5.2.5: Connecting Representations of Functions

Lesson

Let's connect tables, equations, graphs, and stories of functions.

Exercise $\PageIndex{1}$: Which are the Same? Which are Different?

Here are three different ways of representing functions. How are they alike? How are they different? $y=2x$

Figure $\PageIndex{1}$

Table $\PageIndex{1}$

$p$	-2	-1	0	1	2	3
$q$	4	2	0	-2	-4	-6

Exercise $\PageIndex{2}$: Comparing Temperatures

The graph shows the temperature between noon and midnight in City A on a certain day.

Figure $\PageIndex{2}$: Graph in a coordinate plane, horizontal, time in hours after noon, 0 to 12 by ones, vertical, temperature in degrees Farenheit, 50 to 60 by ones. The graph starts at 0 comma 50 and climbs slowly, the more quickly as it moves right, going through 2 comma 52 and 4 comma 57 before peaking at 5 point 8 comma 59. The graph then steadily declines as it moves right until it reaches the point 12 comma 52 point 5.

The table shows the temperature, $T$, in degrees Fahrenheit, for $h$ hours after noon, in City B.

Table $\PageIndex{2}$

$h$	1	2	3	4	5	6
$T$	82	78	75	62	58	59

1. Which city was warmer at 4:00 p.m.?
2. Which city had a bigger change in temperature between 1:00 p.m. and 5:00 p.m.?
3. How much greater was the highest recorded temperature in City B than the highest recorded temperature in City A during this time?
4. Compare the outputs of the functions when the input is 3.

Exercise $\PageIndex{3}$: Comparing Volumes

The volume, $V$, of a cube with edge length $s$ cm is given by the equation $V=s^{3}$. The volume of a sphere is a function of its radius (in centimeters), and the graph of this relationship is shown here.

1. Is the volume of a cube with edge length $s=3$ greater or less than the volume of a sphere with radius 3?
2. If a sphere has the same volume as a cube with edge length 5, estimate the radius of the sphere.
3. Compare the outputs of the two volume functions when the inputs are 2.

Here is an applet to use if you choose.

Are you ready for more?

Estimate the edge length of a cube that has the same volume as a sphere with radius 2.5.

Exercise $\PageIndex{4}$: It's Not a Race

Elena’s family is driving on the freeway at 55 miles per hour.

Andre’s family is driving on the same freeway, but not at a constant speed. The table shows how far Andre's family has traveled, $d$, in miles, every minute for 10 minutes.

Table $\PageIndex{3}$

$t$	1	2	3	4	5	6	7	8	9	10
$d$	0.9	1.9	3.0	4.1	5.1	6.2	6.8	7.4	8	9.1

1. How many miles per minute is 55 miles per hour?
2. Who had traveled farther after 5 minutes? After 10 minutes?
3. How long did it take Elena’s family to travel as far as Andre’s family had traveled after 8 minutes?
4. For both families, the distance in miles is a function of time in minutes. Compare the outputs of these functions when the input is 3.

Summary

Functions are all about getting outputs from inputs. For each way of representing a function—equation, graph, table, or verbal description—we can determine the output for a given input.

Let's say we have a function represented by the equation $y=3x+2$ where $y$ is the dependent variable and $x$ is the independent variable. If we wanted to find the output that goes with 2, we can input 2 into the equation for $x$ and finding the corresponding value of $y$. In this case, when $x$ is 2, $y$ is 8 since $3\cdot 2+2=8$.

If we had a graph of this function instead, then the coordinates of points on the graph are the input-output pairs. So we would read the $y$-coordinate of the point on the graph that corresponds to a value of 2 for $x$. Looking at the graph of this function here, we can see the point $(2,8)$ on it, so the output is 8 when the input is 2.

Figure $\PageIndex{3}$: The graph of a line in the coordinate plane with the origin labeled “O”. The horizontal axis has the numbers negative 1 through 2 indicated and there are vertical gridlines between each integer. The vertical axis has the numbers negative 2 through 8, in increments of 2, indicated, and there are horizontal grid lines in between each integer. The line begins to the right of the y axis and below the x axis. It slants upward and to the right passing through the point with coordinates negative 1 comma negative 1, crosses the y axis at 2, and passes through the indicated point labeled 2 comma 8.

A table representing this function shows the input-output pairs directly (although only for select inputs).

Table $\PageIndex{4}$

$x$	-1	0	1	2	3
$y$	-1	2	5	8	11

Again, the table shows that if the input is 2, the output is 8.

Glossary Entries

Definition: Dependent Variable

A dependent variable represents the output of a function.

For example, suppose we need to buy 20 pieces of fruit and decide to buy apples and bananas. If we select the number of apples first, the equation $b=20-a$ shows the number of bananas we can buy. The number of bananas is the dependent variable because it depends on the number of apples.

Definition: Independent Variable

An independent variable represents the input of a function.

For example, suppose we need to buy 20 pieces of fruit and decide to buy some apples and bananas. If we select the number of apples first, the equation $b=20-a$ shows the number of bananas we can buy. The number of apples is the independent variable because we can choose any number for it.

Definition: Radius

A radius is a line segment that goes from the center to the edge of a circle. A radius can go in any direction. Every radius of the circle is the same length. We also use the word radius to mean the length of this segment.

For example, $r$ is the radius of this circle with center $O$.

Figure $\PageIndex{4}$

Definition: Volume

Volume is the number of cubic units that fill a three-dimensional region, without any gaps or overlaps.

For example, the volume of this rectangular prism is 60 units³, because it is composed of 3 layers that are each 20 units³.

Figure $\PageIndex{5}$

Practice

Exercise $\PageIndex{5}$

The equation and the tables represent two different functions. Use the equation $b=4a-5$ and the table to answer the questions. This table represents $c$ as a function of $a$.

Table $\PageIndex{5}$

$a$	-3	0	2	5	10	12
$c$	-20	7	3	21	19	45

1. When $a$ is -3, is $b$ or $c$ greater?
2. When $c$ is 21, what is the value of $a$? What is the value of $b$ that goes with this value of $a$?
3. When $a$ is 6, is $b$ or $c$ greater?
4. For what values of $a$ do we know that $c$ is greater than $b$?

Exercise $\PageIndex{6}$

Elena and Lin are training for a race. Elena runs her mile at a constant speed of 7.5 miles per hour.

Lin’s total distances are recorded every minute:

Table $\PageIndex{6}$

time (minutes)	1	2	3	4	5	6	7	8	9
distance (miles)	0.11	0.21	0.32	0.41	0.53	0.62	0.73	0.85	1

1. Who finished their mile first?
2. This is a graph of Lin's progress. Draw a graph to represent Elena's mile on the same axes.

Figure $\PageIndex{6}$

3. For these models, is distance a function of time? Is time a function of distance? Explain how you know.

Exercise $\PageIndex{7}$

Match each function rule with the value that could not be a possible input for that function.

1. 3 divided by the input
2. Add 4 to the input, then divide this value into 3
3. Subtract 3 from the input, then divide this value into 1

1. 3
2. 4
3. -4
4. 0
5. 1

(From Unit 5.1.2)

Exercise $\PageIndex{8}$

Find a value of $x$ that makes the equation true. Explain your reasoning, and check that your answer is correct.

$-(-2x+1)=9-14x$

(From Unit 4.2.3)