, 4) f(x) = 2x + 4.

x-intercept and y-intercept coincide: (0,0). x = 1.

x y (0,0) (1,-2) f(x) = -2x. 11 Example.

At one time 100 units of that item cost 200 TL, at another time 150 units cost 275 TL. Find the equation that defines the cost function C.C.

So b = 50 and C(x) = (1.5)x + 50.

A car dealer plans to launch a new model car. He decides to start the launch if the price of a car is over 30000 TL.

Price – suply function is known to be linear. a) Find the equation defining the price – suply function.

a)p(x) = mx + b p(x) = 200x + 30000. b = p(0) = 30000 p(5) = 5m + b = + 30000=31000  m =200 b) 35000 = 200x + 30000  x = 25.

13 Lines in the Plane. x y (0,0) (x,b)(x,b) (0,b) Horizontal Line : y = b constant function x y (0,0) (a,0) (a,y)(a,y) Vertical Line : x= a Not a function.

14 x y (0,0) (0,b) Using similar triangles Inclined Line d slope of dSlope – Intercept FormPoint – Slope Form. 15 Slope – Intercept FormPoint – Slope Form Line with slope m = 3 and y – intercept b = 4:4: Line with slope m = 3 and passing through (1,2): If two points (x1 (x1, y1 y1 ), ( x2 x2, y2 y2 ) of a line are known, the slope of the line is computed as And the equation of the line can be written in sıope-intercept form.

16 Linear Equations. Ax + By = C İs called a linear equation.

The symbols x and y are called the variables. Graph of Ax + By = C : Linear function, inclined line Constant function, horizontal line Not a function, vertical line Let A, B, C be real numbers such that A  0 or B  0.

17 Quadratic Functions. f(x) = ax 2 + bx + c is called a quadratic function.

The function f defined by the equation. 18 Summarizing what we have obtained, The graph of the quadratic function is a parabola.

19 Let us skech the graph of f(x) f(x) = ax 2 + bx + c when a > 1, h > 0, k > 0 : x y (0,0) x y x y (h,0) (h,k) minimum value of f is f(h) = k. Horizontal shift to the right Strech Vertical shift upwards.

21 f(x) = ax 2 + bx + c = a(x-h)2 a(x-h)2 + k, f(h) = k The point (h,k) is called the vertex of the parabola which is the graph of the quadratic function f(x) = ax 2 + bx + c = a(x-h)2 a(x-h)2 + k.k. Thus the shape of the quadratic function f(x) = ax 2 + bx + c is determined by the sign of a,a, the vertex, the x-intercepts and the y-intercept.

f(x) has no maximum value in this case. If a < 0, f(h) = k is the maximum value of f(x) ).

When a > 0 is the lowest point of the parabola and the parabola opens upwards. When a < 0 is the highest point of the parabola and the parabola opens downwards.

22 Example. x y (0,0) (6,3) x-intercept : none y-intercept : f (0) = 21, (0, 21) (0,21) Vertex : (6, 3) Parabola upwards ( a > 0).

x y (0,0) (4,8) x-intercepts : f (x) = 0  -2x 2 + 16x –24 = 0  x = 2, 6  (2, 0), (6, 0) y-intercept : f (0) =- 24, (0, -24) (0,-24) Vertex : (4, 8) Parabola downwards( a < 0) (2, 0) (6, 0). 24 Applications of Linear and Quadratic Functions.

Let the number of subscribers be 1200 + x, x  0. Then the profit of the company is This quadratic function reaches its maximum value for Example.

After 1200, each new subscriber causes a decrease of 0.01 TL in the profit. What should be the number of subscribers for the profit of the company be maximum.

Thus the profit is maximum for 1200 + 400 = 1600 subscribers. Maximum profit is P(400)=25600 TL.

25 Solution. Example.

If the company aims to have a revenue of 150000 TL at the end of the year, how many teapots should be sold and what should be the price of a teapot. The revenue when x teapots are sold: R(x)=xp=(4000–20p)p=4000p–20p 2 Thus for a revenue of 150000 TL,  (p–150)(p–50) = 0 150000 = 4000p–20p 2  p 2 –200p +7500 = 0 The revenue will be 150000 TL if the price is 50 TL per teapot and x = (4000–20.

150) = 1000 teapots are sold.  p = 50 or p = 150.

26 Example. The revenue and cost functions for a company are given as thousand TL, where x denotes the number of items produced and sold.

b) Find the values of x for which the company will have a profit and the values of x for which the company will have a loss. Solution.

27 10 200 400 y x (0,0) 1 15 6 12 PROFİT loss revenue = cost (break even) Break even for 6 or 12 items. Profit in the interval (6,12), that is, the company will have a profit if the number of items produced and sold is more than 6 or less than 12.

