Standard
MCC912.F.IF.7a Graph linear functions and show intercepts. ^{ }
Suppose the linear function f(x) = 0.25x + 10 represents the amount of money you have left to play video games, where f(x) is the amount of money you have left and x is the number of video games that you have played so far. Do you know how to graph this function? What would be the slope and intercept of the graph? In this Concept, you'll learn how to graph linear functions like this one by finding the graph's slope and intercept.
Guidance
You can see that the notations are interchangeable. This means you can substitute the notation for and use all the concepts you have learned on linear equations.
Example A
\begin{align*}\text{Graph} \ f(x)& =\frac{1}{3}x+1.\\ \text{Replace} \ f(x)& = \text{with} \ y=.\\ y& =\frac{1}{3} x+1\end{align*}
This equation is in slopeintercept form. You can now graph the function by graphing the intercept and then using the slope as a set of directions to find your second coordinate.
Watch This
CK12 Foundation: 0408S Graphs Using SlopeIntercept Form (H264)
Try This
To get a better understanding of what happens when you change the slope or the
\begin{align*}y\end{align*}
Identify Slope and
\begin{align*}y\end{align*}
So far, we’ve been writing a lot of our equations in
slopeintercept form—
that is, we’ve been writing them in the form
\begin{align*}y = mx + b\end{align*}
Example A
Identify the slope and
\begin{align*}y\end{align*}
a)
\begin{align*}y = 3x + 2\end{align*}
b)
\begin{align*}y = 0.5x  3\end{align*}
c)
\begin{align*}y = 7x\end{align*}
d)
\begin{align*}y = 4\end{align*}
Solution
a) Comparing
, we can see that
\begin{align*}m = 3\end{align*}
b)
has a
slope of 0.5
and a
\begin{align*}y\end{align*}
Notice that the intercept is
negative
. The
\begin{align*}b\end{align*}
c) At first glance, this equation doesn’t look like it’s in slopeintercept form. But we can rewrite it as
\begin{align*}y = 7x + 0\end{align*}
d) We can rewrite this one as
\begin{align*}y = 0x  4\end{align*}
G raph an Equation in SlopeIntercept Form
Once we know the slope and intercept of a line, it’s easy to graph it.
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
\begin{align*}x\end{align*}
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
\begin{align*}y\end{align*}
Here’s another problem where we can use the same method.
Example C
Graph the following function:
\begin{align*}y=3x+5\end{align*}
Solution
To graph the function without making a table, follow these steps:

Identify the
\begin{align*}y\end{align*}
y− intercept: \begin{align*}b = 5\end{align*}b=5  Plot the intercept: (0, 5)

Identify the slope:
\begin{align*}m = 3\end{align*}
m=−3 . (This is equal to \begin{align*}\frac {3}{1}\end{align*}−31 , so the rise is 3 and the run is 1.)  Move down 3 units and right one unit 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 D
Find integer values for the rise and run of the following slopes, then graph lines with corresponding slopes.
a)
\begin{align*}m=3\end{align*}
b)
\begin{align*}m=2\end{align*}
Solution
a)
b)
Vocabulary

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.
Guided Practice
Find integer values for the rise and run of the following slopes, then graph lines with corresponding slopes.
a) \begin{align*}m=0.75\end{align*}
b) \begin{align*}m=0.375\end{align*}
Solution:
a)
b)
Practice
Identify the slope and \begin{align*}y\end{align*} intercept for the following equations.
 \begin{align*}y=2x+5\end{align*}
 \begin{align*}y=0.2x+7\end{align*}
 \begin{align*}y=x\end{align*}
 \begin{align*}y=3.75\end{align*}
Identify the slope of the following lines.
Identify the slope and \begin{align*}y\end{align*} intercept for the following functions.
For 710, plot the following functions on a graph.
 \begin{align*}y=2x+5\end{align*}
 \begin{align*}y=0.2x+7\end{align*}
 \begin{align*}y=x\end{align*}
 \begin{align*}y=3.75\end{align*}

Which two of the following lines are parallel?
 \begin{align*}y=2x+5\end{align*}
 \begin{align*}y=0.2x+7\end{align*}
 \begin{align*}y=x\end{align*}
 \begin{align*}y=3.75\end{align*}
 \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*}
 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.)