I'll fan
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:

graph of quadratic function f of x having x intercepts at ordered pairs 0, 0 and 0, 20. the vertex is at 10, 15.

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 = 12 to x = 15, and what does this rate represent? (4 points)

Remark: the x-intercepts are clearly (0, 0) and (20, 0), not (0, 0), (0, 20).

having the 2 x intercepts and the vertex we can draw the graph, check picture 1 attached.

Moreover, we can also determine the quadratic function as follows.

The quadratic function with roots a and b, is given by the equation:

f(x)=c(x-a)(x-b).

[ since f(a)=0, f(b)=0, f(x) must have one factor (x-a) and one (x-b)]

The x coordinates of the x intercepts are the roots of the function so with the roots given:

f(x)=c(x-0)(x-20)=cx(x-20), the vertex is at (10, 15) so

15=f(10)=c*10*(10-20)=c*10*(-10)=-100c
c=-15/100=-3/20

the quadratic function (expressed in factorized form) is

f(x)=-3/20x(x-20)

part A: the x intercept 0 is the point where the ball was hit, the x intercept 20 is the point where the ball fell back to the ground, 20 feet away from the kicker.

the function is increasing in the interval x∈(0, 10) and decreasing in x∈(10, 20) 
this means that the height is increasing in the interval  (0, 10) and decreasing as x goes through the interval (10,20)
the distance from the kicker is increasing during the whole interval (0, 20)

Part B: at x=15, f(x)=f(15)=-3/20*15(15-20)=

at x=12, f(x)=f(12)=-3/20*12(12-20)=

the average rate of change is the ratio of the differences in y to the differences in x : 

so the average rate of change is -1.05. The minus, tells us that the height is decreasing in the following way: for 1 feet in distance, we are loosing 1.05 feet in height.  

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