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Graphs of Linear Functions

Graph f(x) = ax +b

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Graphs of Linear Functions

Suppose the linear function \begin{align*}f(x)= -0.25x + 10\end{align*} represents the amount of money you have left to play video games, where \begin{align*}f(x)\end{align*} is the amount of money you have left and \begin{align*}x\end{align*} 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 \begin{align*}y-\end{align*}intercept of the graph? 

Linear Function Graphs

You can see that the notation \begin{align*}f(x)=\end{align*} and \begin{align*}y=\end{align*} are interchangeable. This means you can substitute the notation \begin{align*}y=\end{align*} for \begin{align*}f(x)=\end{align*} and use all the concepts you have learned on linear equations.

Let's rewrite the following function in slope-intercept form:

\begin{align*}f(x)& =\frac{1}{3}x+1\end{align*}

\begin{align*}\text{Replace} \ f(x)& = \text{with} \ y=.\\ y =\frac{1}{3} x+1\end{align*} 

This equation is in slope-intercept form. You can now graph the function by graphing the \begin{align*} y-\end{align*}intercept and then using the slope as a set of directions to find your second coordinate.

Now, let's graph the following function:


The first step is to rewrite the single fraction as two separate fractions.

\begin{align*}f(x)=\frac{3x+5}{4}=\frac{3}{4} x+\frac{5}{4}\end{align*}

This equation is in slope-intercept form. The \begin{align*}y-\end{align*}intercept is at the ordered pair (0, \begin{align*} \frac{5}{4}\end{align*}) and the slope is \begin{align*}\frac{rise}{run}=\frac{3}{4}\end{align*}. Beginning at the \begin{align*}y-\end{align*}intercept and using the slope to find a second coordinate, you get the graph:

Analyzing Graphs of Real-World Linear Functions

The previous graph, created by T. Barron and S. Katsberg from the University of Georgia (http://jwilson.coe.uga.edu/emt668/EMAT6680.Folders/Barron/unit/Lesson%204/4.html), shows the relationship between the salary (in thousands of dollars) and the taxes paid (in thousands of dollars) in red. The blue function represents a direct variation situation in which the constant of variation (or the slope) is 0.30, or a 30% tax rate. This direct variation represents a flat tax of 30%.

The red line has three slopes. The first portion of the line from $0 to $15,000 has a slope of 0.20, or 20%. The second portion of the line from $15,000 to $45,000 has a slope of 0.25, or 25%. The slope of the portion of the line representing greater than $45,000 of salary is 0.35, or 35%.

Suppose you wanted to compare the amount of taxes you would pay if your salary was $60,000. If the blue line was \begin{align*}blue(s)\end{align*} and the red line was \begin{align*}red(s)\end{align*}, then you would evaluate each function for \begin{align*}s=60,000\end{align*}.

Using the graph, \begin{align*}blue(60)=18\end{align*} and \begin{align*}red(60)=15\end{align*}. Therefore, you would pay more taxes with the blue line tax rate than the red line tax rate. 




Example 1

Earlier, you were told that the function \begin{align*}f(x)= -0.25x + 10\end{align*} represents the amount of money you have left to play video games, where \begin{align*}f(x)\end{align*} is the amount of money you have left and \begin{align*}x\end{align*} is the number of video games you have played so far. How do you graph this function? What would be the slope and the \begin{align*}y-\end{align*}intercept of the graph?

To graph this function, you need to replace \begin{align*}f(x)\end{align*} with \begin{align*}y\end{align*}. Then, the equation will be \begin{align*}y = -0.25x + 10\end{align*}. The slope of the graph is -0.25 and the  \begin{align*}y-\end{align*}intercept of the graph is 10. The graph will look something like the below:

License: CC BY-NC 3.0

Example 2

Car A uses 15 gallons of gasoline to drive 2.5 hours. Write an equation for this function, graph it and use it to answer: How much gas would this car use if it were driving 30 minutes?

The car uses a certain number of gallons of gasoline per hour. That is a rate, and if you multiply it by a certain number of hours, it will tell you how many gallons are needed to drive that many hours. This can be written as a linear function, where the dependent variable \begin{align*}f(x)\end{align*} is the number of gallons. In other words, the number of gallons needed depends on what length of time the car drives. Time in hours is the independent variable, \begin{align*}x\end{align*}.

The rate, which is the slope, is:


The function is then:


The graph looks like the following:

You can see in the graph that \begin{align*}f(0.5)=3\end{align*}. Thus, to drive a half hour will require 3 gallons of gasoline.

You can also check that your graph was correct in the first place by seeing that one of the coordinate pairs is 2.5 hours and 15 gallons, which was originally given in the problem.


  1. The roasting guide for a turkey suggests cooking for 100 minutes plus an additional 8 minutes per pound.
    1. Write a function for the roasting time, given the turkey weight in pounds \begin{align*}(x)\end{align*}.
    2. Determine the time needed to roast a 10-lb turkey.
    3. Determine the time needed to roast a 27-lb turkey.
    4. Determine the maximum size turkey you could roast in \begin{align*}4\frac{1}{2}\end{align*} hours.
  2. \begin{align*}F(C)=1.8C+32\end{align*} is the function used to convert Celsius to Fahrenheit. Find \begin{align*}F(100)\end{align*} and explain what it represents.
  3. A prepaid phone card comes with $20 worth of calls. Calls cost a flat rate of $0.16 per minute. Write the value of the card as a function of minutes per call. Use a function to determine the number of minutes of phone calls you can make with the card.
  4. You can burn 330 calories during one hour of bicycling. Write this situation using \begin{align*}b(h)\end{align*} as the function notation. Evaluate \begin{align*}b(0.75)\end{align*} and explain its meaning.
  5. Sadie has a bank account with a balance of $650.00. She plans to spend $55 per week.
    1. Write this situation using function notation.
    2. Evaluate her account after 10 weeks. What can you conclude?

Mixed Review

  1. Simplify \begin{align*}-120\left (\frac{1}{2}\right )\left (\frac{3}{5}\right )\end{align*}.
  2. Find the sum: \begin{align*}7\frac{1}{4}+3\frac{2}{3}+5\frac{3}{4}\end{align*}.
  3. Simplify \begin{align*}-3(4m+11)\end{align*}.
  4. Is the following situation an example of a function? Let \begin{align*}x=\end{align*}salary and \begin{align*}y=\end{align*}taxes paid.
  5. \begin{align*}y\end{align*} varies directly as \begin{align*}z\end{align*}, and \begin{align*}y=450\end{align*} when \begin{align*}z=6\end{align*}. Find the constant of variation.

Review (Answers)

To see the Review answers, open this PDF file and look for section 4.14. 

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  1. [1]^ License: CC BY-NC 3.0

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