The graph below shows the height of a kicked soccer ball f(x), in feet, depending on the distance from the kicker x, in feet: 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 = 22 to x = 26, and what does this rate represent 

Part A:

a) What do the x-intercepts and maximum value of the graph represent?

The x-intercepts are the distances at which the ball is on the ground.

First, at x = 0, that is when the ball is kicked; second, at x = 30, when the ball falls (return) to the ground.

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

The function is increasing in the interval (0, 15) and is decreasing in the interval (15,30)

The increasing interval (0,15) is the horizontal distance from the point the the ball was kicked until it reached its highest altitude, this is where the ball was going upward.

The decreasing interval (15,30) is the horizontal distance from the point where the ball reached its highest altitude until it landed on the ground, this is where the ball was falling down.

Part B: What is an approximate average rate of change of the graph from x = 22 to x = 26, and what does this rate represent 

On the graph you can read that at x = 22, f(x) ≈ 12, and at x = 26 f(x) ≈ 7.

So, an approximate rate of change from x = 22 to x = 26 is given by the equation below:

                                            change on f(x)        7 - 12
average rate of change = --------------------- =    ----------- = -5/4
                                            change of x           26 - 22

That rate represents that the ball fell about 5 ft per 4 ft in that interval.