4.7: Linear Function Graphs
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
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
If the function looks like this:
Using Function Notation
Function notation allows you to easily see the input value for the independent variable inside the parentheses.
Example: Consider the function
Evaluate
Solution: The value inside the parentheses is the value of the variable
To use function notation, the equation must be written in terms of
Example: Rewrite
Solution: The goal is to rearrange this equation so the equation looks like
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,
When you use the function machine to evaluate
Example 1: A function is defined as
a)
b)
Solution:
a) Substitute
b) Substitute
Graphing Linear Functions
You can see that the notation
This equation is in slopeintercept form. You can now graph the function by graphing the
Example: Graph
Solution: The first step is to rewrite the single fraction as two separate fractions.
This equation is in slopeintercept form. The
Analyzing Graphs of RealWorld 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
Using the graph,
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. CK12 Basic Algebra: Linear Function Graphs (11:49)
 How is
f(x) read?  What does function notation allow you to do? Why is this helpful?
 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.

y=7x−21 
6x+8y=36 
x=9y+3 
y=6 
d=65t+100 
F=1.8C+32 
s=0.10(m)+25,000
In 15 – 19, evaluate

f(x)=−2x+3 
f(x)=0.7x+3.2 
f(x)=5(2−x)11 
f(t)=12t2+4 
f(x)=3−12x  The roasting guide for a turkey suggests cooking for 100 minutes plus an additional 8 minutes per pound.
 Write a function for the roasting time, given the turkey weight in pounds
(x) .  Determine the time needed to roast a 10lb turkey.
 Determine the time needed to roast a 27lb turkey.
 Determine the maximum size turkey you could roast in
412 hours.
 Write a function for the roasting time, given the turkey weight in pounds

F(C)=1.8C+32 is the function used to convert Celsius to Fahrenheit. FindF(100) and explain what it represents.  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.
 You can burn 330 calories during one hour of bicycling. Write this situation using
b(h) as the function notation. Evaluateb(0.75) and explain its meaning.  Sadie has a bank account with a balance of $650.00. She plans to spend $55 per week.
 Write this situation using function notation.
 Evaluate her account after 10 weeks. What can you conclude?
Mixed Review
 Simplify
−120(12)(35) .  Find the sum:
714+323+534 .  Simplify
−3(4m+11) .  Is the following situation an example of a function? Let
x= salary andy= taxes paid. 
y varies directly asz , andy=450 whenz=6 . Find the constant of variation.  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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