Connecting Graphical and Verbal Representations of a Function

4.6.3 Connecting Graphical and Verbal Representations of a Function

4.6.3 • Connecting Graphical and Verbal Representations of a Function

Activity

In a bungee jump, the height of the jumper is a function of time since the jump begins.

Line graph showing height (h, in meters) versus time (t, in seconds), with oscillations decreasing in amplitude over time, starting high at t equals 0 and gradually leveling off around 30 meters by t equals 30 seconds.

Expressions and Equations:

$h (0)$
$h (t) = 0$
$h (4)$
$h (t) = 80$
$h (t) = 45$

Features:

first dip in the graph
vertical intercept
first peak in the graph
horizontal intercept
maximum

1. Match each description about the jump to a corresponding expression or equation and to a feature on the graph.

One expression or equation does not have a matching verbal description. Its corresponding graphical feature is also not shown on the graph. Interpret that expression or equation in terms of the jump and in terms of the graph of the function. Record your interpretation in the last row of the table.

Description of Jump	Expression or Equation	Feature of Graph
A. the greatest height that the jumper is from the river
B. the height from which the jumper was jumping
C. the time at which the jumper reached the highest point after the first bounce
D. the lowest point that the jumper reached in the entire jump
E. the time at which the jumper hits the surface of the river

This table outlines key descriptive features of a jump, providing prompts for corresponding mathematical expressions and graph characteristics.

A.Expression or Equation

Compare your answer:

$h (t) = 80$

Feature of Graph

Compare your answer:

maximum

B. Expression or Equation

Compare your answer:

$h (0)$

Feature of Graph

Compare your answer:

vertical intercept

C. Expression or Equation

Compare your answer:

$h (t) = 45$

Feature of Graph

Compare your answer:

first peak in the graph

D. Expression or Equation

Compare your answer:

$h (4)$

Feature of Graph

Compare your answer:

first dip in the graph; also the minimum of the graph shown

E. Description of Jump

Compare your answer:

the time at which the jumper hits the surface of the river

Expression or Equation

Compare your answer:

$h (t) = 0$

Feature of Graph

Compare your answer:

horizontal intercept

2. Use the graph to:

A. estimate $h (0)$ and $h (4)$ .

Compare your answers:

$h (0)$ is 80, and $h (4)$ is about 10.

2. If $t = 4$ , then $h (4)$ equals what value?

Compare your answer:

$h (4) \approx 10$

3. If $h (t) = 45$ , then $t$ equals what value?

Compare your answer:

The approximate solutions to $h (t) = 45$ are $t = 2$ and $t = 9$ .

4. If $h (t) = 0$ , then $t$ equals what value?

Compare your answer:

There is no solution to $h (t) = 0$ when $t$ is between 0 and 35, which is what is shown on the graph. We don't know if the function will have a solution when $t$ is greater than 35.

Are you ready for more?

Extending Your Thinking

Based on the information available, how long do you think the bungee cord is? Make an estimate and explain your reasoning.

Compare your answers:

About 50 meters. The low points on the graph increase with each bounce, and the high points decrease. They look like they are approaching about 30 meters in height above the river. Since the jump started from 80 meters above the river, this means that the bungee cord is about 50 meters long.

Video: The Meaning of Function Notation

Watch the following video to learn more about connecting graphical and verbal representations of a function.

Access multimedia content

The horizontal intercept is also called the $x$ -intercept, or zero .

The vertical intercept is also called the $y$ -intercept.

Self Check

The graph models the height, in feet, of a kickball over time. When is the ball at its maximum height?

$GRAPH THAT SHOWS THE HEIGHT OF A KICKBALL IN FEET AS A FUNCTION OF TIME. THE FUNCTION IS MODELED BY AN UPSIDE DOWN PARABOLA WITH A $y$-intercepts OF 0, $x$-intercepts OF 0 AND 5, AND MAXIMUM AT THE POINT (2.5, 31).$

$h(2.5) = 31$ ; The maximum height is about 2.5 feet. It occurred about 31 seconds after the ball was kicked.
$h(1)=20$ ; The maximum height occurred 1 second after the ball was kicked.
$h(5)=0$ ; The maximum height occurs about 6 seconds after the ball was kicked.
$h(2.5) = 31$ ; The maximum height is about 31 feet. It occurred about 2.5 seconds after the ball was kicked.

Additional Resources

More Real-Life Graphs

The graph below represents the total number of smartphones that are shipped to a retail store over the course of 50 days.

Graph that shows number of smartphones as a function of time in days x-axis goes from 0 to 50 in increments of 10 y-axis goes from 0 to 4000 in increments of 1000 the graph increases linearly from x equals 0 to x equals 20, (section a), remains constant from x equals 20 to x equals 25 (section b), and increases linearly again but with a less steep slope (section c).

Why does $f (20) = f (22)$ ?

Since section B is not changing, it represents the period of time where the store did not receive new shipments, so $f (20) = f (22)$ .

Try it

More Real-Life Graphs

The relationship between Jameson’s account balance $B$ , in dollars, and time $t$ , in days, is modeled by the graph below.

Graph that shows account balance in dollars as a function of time in days the x-axis goes from 0 to 14 in intervals of 2 the y-axis goes from 0 to 100 in intervals of 20 the graph is constant from x equals 0 to x equals 6, increases from x equals 6 to x equals 9, reaching a maximum at the point (9, 100), and then decreases.

What does the point $B (9) = 98$ represent in the context of the problem?

Here is how to describe what $B (9) = 98$ represents in the context of the problem:

$B (9) = 98$ represents the maximum amount of money Jameson had in his account over the time period shown. It also represents that after 9 days, Jameson had $98 in his account. It is the maximum because it is the highest point on the graph provided.