28 Look at the profit function: Maximum profit : 36000 TL. (thousand TL).

29 Rational Functions. where p(x) ande d(x) are polynomials.

When learning how to graph lines and linear functions—notably in y=mx+b form—and deal with points on the coordinate plane, it is incredibly important to understand slope and how to use the formula for slope to determine the slope of a line that passes through two known points. This short guide on slope will cover the following topics:

What is the formula for a slope of a line.

In math, the formula for slope is used to determine the steepness of a line that passes through two or more points. One of the most basic characteristics of slopes is that they can be positive (↗ increasing from left to right), negative (↘ decreasing from left to right), zero (↔ a horizontal line), or undefined (↕ a vertical line).

In other words, the formula for slope is the change in the y-coordinates over the change in the x-coordinates. To find the slope of the lines that passes through the points (x1, y1) and (x2,y2), you can use the formula for slope as follows:

In this lesson, you’ll take a look at some oddball slopes. But first, let’s review the basics.

In other words, as a line moves one unit to the right, how many units does it go up or down.

You also know how to identify whether the slope is positive or negative when the line is written in y = mx + b form. M represents the slope of the line, so if m is positive, the slope is positive, and the line will be slanting upwards.

But what if you get one of these.

In this lesson, you’ll learn how to deal with both of those cases. They might look tricky when you first start out, but they’re not actually that bad once you get to know them – in fact, they’re some of the easiest slopes to handle.

A linear regression model attempts to explain the relationship between a dependent (output variables) variable and one or more independent (predictor variable) variables using a straight line. This straight line is represented using the following formula:

Where, y: dependent variable. x: independent variable.

c: y intercept (The value of Y is c when the value of X is 0). The first step in finding a linear regression equation is to determine if there is a relationship between the two variables.

When a correlation coefficient shows that data is likely to be able to predict future outcomes and a scatter plot of the data appears to form a straight line, we can use simple linear regression to find a predictive function. Let us consider an example.

The next step is to find a straight line between Sales and Marketing that explain the relationship between them. But there can be multiple lines that can pass through these points.

That’s the problem that we will solve in this article. For this, we will first look at the cost function.

The cost is the error in our predicted value. We will use the Mean Squared Error function to calculate the cost.

We are not going to try all the permutation and combination of m and c (inefficient way) to find the best-fit line. For that, we will use Gradient Descent Algorithm.

Gradient Descent is an algorithm that finds the best-fit line for a given training dataset in a smaller number of iterations. If we plot m and c against MSE, it will acquire a bowl shape (As shown in the diagram below).

That combination of m and c will give us our best fit line. The algorithm starts with some value of m and c (usually starts with m=0, c=0).

Let say the MSE (cost) at m=0, c=0 is 100. Then we reduce the value of m and c by some amount (Learning Step).

We will continue doing the same until our loss function is a very small value or ideally 0 (which means 0 error or 100% accuracy).

Let m = 0 and c = 0. Let L be our learning rate.

Learning rate gives the rate of speed where the gradient moves during gradient descent. Setting it too high would make your path instable, too low would make convergence slow.

Calculate the partial derivative of the Cost function with respect to m. Let partial derivative of the Cost function with respect to m be Dm (With little change in m how much Cost function changes).

Let partial derivative of the Cost function with respect to c be Dc (With little change in c how much Cost function changes).

Now update the current values of m and c using the following equation:.

We will repeat this process until our Cost function is very small (ideally 0). Gradient Descent Algorithm gives optimum values of m and c of the linear regression equation.

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1 What does y=mx mean. Where does it come from.

2 In this lesson you will learn to derive the equation y=mx by using similar triangles.

We call this ratio the slope of the line. 4 6 6 9.

5 Let’s Review Core Lesson (0, 0) 2 3 (x, y) y x. 6 Let’s Review Core Lesson Slope: m (0, 0) m 1 (x, y) y x y = mx.

y=mx is the equation of a line through the origin with slope m.

9 Let’s Review Guided Practice Use similar triangles to demonstrate that the equation of a line through the origin with slope 3 is y=3x.

11 Let’s Review Extension Activities A line through the origin passes through (5,6). What is the line’s equation.

12 Let’s Review Quick Quiz.

In Maths, an intercept is a point on the y-axis, through which the slope of the line passes. It is the y-coordinate of a point where a straight line or a curve intersects the y-axis.

There are basically two intercepts, x-intercept and y-intercept. The point where the line crosses the x-axis is the x-intercept and the point where the line crosses the y-axis is the y-intercept.

The point where the line or curve crosses the axis of the graph is called intercept. If a point crosses the x-axis, then it is called the x-intercept.

