A graph, origin O. The horizontal axis is labeled time. The vertical axis is labeled temperature. A curve, shaped like a U, opens up. The curve has a minimum below the horizontal axis.
D
A graph, origin O. Horizontal axis is labeled time. Vertical axis is labeled temperature. The curve begins on the vertical axis and trends up, then down, then up, then down forming U shapes, alternating opening up and down.
8.2
Activity
A flag ceremony is held at a Fourth of July event. The height of the flag is a function of time.
Here are some graphs that could each be a possible representation of the function.
A
A graph, origin O. Horizontal axis, time, scale from 0 to 6 by 1s. Vertical axis, height, scale from 0 to 6 by 1s. A line passes through the points 0 comma 0 and 4 comma 5.
B
A graph, origin O. Horizontal axis, time, scale from 0 to 6 by 1s. Vertical axis, height, scale from 0 to 6 by 1s. A horizontal line passes through 3 on the vertical axis.
C
A graph, origin O. Horizontal axis, time, scale from 0 to 6 by 1s. Vertical axis, height, scale from 0 to 6 by 1s. The curve begins at the point 0 comma 0 and trends upwards towards the point 1 comma 2 then levels out. As it approaches 3 on the horizontal axis, the graph shifts and trends up and to the right at a fast pace.
D
A graph, origin O. Horizontal axis, time, scale from 0 to 6 by 1s. Vertical axis, height, scale from 0 to 6 by 1s. In each horizontal interval of 1.25, the line trends upwards and downward in the shape of a U, opening downward. This pattern repeats for all 6 intervals.
E
A graph, origin O. Horizontal axis, time, scale from 0 to 6 by 1s. Vertical axis, height, scale from 0 to 6 by 1s. In each horizontal interval of 1, the line trends upwards and then levels out before continuing upward in the next interval. This pattern repeats for all 6 intervals.
F
A graph, origin O. Horizontal axis, time, scale from 0 to 6 by 1s. Vertical axis, height, scale from 0 to 6 by 1s. A line passes through the points 0 comma 0, 3 comma 6, 5 comma 3, and the line continues horizontally off the grid.
For each graph assigned to you, explain what it tells us about the flag.
Graph:
Decide as a group which graph(s) appear to be most realistic and which ones least realistic.
Here is another graph that relates time and height.
A graph, origin O. Horizontal axis, time, scale from 0 to 6 by 1s. Vertical axis, height, scale from 0 to 6 by 1s. A vertical line passes through 3 on the horizontal axis.
Can this graph represent the time and height of the flag? Explain your reasoning.
Is this a graph of a function? Explain your reasoning.
8.3
Activity
Your teacher will show a video of a flag being raised. Function gives the height of the flag over time. Height is measured in feet. Time is measured in seconds since the flag is fully secured to the string, which is when the video clip begins.
On the coordinate plane, sketch a graph that could represent function . Be sure to include a label and a scale for each axis.
Use your graph to estimate the average rate of change from the time the flag starts moving to the time it reaches the top. Be prepared to explain what the average rate of change tells us about the flag.
8.4
Activity
To prepare for a backyard party, a parent uses two identical hoses to fill a small pool that is 15 inches deep and a large pool that is 27 inches deep.
The height of the water in each pool is a function of time since the water is turned on.
An image with two pools. The larger pool has a ladder and a girl filling it with a hose. The smaller pool is smaller and inflatable and being filled with a hose.
Here are descriptions of three situations. For each situation, sketch the graphs of the two functions on the same coordinate plane so that is the height of the water in the small pool after minutes and is the height of the water in the large pool after minutes.
In both functions, the height of the water is measured in inches.
Situation 1: Each hose fills one pool at a constant rate. When the small pool is full, the water for that hose is shut off. The other hose keeps filling the larger pool until it is full.
Situation 2: Each hose fills one pool at a constant rate. When the small pool is full, both hoses are shut off.
Situation 3: Each hose fills one pool at a constant rate. When the small pool is full, both hoses are used to fill the large pool until it is full.
8.5
Activity
Your teacher will show you one or more videos of a tennis ball being dropped from 6 feet off the ground. Here are some still images of the situation.
The height of the ball is a function of time. Suppose the height is feet, seconds after the ball is dropped.
Use the blank coordinate plane to sketch a graph of the height of the tennis ball as a function of time.
To help you get started, here are some pictures and a table. Complete the table with your estimates before sketching your graph.
0 seconds
0.28 seconds
0.54 seconds
0.74 seconds
1.03 seconds
1.48 seconds
1.88 seconds
2.25 seconds
time
(seconds)
height
(feet)
0
0.28
0.54
0.74
1.03
1.48
1.88
2.25
Identify horizontal and vertical intercepts of the graph. Explain what the coordinates tell us about the tennis ball.
Find the maximum and minimum values of the function. Explain what they tell us about the tennis ball.
Student Lesson Summary
We can use graphs to help visualize the relationship between quantities in a situation, even if we have only a general description.
Here is a description of a hiker’s journey on a trail:
A hiker walked briskly and steadily for about 30 minutes and then took a 10-minute break. Afterward, she jogged all the way to the end of the trail, which took about 20 minutes. There, she took a 15-minute break, and then started walking back leisurely, stopping twice to enjoy the scenery. Her return trip along the same trail took 105 minutes.
We can sketch a graph of the distance the hiker has traveled as a function of time based on this description.
A graph. The horizontal axis, time in minutes, scale from 0 to 210 by 15s. The vertical axis, total distance hiked, scale is 8 units. The curve passes through the points 0 comma 0, 6 comma 16, 24 comma 16, 30 comma 20, 40 comma 20, 60 comma 55, 75 comma 55, 105 comma 35, 110 comma 35, 145 comma 15, 150 comma 15, 180 comma 0.
Even though we don’t know the specific distances she has traveled or the length of the trail, we can show some important features of the situation in the graph. For example:
The intervals in which the distance increased or stayed constant
How quickly the distance was increasing
The amount of time the hiker was hiking
If we are looking at distance from the trailhead (the start of the trail) as a function of time, the graph of the function might look something like this:
A graph. The horizontal axis, time in minutes, scale from 0 to 210 by 15s. The vertical axis, distance from trailhead, scale is 8 units. The curve passes through the points 0 comma 0, 6 comma 16, 24 comma 16, 30 comma 20, 40 comma 20, 60 comma 55, 75 comma 55, 105 comma 35, 110 comma 35, 145 comma 15, 150 comma 15, 180 comma 0.
It shows the distance increasing as the hiker was walking away from the trailhead, then decreasing as she was returning to the trailhead.
