# Lesson 7Connecting Representations of Functions

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

### Learning Targets:

• I can compare inputs and outputs of functions that are represented in different ways.

## 7.1Which are the Same? Which are Different?

Here are three different ways of representing functions. How are they alike? How are they different?

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

## 7.2Comparing Temperatures

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

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

 h T 1 2 3 4 5 6 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.

## 7.3Comparing Volumes

The volume, , of a cube with side length is given by the equation . The graph of the volume of a sphere as a function of its radius is shown.

1. Is the volume of a cube with side length greater or less than a sphere with radius 3?

2. Estimate the radius of a sphere that has the same volume as a cube with side length 5.

3. Compare the outputs of the two volume functions when the inputs are 2.

Here is an applet to use if you choose. Note: If you want to graph an equation with this applet, it expects you to enter as a function of , so you need to use instead of and instead of

### Are you ready for more?

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

## 7.4It’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, , in miles, every minute for 10 minutes.

 t d 1 2 3 4 5 6 7 8 9 10 0.9 1.9 3 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.

## Lesson 7 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 where is the dependent variable and is the independent variable. If we wanted to find the output that goes with 2, we can input 2 into the equation for and finding the corresponding value of . In this case, when is 2, is 8 since .

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 -coordinate of the point on the graph that corresponds to a value of 2 for . Looking at the graph of this function here, we can see the point on it, so the output is 8 when the input is 2.

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

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

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

## Lesson 7 Practice Problems

1. The equation and the tables represent two different functions. Use the equation and the table to answer the questions. This table represents as a function of

 a c -3 0 2 5 10 12 -20 7 3 21 19 45
1. When is -3, is or greater?
2. When is 21, what is the value of ? What is the value of that goes with this value of ?
3. When is 6, is or greater?
4. For what values of do we know that is greater than ?
2. 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
3. Elena and Lin are training for a race. Elena runs her mile a constant speed of 7.5 miles per hour.

Lin’s times are recorded every minute:

 time (minutes) distance (miles) 1 2 3 4 5 6 7 8 9 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.

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

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