The meaning of intercept of a line is the point at which it intersects either the x-axis or y-axis. If the axis is not specified, usually the y-axis is considered.

Except that line is accurately vertical, it will constantly cross the y-axis somewhere, even if it is way off the top or bottom of the chart. Also read: Equation of plane in intercept form.

Thus, the equation becomes: y = mx + b Hence, the formula for the y-intercept of a line is given by: b = y – mx Where, b is the intercept, m is the slope of the line and y and x indicate the points on the y-axis and x-axis respectively.

x/a + y/b = 1 Here, a and b are the intercepts of the line which intersect the x-axis and y-axis, respectively. The values of a and b can be positive, negative or zero and explain the position of the points at which the line cuts both axes, relative to the origin.

Consider a straight line equation Ax + By = C. Divide the equation by C,.

Comparing this equation with the equation of a line in intercept form, (x/a) + (y/b) = 1,. We get, x-intercept = a = C/A.

Alternatively,. To find the x-intercept, substitute y = 0 and solve for x.

Ax + B(0) = C. Ax = C.

To find the y-intercept, substitute x =0 and solve for y. i.e.

By = C. y = C/B.

Example: Let us assume the straight-line equation 5x +2y =10. To find x-intercept:.

5x + 2(0) = 10. 5x =10.

To find y-intercept. Substitute x =0 in the given equation.

2y = 10. y = 5.

y -intercept is (0, 5).

y-y1/y2-y1 = x-x1/x2-x1.

Then the formula becomes: => y – 0 / b – 0 = x – a/ 0 – a.

=> x/a + y/b = 1. Hence, proved.

The equation of the line making an intercept c on the y-axis and having slope m is given by: y = mx + c.

Also, check: Slope intercept form. The intercepts are the points on a graph at which the graph crosses the two axes (x-axis and y-axis).

In the above intercept graph, where a line L makes x-intercept a and y-intercept b on the axes. Thus, the equation of the line making intercepts a and b on the x-and y-axis, respectively, is:

Example 1: Let two intercepts P(2,0) and Q(0,3) intersect the x-axis and y-axis, respectively. Find the equation of the line.

From the equation of the line we know,. x/a + y/b = 1 ……….

Here, a = 2 and b = 3. Therefore, putting the values of intercepts a and b, in equation 1, we get:

=> 3x + 2y = 6. => 3x + 2y – 6 = 0,.

Example 2: Find the equation of the line, which makes intercepts –3 and 2 on the x- and y-axes respectively. Solution: Given, a = –3 and b = 2.

x/a + y/b = 1. x/-3 + y/2 = 1.

2x – 3y + 6 = 0. Hence, this is the required equation.

What is the equation of the line.

x/a + y/b = 1. 1 = (a+0)/2 ⇒ a = 2.

Therefore, the required equation of line is.

⇒ 2x + y – 4 = 0. Stay tuned with BYJU’S – The Learning App and download the app to explore more videos.

First, let’s see it in action. Here are two points (you can drag them) and the equation of the line through them.

We use Cartesian Coordinates to mark a point on a graph by how far along and how far up it is: Example: The point (12,5) is 12 units along, and 5 units up.

What is the slope (or gradient) of this line.

We know two points: The slope is the change in height divided by the change in horizontal distance.

Slope m = change in ychange in x = yA − yBxA − xB. In other words, we:

m = change in y change in x = 4−3 6−2 = 1 4 = 0.25. It doesn’t matter which point comes first, it still works out the same.

m = change in y change in x = 3−4 2−6 = −1 −4 = 0.25. Same answer.

Start with the “point-slope” formula (x1 and y1 are the coordinates of a point on the line): y − y1 = m(x − x1).

y − 3 = m(x − 2). We already calculated the slope “m”:

And we have: y − 3 = 14(x − 2).

And we get: y = x4 + 52.

Let us confirm by testing with the second point (6,4): y = x/4 + 5/2 = 6/4 + 2.5 = 1.5 + 2.5 = 4.

Start with the “point-slope” formula: y − y1 = m(x − x1).

And we get: y − 6 = −2(x − 1).

y − 6 = −2x + 2. y = −2x + 8.

The previous method works nicely except for one particular case: a vertical line:.

m = yA − yBxA − xB = 4 − 12 − 2 = 30 = undefined. But there is still a way of writing the equation: use x= instead of y=, like this:

A linear relationship (or linear association) is a statistical term used to describe a straight-line relationship between two variables. Linear relationships can be expressed either in a graphical format where the variable and the constant are connected via a straight line or in a mathematical format where the independent variable is multiplied by the slope coefficient, added by a constant, which determines the dependent variable.

