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4.7: Linear Function Graphs

Difficulty Level: At Grade Created by: CK-12

So far, the term function has been used to describe many of the equations we have been graphing. The concept of a function is extremely important in mathematics. Not all equations are functions. To be a function, for each value of \begin{align*}x\end{align*} there is one and only one value for \begin{align*}y\end{align*}.

Definition: A function is a relationship between two variables such that the input value has ONLY one unique output value.

Recall from Lesson 1.5 that a function rule replaces the variable \begin{align*}y\end{align*} with its function name, usually \begin{align*}f(x)\end{align*}. Remember that these parentheses do not mean multiplication. They separate the function name from the independent variable, \begin{align*}x\end{align*}.

\begin{align*}& \quad \ input\\ & \quad \ \ \ \downarrow\\ & \quad \underbrace{f(x)}= y \leftarrow output\\ & \ function\\ & \quad \ \ box\end{align*}

\begin{align*}f(x)\end{align*} is read “the function \begin{align*}f\end{align*} of \begin{align*}x\end{align*}” or simply “\begin{align*}f\end{align*} of \begin{align*}x\end{align*}.”

If the function looks like this: \begin{align*}h(x)=3x-1\end{align*}, it would be read \begin{align*}h\end{align*} of \begin{align*}x\end{align*} equals 3 times \begin{align*}x\end{align*} minus 1.

Using Function Notation

Function notation allows you to easily see the input value for the independent variable inside the parentheses.

Example: Consider the function \begin{align*}f(x)=-\frac{1}{2} x^2\end{align*}.

Evaluate \begin{align*}f(4)\end{align*}.

Solution: The value inside the parentheses is the value of the variable \begin{align*}x\end{align*}. Use the Substitution Property to evaluate the function for \begin{align*}x=4\end{align*}.

\begin{align*}f(4)& =-\frac{1}{2}(4^2)\\ f(4)& = -\frac{1}{2} \cdot 16\\ f(4)& =-8\end{align*}

To use function notation, the equation must be written in terms of \begin{align*}x\end{align*}. This means that the \begin{align*}y-\end{align*}variable must be isolated on one side of the equal sign.

Example: Rewrite \begin{align*}9x+3y=6\end{align*} using function notation.

Solution: The goal is to rearrange this equation so the equation looks like \begin{align*}y=\end{align*}. Then replace \begin{align*}y=\end{align*} with \begin{align*}f(x)=\end{align*}.

\begin{align*}9x+3y& =6 && \text{Subtract} \ 9x \ \text{from both sides}.\\ 3y& =6-9x && \text{Divide by} \ 3.\\ y& =\frac{6-9x}{3}=2-3x\\ f(x)& =2-3x\end{align*}

Functions as Machines

You can think of a function as a machine. You start with an input (some value), the machine performs the operations (it does the work), and your output is the answer. For example, \begin{align*}f(x)=3x+2\end{align*} takes some number, \begin{align*}x\end{align*}, multiplies it by 3 and adds 2. As a machine, it would look like this:

When you use the function machine to evaluate \begin{align*}f(2)\end{align*}, the solution is \begin{align*}f(2)=8\end{align*}.

Example 1: A function is defined as \begin{align*}f(x)=6x-36\end{align*}. Determine the following:

a) \begin{align*}f(2)\end{align*}

b) \begin{align*}f(p)\end{align*}


a) Substitute \begin{align*}x = 2\end{align*} into the function \begin{align*}f(x): \ f(2)=6 \cdot 2 - 36 = 12-36=-24\end{align*}.

b) Substitute \begin{align*}x = p\end{align*} into the function \begin{align*}f(x): \ f(p)=6p-36\end{align*}.

Graphing Linear Functions

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 in this chapter.

\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 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.

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

Solution: 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, written 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 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% tax rate. The slope of the line 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. We will look at how to use graphs as a problem-solving strategy in the next lesson.

Practice Set

Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set.  However, the practice exercise is the same in both. CK-12 Basic Algebra: Linear Function Graphs (11:49)

  1. How is \begin{align*}f(x)\end{align*} read?
  2. What does function notation allow you to do? Why is this helpful?
  3. Define function. How can you tell if a graph is a function?

In 4 – 7, tell whether the graph is a function. Explain your reasoning.

Rewrite each equation using function notation.

  1. \begin{align*}y=7x-21\end{align*}
  2. \begin{align*}6x+8y=36\end{align*}
  3. \begin{align*}x=9y+3\end{align*}
  4. \begin{align*}y=6\end{align*}
  5. \begin{align*}d=65t+100\end{align*}
  6. \begin{align*}F=1.8C+32\end{align*}
  7. \begin{align*}s=0.10(m)+25,000\end{align*}

In 15 – 19, evaluate \begin{align*}f(-3);f(7);f(0)\end{align*}, and \begin{align*}f(z)\end{align*}.

  1. \begin{align*}f(x)=-2x+3\end{align*}
  2. \begin{align*}f(x)=0.7x+3.2\end{align*}
  3. \begin{align*}f(x)=\frac{5(2-x)}{11}\end{align*}
  4. \begin{align*}f(t)=\frac{1}{2} t^2+4\end{align*}
  5. \begin{align*}f(x)=3-\frac{1}{2} x\end{align*}
  6. 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.
  7. \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.
  8. 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.
  9. 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.
  10. 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.
  6. Car A uses 15 gallons of gasoline to drive 2.5 hours. How much gas would this car use if it were driving 30 minutes?

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