Analyzing Graphs of Functions

4.6.2 Analyzing Graphs of Functions

4.6.2 • Analyzing Graphs of Functions

Activity

A toy rocket and a drone were launched at the same time.

Here are the graphs that represent the heights of two objects as a function of time since they were launched.

Height is measured in meters above the ground, and time is measured in seconds since launch.

A graph shows height in meters vs. time in seconds. Curve R rises to 45 meters at about 2 seconds, then falls to 0 by 5 seconds. Line D rises to 20 meters at 2 seconds, stays constant, then at 5 seconds decreases to be 0 at 7 seconds.

1. Analyze the graph of the toy rocket and describe as precisely as you can what was happening. Your description should be complete and precise enough that someone who is not looking at the graph could visualize how the object was behaving. Include where the toy rocket starts and when it stops.

Compare your answers:

The toy rocket was launched from a height of 25 meters above ground. It went up quickly, but it slowed down and then stopped getting higher when it reached 45 meters, which was 2 seconds after launch. At that point, it started to come back down. It hit the ground 5 seconds after it was launched.

2. Analyze the graph of the drone and describe as precisely as you can what was happening. Your description should be complete and precise enough that someone who is not looking at the graph could visualize how the object was behaving. Include where the drone starts and when it stops.

Compare your answers:

The drone took off from the ground. It went up at a constant speed for 2 seconds, up to a height of 20 meters. Then, it hovered at that height for 3 seconds. Afterward, it started descending at a constant speed for 2 seconds. It landed back on the ground 7 seconds after launch.

3. At what point were the drone and the toy rocket at the same height?

Compare your answers:

A little after 4 seconds, the drone and the toy rocket were at the same height (20 meters).

4. Which parts or features of the graphs show important information about each object’s movement? List the features or mark them on the graphs.

Compare your answers:

the points where the graphs begin (when $t$ is 0)
the points where the graphs cross the horizontal axis
the point where the graphs change direction
the shape of each graph (curved or straight)
the intervals when the graphs behave certain ways
the parts highlighted in the graph

An annotated graph shows height in meters vs. time in seconds. Curve R rises to 45 meters at about 2 seconds, then falls to 0 by 5 seconds. Line D rises to 20 meters at 2 seconds, stays constant, then at 5 seconds decreases to be 0 at 7 seconds. Arrows are at the x- and y-intercept, and the peak. The constant, increase, and decrease are circled.

Self Check

The graph below represents the temperature, $t(m)$ , in degrees Fahrenheit of water used for a cup of hot tea over time in minutes, $m$ . What does $t(0)$ represent?

$GRAPH THAT SHOWS TEMPERATURE IN DEGREES FAHRENHEIT AS A FUNCTION OF TIME IN MINUTES WITH A $y$-intercepts OF 70. THE GRAPH INCREASES, THEN DECREASES, AND THEN REMAINS CONSTANT.$

$t(0)= 180$ ; After 180 minutes, the temperature is 0 degrees.
$t(0)=70$ ; After 0 minutes, the temperature is 70 degrees.
$t(0)=180$ ; After 0 minutes, the temperature is 180 degrees.
$t(0)=70$ ; After 70 minutes, the temperature is 0 degrees.

Additional Resources

Real-Life Graphs of Functions

A ball is bouncing across the school yard. The vertical or $y$ -axis represents the height of the ball in feet, $h (t)$ , while the horizontal or $x$ -axis represents the time, $t$ , that has passed in seconds.

Graph that shows the height of a ball in feet as a function of time in seconds with a y-intercepts of 0 and x-intercepts of 0 and 1 from x equals 0 to x equals 1, the height of the ball is modeled by an upside down parabola with a maximum at the point (0.5, 4).

On the graph, $t = 0$ represents the first time the ball touches the ground.

What is $h (1)$ , and what does it represent in the context of the problem?

$h (1) = 0$ , and it represents the height of the ball after 1 second, or the ball has a height of zero after 1 second. The value inside of the parentheses is the value from the $x$ -axis or horizontal axis. The solution is the value of the graph at that point.

Try it

Real-Life Graphs of Functions

A ball is thrown across the field from point $A$ to point $B$ . It hits the ground at point $B$ . The path of the ball is shown in the diagram below. The $x$ -axis shows the horizontal distance the ball travels in feet, and the $y$ -axis shows the height of the ball in feet.

A graph that shows the height of a ball in feet as a function of the horizontal distance the ball travels in feet with a y-intercepts of A and x-intercepts of B.

Explain what $f (C)$ means in this graph.

Here is how to explain $f (C)$ in the graph:

$f (C)$ tells you to look at the height of the graph when $x = C$ .

This point on the graph is the highest point, or maximum.