A linear relationship may be contrasted with a polynomial or non-linear (curved) relationship.

 y = m x + b where: m = slope b = y-intercept \begin{aligned} &y = mx + b \\ &\textbf{where:}\\ &m=\text{slope}\\ &b=\text{y-intercept}\\ \end{aligned} ​y=mx+bwhere:m=slopeb=y-intercept​. In this equation, “x” and “y” are two variables which are related by the parameters “m” and “b”.

The slope “m” is calculated from any two individual points (x1, y1) and (x2, y2) as:.  m = ( y 2 − y 1 ) ( x 2 − x 1 ) m = \frac{(y_2 – y_1)}{(x_2 – x_1)} m=(x2​−x1​)(y2​−y1​)​.

A commonly used linear relationship is a correlation, which describes how close to linear fashion one variable changes as related to changes in another variable.

It is commonly used in extrapolating events from the past to make forecasts for the future. Not all relationships are linear, however.

Mathematically similar to a linear relationship is the concept of a linear function. In one variable, a linear function can be written as follows:.

This is identical to the given formula for a linear relationship except that the symbol f(x) is used in place of y. This substitution is made to highlight the meaning that x is mapped to f(x), whereas the use of y simply indicates that x and y are two quantities, related by A and B.

In the study of linear algebra, the properties of linear functions are extensively studied and made rigorous. Given a scalar C and two vectors A and B from RN, the most general definition of a linear function states that:  c × f ( A + B ) = c × f ( A ) + c × f ( B ) c \times f(A +B) = c \times f(A) + c \times f(B) c×f(A+B)=c×f(A)+c×f(B).

Let’s take the concept of speed for instance. The formula we use to calculate speed is as follows: the rate of speed is the distance traveled over time.

While there are more than two variables in this equation, it’s still a linear equation because one of the variables will always be a constant (distance).

Because distance is a positive number (in most cases), this linear relationship would be expressed on the top right quadrant of a graph with an X and Y-axis.

Represented graphically with the distance on the Y-axis and time on the X-axis, a line tracking the distance over those 20 hours would travel straight out from the convergence of the X and Y-axis.

These equations express a linear relationship on a graph:.  ° C = 5 9 ( ° F − 3 2 ) \degree C = \frac{5}{9}(\degree F – 32) °C=95​(°F−32).

Assume that the independent variable is the size of a house (as measured by square footage) which determines the market price of a home (the dependent variable) when it is multiplied by the slope coefficient of 207.65 and is then added to the constant term $10,500. If a home’s square footage is 1,250 then the market value of the home is (1,250 x 207.65) + $10,500 = $270,062.50.

In this example, as the size of the house increases, the market value of the house increases in a linear fashion.

 Y = k × X where: k = constant Y , X = proportional quantities \begin{aligned} &Y = k \times X \\ &\textbf{where:}\\ &k=\text{constant}\\ &Y, X=\text{proportional quantities}\\ \end{aligned} ​Y=k×Xwhere:k=constantY,X=proportional quantities​. When analyzing behavioral data, there is rarely a perfect linear relationship between variables.

For example, you could look at the daily sales of ice-cream and the daily high temperature as the two variables at play in a graph and find a crude linear relationship between the two.

Any point in a 2D space is represented by x and y coordinates as an ordered pair, either of which can be zero, positive or negative. If either value is zero, the point is represented as the following:

If both x and y coordinates are non-zero, the point lies somewhere on the 2D coordinate plane in one of its four quadrants. Consider point M in the coordinate plane here.

So, its x coordinate is (1), and its y coordinate is (2). Together, its (x, y) coordinates are represented on the 2D coordinate plane as the following:

Point M is in Quadrant 1. Consider point N in the coordinate plane here.

So, its x coordinate is (-3), and its y coordinate is (-4). Together, its (x, y) coordinates are represented on the 2D coordinate plane as the following:

Point N is in Quadrant 3.

We know that there are infinite points in the coordinate plane. Consider an arbitrary point P(x,y) on the XY plane and a line L.

This is where the importance of the equation of a straight line comes into the picture in two-dimensional geometry. The equation of a straight line contains terms in x and y.

Equations of horizontal and vertical lines. Equation of the lines which are horizontal or parallel to the X-axis is y = a, where a is the y – coordinate of the points on the line.

For example, the equation of the line which is parallel to the X-axis and contains the point (2,3) is y= 3. Similarly, the equation of the line which is parallel to the Y-axis and contains the point (3,4) is x = 3.

Point-slope form equation of a line. Consider a non-vertical line L whose slope is m, A(x,y) be an arbitrary point on the line and P(x1, y1) be the fixed point on the same line.

Slope of the line by the definition is,. For example, equation of the straight line having a slope m = 2 and passes through the point (2, 3) is.

y= 2x-4+3. 2x-y-1 = 0.

