Question

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 100, 0. The vertex is at 75, 50.

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 = 15 to x = 35, and what does this rate represent? (4 points)

Answer 1

Part A) x-intercepts simply show that when the value of the function is zero. Vertex coordinates show that when the function obtains its maximum value. When x=50, function obtains its maximum value and it's 75. The function is increasing in the interval (0, 50) and decreasing in the interval (50, 100). In regard to the height and distance of the tunnel, these numbers show that decreasing and increasing intervals are symmetric. Each number from the intervals has its own pair in the corresponding interval and they are located in the same distance from the midpoint (50,75)

Part B) In order to calculate the average rate of change, we can first write the function. Using the information about the x-intercept and the vertex coordinates, we find that our function is
`f(x)=-0.03x^(2)+3x`.
Plugging 15 and 35 in x, we can find the values of the function, i.e.

`f(15)=38.25` and
`f(35)=68.25`.
Then, the average change is
`(68.25-38.25)/(35-15)=1.5`