4.7: Linear Function Graphs
Learning Objectives
 Recognize and use function notation.
 Graph a linear function.
 Analyze arithmetic progressions.
Introduction
The highly exclusive Fellowship of the Green Mantle allows in only a limited number of new members a year. In its third year of membership it has 28 members, in its fourth year it has 33, and in its fifth year it has 38. How many members are admitted a year, and how many founding members were there?
Functions
So far we’ve used the term function to describe many of the equations we’ve been graphing, but in mathematics it’s important to remember that not all equations are functions. In order to be a function, a relationship between two variables,
Visually this means the graph of
Use Function Notation
When we write functions we often use the notation “
Example 1
Rewrite the following equations so that
a)
b)
c)
d)
Solution
a) Simply replace
b) Again, replace
c) First we need to solve for
d) Solve for
Using the functional notation in an equation gives us more information. For instance, the expression
We can also plug in expressions rather than just numbers. For example, if our function is
Example 2
A function is defined as
a)
b)
c)
d)
e)
Solution
a) Substitute
b) Substitute
c) Substitute
d) Substitute
e) Substitute
Graph a Linear Function
Since the notations “
Example 3
Graph the function
Solution
We can write this function in slopeintercept form:
So our graph will have a
Example 4
Graph the function
Solution
This time we’ll solve for the
To solve for the
To solve for the
We can graph the function from those two points:
Arithmetic Progressions
You may have noticed that with linear functions, when you increase the



0  3 
1  5 
2  7 
3  9 
4  11 
Notice that the values for
Example 5
Find the common difference for the following arithmetic progressions:
a) {7, 11, 15, 19, ...}
b) {12, 1, 10, 21, ...}
c) {7, __, 12, __, 17, ...}
Solution
a)
b)
c) There are not 2 consecutive terms here, but we know that to get the term after 7 we would add the common difference, and then to get to 12 we would add the common difference again. So twice the common difference is
Arithmetic sequences and linear functions are very closely related. To get to the next term in a arithmetic sequence, you add the common difference to the last term; similarly, when the
The graph below shows the arithmetic progression {2, 0, 2, 4, 6...} along with the function
We can write a formula for an arithmetic progression: if we define the first term as
The online calculator at http://planetcalc.com/177/ will tell you the
Lesson Summary
 In order for an equation to be a function, the relationship between the two variables,
x andy , must map eachx− value to exactly oney− value.  The graph of a function of
y versusx must pass the vertical line test: any vertical line will only cross the graph of the function in one place.  Functions can be expressed in function notation using
f(x)= in place ofy= .  The sequence of
f(x) values for a linear function form an arithmetic progression. Each number is greater than (or less than) the preceding number by a fixed amount, or common difference.
Review Questions
 When an object falls under gravity, it gains speed at a constant rate of 9.8 m/s every second. An item dropped from the top of the Eiffel Tower, which is 300 meters tall, takes 7.8 seconds to hit the ground. How fast is it moving on impact?
 A prepaid phone card comes with $20 worth of calls on it. Calls cost a flat rate of $0.16 per minute.
 Write the value left on the card as a function of minutes used so far.
 Use the function to determine how many minutes of calls you can make with the card.
 For each of the following functions evaluate:

f(x)=−2x+3 
f(x)=0.7x+3.2 
f(x)=5(2−x)11 
f(−3) 
f(0) 
f(z) 
f(x+3) 
f(2n) 
f(3y+8) 
f(q2)


 Determine whether the following could be graphs of functions.
 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 the given the turkey weight in pounds
(x) .  Determine the time needed to roast a 10 lb turkey.
 Determine the time needed to roast a 27 lb turkey.
 Determine the maximum size turkey you could roast in 4.5 hours.
 Write a function for the roasting time the given the turkey weight in pounds
 Determine the missing terms in the following arithmetic progressions.
 {11, 17, __, 73}
 {2, __, 4}
 {13, __, __, __, 0}
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