Two-point form equation of line. Let P(x,y) be the general point on the line L which passes through the points A(x1, y1) and B(x2, y2).

Since the three points are collinear,. slope of PA = slope of AB.

Slope-intercept form equation of line. Consider a line whose slope is m which cuts the Y-axis at a distance ‘a’ from the origin.

The point at which the line cuts the y-axis will be (0, a).

y-a = m(x-0). y = mx+a.

The distance b is called x- intercept of the line. Equation of the line will be:

Intercept form. Consider a line L having x– intercept a and y– intercept b, then the line touches X– axis at (a,0) and Y– axis at (0,b).

By two-point form equation,. For example, equation of the line which has x– intercept 3 and y– intercept 4 is,.

Normal form. Consider a perpendicular from the origin having length l to line L and it makes an angle β with the positive X-axis.

Let OP be perpendicular from the origin to line L. Then,.

The slope of the line OP is tan β. Therefore,.

You have learnt about the different forms of the equation of a straight line. To know more about straight lines and their properties, register with BYJU’S – The Learning App.

What is slope-intercept form. This article is here to help.

Read below to learn how to write equations in slope-intercept form. We’ll also discover how to find slope-intercept form from two points, from a slope and a point, and from a graph.

Changing an equation into a certain form can help us to identify useful information. Just like we can change play-dough or clay from one shape to another, we can change equations to reveal the information we need.

What We Review. The slope-intercept form of an equation is:

This is where the line crosses the y-axis. The variable m represents the slope.

For more info on slope, visit our review article on how to find slope. Return to the Table of Contents.

First, we will calculate the slope. This will be the value of m in the slope-intercept form equation:

Let’s answer the question, “What is the slope-intercept form of a line going through the points (1,5) and (-4,7). ”.

\dfrac{y_2-y_1}{x_2-x_1}. First, we can label the points.

\dfrac{y_2-y_1}{x_2-x_1}. \dfrac{7-5}{-4-1}.

\dfrac{-2}{5}. Therefore, the slope of the line is \frac{-2}{5}.

y=mx+b. y=\dfrac{-2}{5}x+b.

To do so, we will use one point and substitute the values of x and y. Let us use the point (1,5).

y=\dfrac{-2}{5}x+b. 5=\dfrac{-2}{5}1+b.

5+\dfrac{2}{5}=b. \dfrac{25}{5}+\dfrac{2}{5}=b.

Therefore, the value of b is \frac{27}{5}, which is the y-intercept. We will now substitute \frac{27}{5} for b.

y=\dfrac{-2}{5}x+\dfrac{27}{5}. For an additional example of writing an equation from two points, watch the video below:

Instead of being given two points, we may need to know how to find slope-intercept form with slope and the y-intercept. In this example, we will use a slope of -4 and a y-intercept of \frac{1}{5}.

We will simply substitute the given values for m and b. We start with form:

y=-4x+\dfrac{1}{5}. Return to the Table of Contents.

When presented with a graph, we must first determine two points on the grid lines and identify those points. For our graph, we will use the points (1,1) and (0,-2).

Slope is \frac{\text{rise}}{\text{run}}. For us, the rise is how far up we must travel to get from (0,-2) to get to (1,1).

Let us determine these values by counting on the graph. This means, the slope of the line is 3.

In this case, the line crosses the y-axis at the point (0,-2), so the y-intercept is -2. Because the slope is 3 and the y-intercept is -2, we will substitute 3 for m and -2 for b to create the slope-intercept form of the equation.

y=3x+(-2). y=3x-2.

Return to the Table of Contents. Consider the slope-intercept form equation y=6x+9.

The x-intercept occurs when y equals 0, and the y-intercept occurs when x equals 0. We can determine the x-intercept by setting the value of y equal to 0.

y=6x+9. 0=6x+9.

\dfrac{-9}{6}=x. \dfrac{-3}{2}=x.

This means the graph will cross the x-axis when x equals \frac{-3}{2}. Because the equation is written in slope-intercept form, we can easily determine the y-intercept.

y=mx+b. y=6x+9.

Return to the Table of Contents. Linear equations can also be written in point-slope form, determined by one point on the line and the slope of the line.

A linear equation can also be written in standard form. This form can be very useful to solve systems of equations.

Return to the Table of Contents. Looking for video summary of this content.

Click here to explore more helpful Albert Algebra 1 review guides. Return to the Table of Contents.

Contents: The graph of a linear function has a straight line.

The equation for a linear function is: y = mx + b, Where: The equation, written in this way, is called the slope-intercept form.

These all represent the same graphs. Examples of linear functions:

The domain and range of a linear function is usually the set of real numbers. There is an exception: if the function is constant (e.g.

