Question

Interpret the key features of the graph in the given context, and answer the following questions.

Part A
What is the vertex of the function? How do you interpret it in the given situation?
Part B
What is the x-intercept of the function? How do you interpret it in the given situation?
Part C
What is the y-intercept of the function? How do you interpret it in the given situation?
Part D
What is the domain of the function in the given situation? Explain your answer.
Part E
Calculate the average rate of change over the intervals [0, 20] and [60, 80]. What does the average rate on these intervals tell you about the balloon’s journey?

Answer 1

Part A

The vertex of the graph lies at (40,18). This is the maximum value on the graph. This means the balloon reached a maximum altitude of 1,800 meters about 40 minutes after its launch.

Part B

One of the x-intercepts of the function lies somewhere between x = 82 and x = 83. This is this point where the balloon is at zero altitude, or sea level. So, the balloon stays in the air for a little over 82 minutes. There is another x-intercept to the left of the origin. But a negative time value doesn’t make sense. So, we won’t considerate it for this scenario.

Part C

The -y-intercept of the function lies at (0,2). This is the starting point of the balloon’s journey. The point the balloon launched from was 200 meters above sea level.

Part D

We can’t interpret negative time values in this situation. So, the domain of the function is restricted to positive values of x. A negative y-value indicates an altitude below sea level. While some locations have negative altitudes, we can consider only positive values for this scenario. So, we should only consider positive values of y. So the domain of the function is restricted to only those values of x that produce positive values of y. Since we can’t see the x-intercept clearly from the graph, we’ll consider the domain of the function to be [0, 82].

Part E

Average rate of change on [0, 20] = f(20)-f(0) / 20-0

= 14-2 / 20

= 12 / 20

=0.6

Average rate of change on [60, 80] = f(80)-f(60) / 80-60

= 2-14 / 20

= -12 / 20

= -0.6

In this case, the rate of change represents the change in the balloon’s altitude within a time interval. The average rate of change on [0, 20] is positive. This indicates that the balloon is ascending, or climbing, during the first 20 minutes of its journey.

The average rate of change on [60, 80] is negative. This indicates that the balloon is descending between 60 and 80 minutes. Although negative, the rate of change in the second interval has the same absolute value as the rate of change for the first interval. So, during the last 20 minutes the balloon is descending at the same rate it was ascending during the first 20 minutes.