4.5: Graphs Using SlopeIntercept Form
Learning Objectives
 Identify the slope and
y− intercept of equations and graphs.  Graph an equation in slopeintercept form.
 Understand what happens when you change the slope or intercept of a line.
 Identify parallel lines from their equations.
Introduction
The total profit of a business is described by the equation
Identify Slope and y− intercept
So far, we’ve been writing a lot of our equations in slopeintercept form—that is, we’ve been writing them in the form
Example 1
Identify the slope and
a)
b)
c)
d)
Solution
a) Comparing , we can see that
b) has a slope of 0.5 and a
Notice that the intercept is negative. The
c) At first glance, this equation doesn’t look like it’s in slopeintercept form. But we can rewrite it as
d) We can rewrite this one as
Example 2
Identify the slope and
The intercepts have been marked, as well as some convenient lattice points that the lines pass through.
Solution
a) The
b) The
c) The
d) The
Graph an Equation in SlopeIntercept Form
Once we know the slope and intercept of a line, it’s easy to graph it. Just remember what slope means. Let's look back at this example from Lesson 4.1.
Ali is trying to work out a trick that his friend showed him. His friend started by asking him to think of a number, then double it, then add five to the result. Ali has written down a rule to describe the first part of the trick. He is using the letter
Help him visualize what is going on by graphing the function that this rule describes.
In that example, we constructed a table of values, and used that table to plot some points to create our graph.
We also saw another way to graph this equation. Just by looking at the equation, we could see that the
Here’s another problem where we can use the same method.
Example 3
Graph the following function:
Solution
To graph the function without making a table, follow these steps:
 Identify the
y− intercept:b=5  Plot the intercept: (0, 5)
 Identify the slope:
m=−3 . (This is equal to−31 , so the rise is 3 and the run is 1.)  Move over 1 unit and down 3 units to find another point on the line: (1, 2)
 Draw the line through the points (0, 5) and (1, 2).
Notice that to graph this equation based on its slope, we had to find the rise and run—and it was easiest to do that when the slope was expressed as a fraction. That’s true in general: to graph a line with a particular slope, it’s easiest to first express the slope as a fraction in simplest form, and then read off the numerator and the denominator of the fraction to get the rise and run of the graph.
Example 4
Find integer values for the rise and run of the following slopes, then graph lines with corresponding slopes.
a)
b)
c)
d)
Solution
a)
b)
c)
d)
Changing the Slope or Intercept of a Line
The following graph shows a number of lines with different slopes, but all with the same \begin{align*}y\end{align*}
You can see that all the functions with positive slopes increase as we move from left to right, while all functions with negative slopes decrease as we move from left to right. Another thing to notice is that the greater the slope, the steeper the graph.
This graph shows a number of lines with the same slope, but different \begin{align*}y\end{align*}
Notice that changing the intercept simply translates (shifts) the graph up or down. Take a point on the graph of \begin{align*}y = 2x\end{align*}
Notice also that these lines all appear to be parallel. Are they truly parallel?
To answer that question, we’ll use a technique that you’ll learn more about in a later chapter. We’ll take 2 of the equations—say, \begin{align*}y = 2x\end{align*}
So what values would satisfy both \begin{align*}y = 2x\end{align*}
But what happens when we try to solve that equation? If we subtract \begin{align*}2x\end{align*}
And we’d find out the same thing no matter which two lines we’d chosen. In general, since changing the intercept of a line just results in shifting the graph up or down, the new line will always be parallel to the old line as long as the slope stays the same.
Any two lines with identical slopes are parallel.
Further Practice
To get a better understanding of what happens when you change the slope or the \begin{align*}y\end{align*}
Lesson Summary
 A common form of a line (linear equation) is slopeintercept form: \begin{align*}y=mx+b\end{align*}
y=mx+b , where \begin{align*}m\end{align*}m is the slope and the point \begin{align*}(0, b)\end{align*}(0,b) is the \begin{align*}y\end{align*}y− intercept  Graphing a line in slopeintercept form is a matter of first plotting the \begin{align*}y\end{align*}
y− intercept \begin{align*}(0, b)\end{align*}(0,b) , then finding a second point based on the slope, and using those two points to graph the line.  Any two lines with identical slopes are parallel.
Review Questions
 Identify the slope and \begin{align*}y\end{align*}
y− intercept for the following equations.
\begin{align*}y=2x+5\end{align*}
y=2x+5 
\begin{align*}y=0.2x+7\end{align*}
y=−0.2x+7 
\begin{align*}y=x\end{align*}
y=x 
\begin{align*}y=3.75\end{align*}
y=3.75

\begin{align*}y=2x+5\end{align*}
 Identify the slope of the following lines.
 Identify the slope and \begin{align*}y\end{align*}
y− intercept for the following functions.  Plot the following functions on a graph.

\begin{align*}y=2x+5\end{align*}
y=2x+5 
\begin{align*}y=0.2x+7\end{align*}
y=−0.2x+7 
\begin{align*}y=x\end{align*}
y=x 
\begin{align*}y=3.75\end{align*}
y=3.75

\begin{align*}y=2x+5\end{align*}
 Which two of the following lines are parallel?

\begin{align*}y=2x+5\end{align*}
y=2x+5 
\begin{align*}y=0.2x+7\end{align*}
y=−0.2x+7 
\begin{align*}y=x\end{align*}
y=x 
\begin{align*}y=3.75\end{align*}
y=3.75  \begin{align*}y= \frac{1}{5}x11\end{align*}
 \begin{align*}y=5x+5\end{align*}
 \begin{align*}y=3x+11\end{align*}
 \begin{align*}y=3x+3.5\end{align*}

\begin{align*}y=2x+5\end{align*}
 What is the \begin{align*}y\end{align*}intercept of the line passing through (1, 4) and (3, 2)?
 What is the \begin{align*}y\end{align*}intercept of the line with slope 2 that passes through (3, 1)?
 Line \begin{align*}A\end{align*} passes through the points (2, 6) and (4, 3). Line \begin{align*}B\end{align*} passes through the point (3, 2.5), and is parallel to line \begin{align*}A\end{align*}
 Write an equation for line \begin{align*}A\end{align*} in slopeintercept form.
 Write an equation for line \begin{align*}B\end{align*} in slopeintercept form.
 Line \begin{align*}C\end{align*} passes through the points (2, 5) and (1, 3.5). Line \begin{align*}D\end{align*} is parallel to line \begin{align*}C\end{align*}, and passes through the point (2, 6). Name another point on line \begin{align*}D\end{align*}. (Hint: you can do this without graphing or finding an equation for either line.)
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Date Created:
Jul 23, 2013Last Modified:
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