Watch this quick video for an explanation and graphical approach:. You can find the limit of a linear function in several ways, including:

Sometimes you have to make a more formal approach, using the definition of a limit. The following example shows how to do this for the function y = 2x + 2.

Example problem: Find the limit of y = 2x + 2 as x tends to 0. The limit for this function is 0 at x = 0, and ∞ for x=∞.

f(x) = 2 x + 2 c = 0 lim f(x) = L = lim 2x + 2 x→c x→0. Step 2: Solve for the limit of the function, using some basic properties of linear functions:

lim(x→0) 2x + 2 = lim(x→0) 2x + lim(x→0) 2 = 0 + 2 = 2. Example problem: Find the limit of 2x + 2 as x tends to 0.

f(x) = 2x + 2 c = ∞ lim(x→&infin) 2x + 2 = lim(x→&infin) 2x + lim(x→&infin) 2 = ∞ = Limit does not exist. Tip: Since the limit goes to infinity when you times infinity by 2, the limit of the function does not exist due to infinity not being a real number.

Since the 0 negates the infinity, the line has a constant limit. This would appear as a horizontal line on the graph.

A nonlinear function is defined as one that isn’t a linear function.

More formally, a straight line produced when the dependent variable (y) changes at a constant rate with the independent variable (y), following the equation y = mx + b. In addition, a linear function has a domain and range of all real numbers.

They are, in a sense, the opposite of linear functions. In other words, a nonlinear function is any function that:

Most polynomial functions are nonlinear functions with one exception: Algebraically, a linear function is a polynomial with a degree (highest exponent) of 1. They are also known as first degree polynomials.

Linear functions are any functions that produce a straight line graph. So by definition, nonlinear functions produce graphs that aren’t a straight line.

Graph created with Desmos.com. In order to figure out if your function is linear or nonlinear, you have several options.

An absolute value function has a sharp dip where it changes direction. Therefore, it isn’t linear, but does appear to have the same slope.

Nonlinear functional analysis is the study of nonlinear functions. It’s the complement of linear functional analysis.

Desmos Graphing Calculator. The Biology Project.

Retrieved January 12, 2021 from: Back to Top. In general, a linear combination of a set of terms is where terms are first multiplied by a constant, then added together.

For example, let’s say you have two terms x and y. You might multiply x by 10, and y by 8, to get: 10x + 8y.

The constants placed in front of the terms (10 and 8 in this example) are sometimes called coefficients. Linear combinations are used frequently because they are easier to conceptualize than some of the more complicated expressions (like those involving division or exponents).

Coefficients in a linear combination can be positive, negative or zero. You can also have one term, or more.

The above definition also extends to vectors. Let’s say that you have two vectors v and w.

The expression av + bw is called a linear combination of v and w.

As an example, the vector (7, 11, 15) is a linear combination of the vectors (1, 1, 1) and (1, 2, 3). The first vector (1, 1, 1) is multiplied by the scalar 3, and the second vector (1, 2, 3) is multiplied by the scalar 4.

A linear relationship is where you represent the relationship between variables as a line (the word comes from the Latin linearis, from linea “a line”). If you graph linear line, you’ll see a perfectly straight line with no curves.

Non linear relationships are (perhaps not surprisingly) everything else. If there’s no straight line, then it’s non linear.

If a set of data has a linear relationship, you can represent it with a linear equation or function.

The slope formula looks like this: y = mx + b Where: Linear functions are similar to linear equations.

A few examples of linear functions that will give a straight line graph: The variables in linear functions have linear relationships.

The easiest way to visualize a linear relationship or recognize a linear function with a small set of data is to make a scatter plot. This scatter plot shows a clear linear relationship.

You can also plug the numbers into a table on the TI-89 and graph a scatter plot that way. A set of points is collinear if you can draw one line through them all.

A general way to write this is: “Points P1, P2 and P3 are collinear”, which can also be written as “point P1 is collinear with points P2 and P3“. It goes without saying that points are non-collinear if they do not fall on the same line.

It seems reasonable that if you can draw a line through a set of points, then those points are collinear. The trouble is, those points may not be exactly on the same line.

For example, if you are given the linear equation y = 4x + 16, you know that the points (-4, 0) and (-1, 12) meet the definition because (plugging the x and y values into the equation) we get: A second way is to find the slope between the points (i.e.

if the slopes are the same then the points are collinear. For example, the set of points in the image below fit the definition if the slope of line segment A equals the slope of line segment B.

Example question: Do the points P1 = (−4, 0), P2 = (−1, 12) and P3 = (4, 32) show collinearity.

Step 2: Find the slope for the line segment between the next two points =(y3 − y2)/(x3 − x2) = (32 − 12)/(4 – (-1))= 20/5 = 4. Step 3: Compare the slopes you calculated in Steps 1 and 2.

