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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?

printer01.jpg (881 bytes)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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