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?

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.

your exact question is kind of difficult to understand, so maybe an example will explain what you're supposed to do? i'll work through the first problem on the worksheet.

suppose you have a tray of 652 brownies, and you have 5 friends that you want to feed without picking favorites. how many brownies can you dole out to each of your friends in equal amounts? how many will be left over for you to eat?

what you're doing here is finding a way to arrange the number 652 as the sum of integer multiple of 5 and some remainder:

here ("q" for quotient) is the number of brownies each of your friends get, and is the remainder. that's the basic idea behind division.

we break up the number 652 into parts so that the leading digit stands on its own:

652 = 600 + 52

first we check if it's possible to hand out exactly 100 brownies to each friend. with 5 friends, this means we want to know if we can take out 500 brownies from the tray. we can do this 1 time because we have 600 brownies to take from, and when we do that we're left with 100 + 52 brownies to divvy up. so we know that

652 = 1*500 + 152

next we want to know if it's possible to hand out exactly 10 brownies per friend. if we can, we have to remove 50 brownies from the tray each time until we can't. we have 152 brownies left, and we know that 3(50) = 150, so

152 = 3*50 + 2

now we have 2 brownies left in the tray. is it possible to hand 1 out to each of your 5 friends? no, you'd need a minimum of 5 brownies to do that. so the last 2 are yours.

putting everything together, we found that

652 = 1*500 + 3*50 + 2

and we can rearrange this to write

652 = 5*100 + 5*30 + 2

652 = 5(100 + 30) + 2

652 = 5*130 + 2

so and .

you might be more accustomed to the table depicting the algorithm. that's just a compact way of organizing what we've done above. you can consult the image i've attached below.