A linear equation graphs a straight line.A linear equation graphs to a straight line and is a degree-1 polynomial. In other words, each term is either:

a1x1 + … anxn + b = 0. Here:

Some physical processes show a direct linear relationship, and even non linear relationships can often be approximated by systems of linear equations. When possible, we like to estimate with them because they are easy to manipulate and calculate with.

The simplest linear equation is the one with one variable: ax + b = 0. A little bit of algebraic manipulation makes it clear that the unique solution to this linear equation is always -b/a.

If the linear equation has two variables, they are usually called x and y. Then the equation can be written as.

Two independent linear equations will define these two variables completely. Reducing them down to an x = d, y = e form usually requires a small amount of algebraic multiplication.

An asymptote is a straight line that constantly approaches a given curve but does not meet at any infinite distance. In other words, Asymptote is a line that a curve approaches as it moves towards infinity.

The curves visit these asymptotes but never overtake them. The method opted to find the horizontal asymptote changes involves comparing the degrees of the polynomials in the numerator and denominator of the function.

An asymptote of the curve y = f(x) or in the implicit form: f(x,y) = 0 is a straight line such that the distance between the curve and the straight line lends to zero when the points on the curve approach infinity. There are three types of asymptotes namely:

When x moves to infinity or -infinity, the curve approaches some constant value b, and is called a Horizontal Asymptote.

When x moves towards infinity (i.e.,∞) , or -infinity (i.e., -∞), the curve moves towards a line y = mx + b, called Oblique Asymptote.

We know that the vertical asymptote has a straight line equation is x = a for the graph function y = f(x), if it satisfies at least one the following conditions:.

For Oblique asymptote of the graph function y=f(x) for the straight-line equation is y=kx+b for the limit x → + ∞, if and only if the following two limits are finite.

The above limit is same for x → – ∞,. Example:

Solution: Given,.

Here, f(x) is not defined for x = -1. Let us find the one-sided limits for the given function at x = -1.

Therefore, the function f(x) has a vertical asymptote at x = -1. Now, let us find the horizontal asymptotes by taking x → ±∞.

Therefore, the function f(x) has a horizontal asymptote at y = 3. \(\begin{array}{l}k=\lim_{x\rightarrow +\infty}\frac{f(x)}{x}\\=\lim_{x\rightarrow +\infty}\frac{3x-2}{x(x+1)}\\ = \lim_{x\rightarrow +\infty}\frac{3x-2}{(x^2+x)}\\=\lim_{x\rightarrow +\infty}\frac{\frac{3}{x}-\frac{2}{x^2}}{1+\frac{1}{x}} \\= \frac{0}{1}\\=0\end{array} \).

As k = 0, there are no oblique asymptotes for the given function. To recall that an asymptote is a line that the graph of a function approaches but never touches.

In the above example, we have a vertical asymptote at x = 3 and a horizontal asymptote at y = 1. The curves approach these asymptotes but never visit them.

Below are the points to remember to find the horizontal asymptotes: Hyperbola contains two asymptotes.

These can be observed in the below figure.

If the centre of the hyperbola is located at the origin, then the pair of asymptotes is given as: y = ±(b/a)x.

y = (b/a)x. y = -(b/a)x.

Example 1: Find the horizontal asymptotes for f(x) = x+1/2x.

Given, f(x) = (x+1)/2x. Since the highest degree here in both numerator and denominator is 1, therefore, we will consider here the coefficient of x.

Example 2: Find the horizontal asymptotes for f(x) = x/x2+3.

Given, f(x) = x/x2+3. Since we can see here the degree of the numerator is less than the denominator, therefore, the horizontal asymptote is located at y = 0.

Find the horizontal asymptotes for f(x) =(x2+3)/x+1. Solution:

As you can see, the degree of the numerator is greater than that of the denominator. Hence, there is no horizontal asymptote.

Types of Functions >. Every function in the Cartesian plane stems from a particular parent function.

Every other possible linear function of the form y = mx + b is a child function of this parent. Together, parent functions and child functions make up families of functions.

For example, the function f(x) = 2x is the linear parent function vertically stretched by a factor of 2. Instead of the function passing through (1, 1) the graph passes through (2, 1):

Graph of the absolute value parent function f(x) = x. Characteristics:

The graph depends on the value of c. For example, the following graph shows two constant functions where c = 3 (red) and c = 2.5 (blue): Two constant functions y = 3 and y = 2.5.

The cube root function is an odd function that has the parent f(x) = ∛x. Graph of the cube root parent function f(x) = ∛x.

Cubic functions are odd functions. The parent is: f(x) = x3.

