Name____________________________ Period______ Date___________ Functions Part 3
Functions Activity 1.3: Graphing Functions
A very effective method of representing a function is graphically. The way in which the output quantity
changes as the input increases can often be more readily identified in visual form. A graph can also show
any trends or patterns in the function situation. In the following problems, you will investigate the
graphical representation of the Park It function in Activity 1.1 and the cost-of-fill-up function in Activity
1.2.
Defining Functions Graphically
You may have seen an ordered pair before as the coordinates of a point in a rectangular coordinate
system, typically using ordered pairs of the form where 𝑥 is the input and 𝑦 is the output. The first value,
the horizontal coordinate, indicates the directed distance (right or left) from the vertical axis. The
second value, the vertical coordinate, indicates the directed distance (up or down) from the horizontal
axis.
The variables may not always be represented by 𝑥 and 𝑦, but the horizontal axis will
always be the input axis, and the vertical axis will always be the output axis.
1. Using a 24-hour clock, convert to ordered pairs
all the values in the Park It table from Actvity 1.1.
Plot each ordered pair on the following grid. Set
your axes and scales by noting the smallest and
largest values for both input and output. Label each
axis by both the variable name and its designation
as input or output. Remember, the scale for the horizontal axis does not
have to be the same as the scale for the vertical axis. However, each
scale (vertical and horizontal) must be divided into equal intervals.
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The preceding graph, which consists of a set of labeled axes and 16 points, presents the same
information that is in the Park It table, but in a different way. It shows the information as a graph and
therefore defines the function graphically.
2. a. Determine the practical domain of the Park It function.
b. Determine the practical range of the function. Assume that the capacity of the parking lot is 600
cars.
c. Describe any patterns or trends that you observe in the graph.
d. Can the Park It function be defined symbolically? Explain.
Note that the graph of the Park It function consists of 16 distinct points that are not connected. The
input variable (time of day) is defined only for the car counts in the parking lot for each hour from 7A.M.
to 10 P.M. The Park It function is said to be discrete because it is defined only at isolated, distinct input
values (practical domain). The function is not defined for input values between these particular values.
Caution. In order to use the graph for a relationship such as the parking lot situation to make predictions
or to recognize patterns, it is convenient to connect the points with line segments. This creates a type of
continuous graph. This changes the domain shown in the graph from “some values” to “all values.”
Therefore, you need to be cautious. Connecting data points may cause confusion when working with
real-world situations.
3. a. In Example 1 on page 2 of Activity 1.1, the
high temperature in Lake Placid is a function of
the date. Plot each ordered pair as a point on
an appropriately scaled and labeled set of
coordinate axes.
b. Determine the practical domain of the
temperature function.
c. Determine the practical range of the
function.
d. Is this function discrete?
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4. The cost-of-fill-up function in Activity 1.2 was defined by the equation 𝑐 = 12.6𝑝, where 𝑐 is the cost
to fill up your car with 12.6 gallons of gas at a price of 𝑝 dollars per gallon.
a. What is the practical domain of this function? Refer to Problem 7 in Activity 1.2.
b. List five ordered pairs of the cost-of-fill-up function in the following table.
Price per Gallon, 𝑝
(dollars)
Cost of Fill-Up, 𝑐
c. Sketch a graph of the cost-of-fill-up function by first plotting the five points from
part b on properly scaled and labeled coordinate axes.
d. Does the graph of the function consist of just the five points from part b? Explain.
e. Describe any patterns or trends in the graph.
The cost-of-fill-up function is defined for all input values in the practical domain in Problem 4a.
The five points determined in part b can be connected to form a continuous graph. Such a graph
is said to be continuous over its practical domain.
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5. a. Consider the gross pay function defined 𝑔 = 9.50ℎ by in Problem 10 of Activity 1.2 where you work
between 0 and 25 hours per week. Plot the ordered pairs based on your salary of $9.50 per hour.
Number of Hours, ℎ
Gross Pay, 𝑔, (dollars)
0
5
10
15
20
25
b. Is the gross pay function discrete or continuous? Explain.
Graphing Functions Using Technology
Following is the graph of the cost-of-fill-up function defined by over its practical domain.
Graph 1
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The domain for the general function defined by 𝑐 = 12.6𝑝, with no connection to the context
of the situation, is the set of all real numbers, since any real number can be substituted for 𝑝
in 12.6𝑝. Following is a graph of 𝑐 = 12.6𝑝 for any real number 𝑝.
Graph 2
Each of these graphs can be obtained using your graphing calculator as demonstrated in Problems 6 and
7.
Recall that the independent variable in your graphing calculator is represented by 𝑥 and the dependent
variable is represented by 𝑓(𝑥). Therefore, the cost-of-fill-up equation 𝑐 = 12.6𝑝 need to be keyed in as
𝑓(𝑥) = 12.6𝑥.
