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 36, 0. the vertex is at 18, 32.

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? (6 points)

part b: what is an approximate average rate of change of the graph from x = 5 to x = 15, and what does this rate represent? (4 points)

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

There are two x-intercepts: x = 0 and x = 36

Those values are the points at which the height of the tunnel is 0.

That means that the width of the tunnel at the ground level is 36 - 0 = 36.

The maximum value of the graph is 32, at x= 18. That, represents that at the center, x= 18,  the tunnel has a height of 32.

What are the intervals where the function is increasing and decreasing, and what do they represent about the distance and height? (6 points)

The interval where the function is increasing is (0,18) and the interval where teh function is decreasing is (18,36)

That represents that the height starts to increase with the distance (from 0 to 18) reaches a maximum of 32, at x = 18, and starts to decrease with the distance until x = 36.

Part B: What is an approximate average rate of change of the graph from x = 5 to x = 15, and what does this rate represent? (4 points)

The average rate of change from x = 5 to x = 15 is the change in height divided by the change in distance

=> average rate of change = Δy / Δx

Δy ≈ 31 -14  = 17

Δx = 15 - 5 = 10

=> Δy / Δx ≈ 17 / 10 = 1.7