Graph of the cubic parent function f(x) = x3. Characteristics:

The parent function is either f(x) = ex or f(x) = 10x. Graph of the exponential parent functions f(x) = ex(red) and f(x) = 10x (black).

The linear function is an odd function with the parent: f(x) = x. The linear parent function f(x) = x is a straight line that goes through the origin (0, 0).

More info: Linear parent functions. Characteristics:

Where b is the base.

The parent function for the natural logarithm function is ln(x). Two logarithmic functions with different bases: f(x) = log10 (blue) and f(x) = log2.

Characteristics: Parent: F(x) = x2.

Desmos Graphing Calculator. UF Library of Functions.

In data analysis and statistics, determining the slope of a line allows you to quantify the relationship and correlation between two variables. The slope value provides a numerical representation of the steepness and direction of the linear trend between data points.

This comprehensive guide will teach you several easy methods to find and interpret slope values using Excel’s built-in tools and formulas. The slope of a straight line measures its steepness and indicates if the line is rising or falling from left to right.

In the linear equation y = mx + b, the slope is represented by m. Slope is hugely helpful for analyzing data correlations.

The higher the positive slope value, the steeper the uptrend line. A negative slope indicates an inverse correlation between the variables, with y decreasing as x increases.

A slope of zero means a perfectly horizontal flat line with no correlation between the variables. By determining the slope of a linear regression trendline fitted to your data, you can measure the strength of correlation and the dynamics of change between data variables.

Excel’s SLOPE function calculates slope automatically from arrays of x- and y-values. Syntax:

Where: For example:

=SLOPE(B2:B11,A2:A11). This returns the slope of the linear regression line for the x-values in A2:A11 and y-values in B2:B11.

The SLOPE function is quick and straightforward for finding slope in Excel. You can also determine slope visually from an Excel scatter chart.

This will show the equation for the trendline on the chart, with the slope value clearly labeled.

It’s also possible to calculate slope yourself using two data points: Slope Formula:

Where: For example:

m = (5 – 3) / (6 – 2). m = 2/4.

While more time-consuming than Excel formulas, manually calculating slope helps you understand what the value represents. To find the y-intercept (b) of a line along with slope, use the INTERCEPT function:

Or use the chart trendline equation or manual calculations with point coordinates. The y-intercept indicates where the line crosses the y-axis.

Understanding slope in context is crucial for proper data analysis and modeling. Note that the SLOPE and trendline methods calculate a linear regression slope.

For non-linear data, consider adding a polynomial or logarithmic trendline to inspect curve slope changes. Or use the LINEST function for linear or non-linear regression statistics.

The built-in Excel tools, formulas, charts, and manual slope calculations discussed here make it easy to find and interpret slope values from your data. Mastering slope allows a much deeper understanding of correlations and trends for improved data analysis and modeling.

1 Direct Variation y = kx. 2 A direct variation is… A linear function Equation can be written in the form.

m  0 ory = kx. k  0 The y-intercept must be zero.

Graphs a line passing through the origin.

The constant of variation is found by dividing “y” by “x” (k = y/x)y/x) The constant of variation is the slope of a linear equation whose y-intercept is zero.

4 Constant of Variation: y = kx y= x k= 1 y= 2x k= 2 y= -x k= -1 y= x k= y= x k=. 5 Which is a direct variation.

y = 3x + 2 2. y = 2x 3.

9x + 3y = -3 5. 2x + 3y = 0 6.

#2) Check out equation: y = kx + 0 or y= kx no yes. 6 Example 1 The distance that you travel at a constant speed varies directly with the time spent traveling.

How long will it take you to travel 400 miles. Direct variation: y varies directly as x y = k x This problem: distance varies directly as time d = k t.

Find “k” “k” first using the given ordered pair (time, distance) …. (2, 100) 100 = k (2) 50 = k Thus the equation for this problem: d = 50 t Example 1 (cont.).

How long will it take you to travel 400 miles. equation : d = 50 t 400 = 50 t 750 = 50 t d = 50 (3.5) How long will it take you to travel 750 miles.

9 Example 2 The money you earn varies directly with the number of lawns you mow. You earn $36 for mowing 3 lawns.

How much money would you earn for mowing seven lawns. How many lawns did you mow for $60.

10 Suppose the ordered pairs in each problem are for the same direct variation. Find the missing value.

(4, 2) and (6, y) 2/4 2/4 = y/6y/6 2. (2, 7) and (x, 3) 7/2 7/2 = 3/x3/x 3.

Slope is constant: 1 st ) given two ordered pairs: (x,y) & 2 nd ) k = y/xy/x 3 rd ) y/x y/x = y/xy/x 4 th ) solve the proportion for the unknown. 11 Write a direct variation equation for the following: y = kx (2,5) (0,0) k = y = x k =.