The graph below shows the height of a tunnel f(x), in feet, depending on the distance from one side of the tunnel x, in feet:

graph of quadratic function f of x having x intercepts at ordered pairs 0, 0 and 120, 0. the vertex is at 60, 35.

part a: what do the x-intercepts and maximum value of the graph represent? what are the intervals where the function is increasing and decreasing, and what do they represent about the distance and height?

part b: what is an approximate average rate of change of the graph from x = 20 to x = 35, and what does this rate represent?

x - intercepts (0, 0) and (120, 0)

Vertex is at (60, 35).

Part A: What do the x-intercepts and maximum value of the graph represent?

The x-intercepts represent the points at the ground, so with that you have the width of the tunnel at ground level is 120 ft - 0 ft = 120 ft..

The vertex represents the maximum height, which is at the center of the tunnel. So, you know that the maximum height is at x = 60 ft and it is 35 ft

What are the intervals where the function is increasing and decreasing, and what do they represent about the distance and height?

The increasing interval is (0,60), and the decreasing interval is (60,120).

That represents that the height increase from the left border (x = 0) until the center of the tunnel (x = 60 ft), reaching a height of 35 ft, and starts to decrease toward the right border until the ground level (x = 120).

Part B: What is an approximate average rate of change of the graph from x = 20 to x = 35, and what does this rate represent?

Find the y -coordinate for x = 20 and for x = 35. Supposse they are (I do not have the figure, because you did not include it) y = 15 and y = 45.

Then the average rate of change is equal to [change in height] / [change in x] = [45 - 15] / [35 - 20] = 30 / 15 = 2.

Remember that you have to use the real approximate values for the height that you can read in the figure. If the number is positive means that the height is increasing in that interval.