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Chapter 12—Graphing

12-4 Graphing Linear Equations

Defining a Line in Terms of Its Slope and Y-Intercept

You can define a line in terms of its slope and y-intercept

Let m = 2 and b = -4

Step 1: Plot the y-intercept point.

Step 2: Draw a straight line through the y-intercept point, making certain the line has the given slope. More Examples

Examples & Exercises

 Sketch a line, given its slope and y-intercept.

The Slope-Intercept Equation for a Straight Line

The slope-intercept equation is a linear equation that completely defines a straight line on the coordinate plane.

 Equation Slope-Intercept Equation y = mx + b Where x  and y are coordinates of any point (x,y) on the line m = slope of the line b = y-intercept of the line

Given the slope and y-intercept  of a line, determine its x-intercept.

In the equation y = mx + b

Given the values for m and b, you can also find the y-intercept

This is done by setting y = 0 and solving for x

y = mx + b

0 = mx + b

x = b/m

 Equation Point-Slope Equation y - y1 = m(x - x1) Where x  and y are coordinates of any point (x,y) on the line x1  and y1 are coordinates of a given point (x1,y1) on the line m = slope of the line

 Example Problem Write the linear equation (slope-intercept form) for a line where the slope is 4 and one of the points on the line is (-2,5). Procedure 1. The equation y - y1 = m(x - x1) 2. Substitute known values y - y1 = m(x - x1) = y - 5 = 4(x + 2) 3. Rewrite in standard slope-intercept form y = 4x + 8 + 5 = 4x + 13 Solution y =  4x + 13

Examples & Exercises

 Given the slope and one point on a line, write the linear equation for the line in slope-intercept form.

Introduction to Linear Equations

 Definition A linear equation is an equation of two variables whose graph is a straight line. (straight line = linear). Example:   y = mx + b where: x and y are variables m is the slope of the line b  is the y-intercept

Here are four examples of linear equations and interpretations of their components:

 y = 2x + 4 slope = 2 y-intercept is (0,4) y = x - 8 slope = 1 y-intercept = (0,-8) y = 2 slope = 0 (horizontal line) y-intercept (0,2) y = x slope = 1 y-intercept = 0

So you can directly determine the slope and y-intercept of a straight line directly from its linear equations. But what about the x-intercept?  That takes just a bit more work.  To determine the x-intercept from a linear equation:

1. set y equal to zero
2. solve the equation for x

Consider the first equation in the examples above: y = 2x + 4

1. Set y equal to zero:  0 = 2x + 4
2. Solve for x: x = -2/4  or -1/2

So the x-intercept is (-1/2, 0)

How about the second equation: y = x - 8

1. Set y equal to zero: 0 = x -8
2. Solve for x: x = 8

So  the x-offset is (8,0)

The equation y = 2 has no  x-intercept because the line is parallel to the x-axis

To determine the x-intercept for equation y = x:

1. Set y equal to zero: 0 = x
2. So the x-intercept is (0,0) -- the origin

Examples & Exercises

 In these Examples & Exercises, you are given a simple linear equation and asked to determine the slope, x-intercept, and y-intercept for the corresponding line on a typical coordinate plane.Work these examples until you are confident you perfectly understand the principles.

slope-intercept form.

More Examples

1. Plot the linear equation y = 2x + 1

The simplest approach is to determine the two intercepts, and this means solving the equations twice: first with x = 0 and then again with y = 0

y = 2x + 1

Substituting 0 for x

y = 2 x 0 + 1

y = 1

One possible  point on the line is its llllll (0,1)  Solve again, but with y set to 0: y = 2x + 1

0 = 2x + 1

2x = -1

x = -0.5

(-0.5,0)

So the intercepts are two possible points on the line

Exercises

 Do you need graph paper? Click here to download a version of the coordinate plane that you can print out for this exercise. It's free!

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