Lesson 4Tables, Equations, and Graphs of Functions

Learning Goal

Let’s connect equations and graphs of functions.

Learning Targets

I can identify graphs that do, and do not, represent functions.
I can use a graph of a function to find the output for a given input and to find the input(s) for a given output.

Lesson Terms

dependent variable
independent variable
radius

Warm Up: Notice and Wonder: Doubling Back

What do you notice? What do you wonder?

A graph with distance from starting line (meters on the x from 0 to 175 and time (seconds) on the y from 0 to 72. The graph make a sideways V with the point in the middle on the right.

Activity 1: Equations and Graphs of Functions

The graphs of three functions are shown.

The graphs of three functions on the coordinate plane labeled A, B, and C. Graph A is the graph of a curve with the origin labeled O. The horizontal axis has the numbers 1 through 3 indicated and the vertical axis has the numbers 10 through 40, in increments of 10, indicated. The curve begins at the origin, extends to the right, and then extends upward and to the right. Graph B is the graph of a line with the origin labeled O. The horizontal axis has the numbers 1 through 3 indicated and the vertical axis has the numbers 50 through 200, in increments of 50, indicated. The line begins at the origin and slants upward and to the right. Graph C is the graph of a line with the origin labeled O. The horizontal axis has the nubers 20 through 100, in increments of 20, indicated and the vertical axis has the numbers 10 through 50, in increments 10, indicated. The line begins on the vertical axis at 50 and slants downward and to the right.

Match each of these equations to one of the graphs.
1. $d = 60 t,$ where $d$ is the distance in miles that you would travel in $t$ hours if you drove at 60 miles per hour.
2. $q = 50 - 0.4 d$ , where $q$ is the number of quarters, and $d$ is the number of dimes, in a pile of coins worth $12.50.
3. $A = π r^{2},$ where $A$ is the area in square centimeters of a circle with radius $r$ centimeters.
1. Graph A
2. Graph B
3. Graph C
Label each of the axes with the independent and dependent variables and the quantities they represent.
For each function: What is the output when the input is 1? What does this tell you about the situation? Label the corresponding point on the graph.
Find two more input-output pairs. What do they tell you about the situation? Label the corresponding points on the graph.

Are you ready for more?

A function inputs fractions $\frac{a}{b}$ between 0 and 1 where $a$ and $b$ have no common factors, and outputs the fraction $\frac{1}{b}$ . For example, given the input $\frac{3}{4}$ the function outputs $\frac{1}{4}$ , and to the input $\frac{1}{2}$ the function outputs $\frac{1}{2}$ . These two input-output pairs are shown on the graph.

Plot at least 10 more points on the graph of this function. Are most points on the graph above or below a height of $0.3$ ? Of height $0.01$ ?

A graph from 0 to 1 on both axis with points (one-half, one-half) and (three-fourths, one-fourth) plotted.

Activity 2: Running around a Track

Problem 1

Kiran was running around the track. The graph shows the time, $t$ , he took to run various distances, $d$ . The table shows his time in seconds after every three meters.

The graph of a curve in the coordinate plane with the origin labeled “O”. The horizontal axis is labeled “distance, in meters” and the numbers 0 through 30, in increments of 3, are indicated. The vertical axis is labeled “time, in seconds” and the numbers 0 through 10 are indicated. The curve begins at the origin and moves steadily upward and to the right. The line passes through the points with approximate coordinates 3 comma 1, 12 comma 4, and 30 comma 10.

$d$	$0$	$3$	$6$	$9$	$12$	$15$	$18$	$21$	$24$	$27$
$t$	$0$	$1.0$	$2.0$	$3.2$	$3.8$	$4.6$	$6.0$	$6.9$	$8.09$	$9.0$

How long did it take Kiran to run 6 meters?
How far had he gone after 6 seconds?
Estimate when he had run 19.5 meters.
Estimate how far he ran in 4 seconds.
Is Kiran’s time a function of the distance he has run? Explain how you know.

Problem 2

Priya is running once around the track. The graph shows her time given how far she is from her starting point.

What was her farthest distance from her starting point?
Estimate how long it took her to run around the track.
Estimate when she was 100 meters from her starting point.
Estimate how far she was from the starting line after 60 seconds.
Is Priya’s time a function of her distance from her starting point? Explain how you know.

Lesson Summary

Here is the graph showing Noah’s run.

The time in seconds since he started running is a function of the distance he has run. The point (18,6) on the graph tells you that the time it takes him to run 18 meters is 6 seconds. The input is 18 and the output is 6.

A graph of a function in the coordinate plane with the origin labeled "O". The horizontal axis is labeled “distance in meters” and the numbers 0 through 30, in increments of 3, are indicated. The vertical axis is labeled “time in seconds” and the numbers 0 through 10 are indicated. The function is approximately linear. It starts at the origin and moves upward and to the right, passing through the points with the coordinates 3 comma 1, 9 comma 3, 18 comma 6, and 30 comma 10.

The graph of a function is all the coordinate pairs, (input, output), plotted in the coordinate plane. By convention, we always put the input first, which means that the inputs are represented on the horizontal axis and the outputs, on the vertical axis.