# Lesson 7: Connecting Representations of Functions

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

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

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

1. $y = 2x$
2. 3.  $p$ $q$ -2 -1 0 1 2 3 4 2 0 -2 -4 -6

## 7.2: Comparing Temperatures

The graph shows the temperature between noon and midnight in City A on a certain day. The table shows the temperature, $T$, in degrees Fahrenheit, for $h$ 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.

## 7.3: Comparing Volumes

The volume, $V$, of a cube with side length $s$ is given by the equation $V = s^3$. 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 $s=3$ 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 $y$ as a function of $x$, so you need to use $y$ instead of $V$ and $x$ instead of $s$.

## 7.4: 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.

 $t$ $d$ 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.

## 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 $3\boldcdot 2 + 2=8$.

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. 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.