6. To graph using the Nspire calculator, follow the procedure below. First you will set up the window for
viewing the graph. Next you will enter the equation.
a. Use your graphing calculator to sketch the graph of this function.
 c1 no, 2 to display a graphing window
 b41 and scroll down to set the values
o 𝑋𝑚𝑖𝑛 = 0, 𝑋𝑚𝑎𝑥 = 5 , 𝑋𝑠𝑐𝑎𝑙𝑒 = 𝑎𝑢𝑡𝑜 𝑌𝑚𝑖𝑛 = 0, 𝑌𝑚𝑎𝑥 = 65, 𝑌𝑠𝑐𝑎𝑙𝑒 = 𝑎𝑢𝑡𝑜 .
and hit OK.
 Hit /G and enter 𝑓(𝑥) = 𝟏𝟐. 𝟔𝒙 and press enter and you will see the graph.
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 The screen should appear as follows:
7. a. Determine reasonable window settings to obtain the graph of the general function defined by for
which the domain is any real number.
b. Change your window to new screen values. Enter b41 and scroll down to set the values
𝑋𝑚𝑖𝑛 = −10,
𝑋𝑚𝑎𝑥 = 10 , 𝑋𝑠𝑐𝑎𝑙𝑒 = 𝑎𝑢𝑡𝑜
𝑌𝑚𝑖𝑛 = −25,
𝑌𝑚𝑎𝑥 = 25, 𝑌𝑠𝑐𝑎𝑙𝑒 = 𝑎𝑢𝑡𝑜 .
The screen should appear as follows.
c. How does the graph compare to Graph 2 on the page 5?
Defining Functions: A Summary
A function can be defined by a written statement (verbally), symbolically, numerically, and graphically.
The following example illustrates the different ways that the gross pay function can be defined.
If you work for an hourly wage, then your gross pay is a function of the
number of hours worked. If you earn $9.50 per hour, then define the
gross pay function verbally, symbolically, numerically, and graphically.
SOLUTION:
Verbal Definition: A Statement of the Definition of the Function:
The gross pay will be the number of hours worked multiplied by $9.50.
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Symbolic Definition: If 𝑔 represents the gross pay and ℎ represents the number of hours worked, then
𝑔 = 9.50ℎ
Numerical Definition: The gross pay is represented by the following table.
Graphical Definition: The gross pay is represented by the following graph.
8. Sales tax is a function of the cost of any item. If the sales tax in your area is 6%, define the cost
function in four different ways.
Verbal Definition:
Symbolic Defintion:
Numerical Defintion:
Cost of item ($) 10
20
Sales Tax
30
40
50
100
150
200
300
500
1000
Graphical Definition:
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Summary - Graphing Functions: Activity 1.3
1. When a function is defined graphically, the input variable will be represented on the horizontal
axis and the output on the vertical axis.
2. Functions are discrete if they are defined only at isolated input values and do not make sense or
are not defined for input values between those values.
3. Functions are continuous if they are defined for all input values, and if there are no gaps between
any consecutive input values.
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Name_________________
Date_________
Functions Activity 1.3: Graphing Functions: Exercises
1. In the following table, the amount of snowfall is a function of the elevation.
a. Plot the ordered pairs on an appropriately scaled and labeled set of coordinate axes.
b. Would you consider this a discrete situation (consisting of separate, isolated points) or a
continuous situation? Explain.
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2. Plot the ordered pairs of each of the following functions on an appropriately scaled and labeled set
of axes. Then determine if a continuous or discrete graph is more appropriate for the situation.
a. In a science experiment, the amount of water displaced in a graduated cylinder is a function
of the number of marbles placed in the cylinder. The results of an experiment are recorded
in the following table.
b. Forensic anthropologists predict the height of a male based on the length of his femur. The
following table demonstrates this functional relationship.
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A primary objective of this course is to help you develop a familiarity with graphs, equations, and
properties of a variety of functions. Later, you will study a type of function called a quadratic function.
a. In Exercises 3–6, sketch a graph of each quadratic function in the standard window of a
graphing calculator. Then match each graph with the corresponding graphs in parts a–d.
Remember to use c1 no, 2 to create the graph.
7. Sketch a graph of the quadratic function 𝑦 = 𝑥 2 + 15 defined by in the standard window of your
graphing calculator.
a. Use your graphing calculator to see the graph c1 no, 2
o Enter 𝑓(𝑥) = 𝒙𝟐 + 𝟏𝟓 and press enter. What do you observe?
b. Complete the following table for this function. You can use cA and c4to go
back and forth on the calculator.
𝒙
𝒚
−𝟑
−𝟐
−𝟏
𝟎
𝟏
𝟐
𝟑
c. Describe how you would use the results in part b to help select an appropriate viewing
window.
d. Change your window to new screen values. Enter b41 and scroll down to set the
values
e. Sketch a graph of the function below.
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8. In Exercise 10 of Activity 1.1, you were assigned the job of supplying paint for the exterior of the
bunk houses at a youth summer camp. You found that 1 gallon of paint will cover 400 square feet of
flat surface. The number of square feet you can cover with the paint is a function of the number of
gallons of paint used. Define the paint coverage function in four different ways.
Verbal Statement:
Symbolic Statement:
𝑛, Number of Gallons
of Paint
𝑠, Square Feet Covered
by the Paint
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