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)

Because the vertex of the parabola is at (10,15), its equation is of the form
y = a(x-10)² + 15

The graph goes through (0,0), therefore
a(0 - 10)² + 15 = 0
100a = -15
a = -0.15

The equation is 
y = f(x) = -0.15(x - 10)² + 15

The graph is shown below.

Part A
Note that y = f(x).

The x-intercepts identify values where the function or  y=0. The x-intercepts occur at x=0 and x=20, or at (0,0) and (20,0).

The maximum value of y occurs at the vertex (10, 15) because the curve is down due to the negative leading coefficient of -0.15.

The curve increases in the interval x = (-∞, 10) and it decreases in the interval x = (10, ∞).

Part B
When x=12, y = -0.15(12 - 10)² + 15 = 14.4
When x=15, y = -0.15(15 - 10)² + 15 = 11.25

The average rate of change between x =12 to x = 15 is
(11.25 - 14.4)/(15 - 12) = -1.05

This rate of change represents the slope of the secant line from A to B. It approximates the rate at which f(x) decreases in the interval x =[12, 15].

The x-intercepts are clearly (0, 0) and (20, 0), not (0, 0), (0, 20). Because if we have (0, 0) and (0, 20) then this is no longer a function since we have two similar values of x for different y’s. It should be similar y for different x’s.

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

Part A. the x intercept = 0 tells us the point wherein the ball was hit, while the x intercept = 20 tells us the point wherein the ball fell back to the ground, that is 20 feet away from the kicker

The function is increasing in the interval x∈(0, 10) and decreasing in x∈(10, 20) . This only tells us that the height  increases in the interval from  (0, 10) and then decreases as x goes through the interval (10,20). While the distance from the kicker is constantly increasing during the whole interval (0, 20)

Part B:

at x1=15,

f(x)=f(15) = y1=(-3/20)*15(15-20)= 45/4

at x2=12

f(x)=f(12)= y2=-3/20*12(12-20)= 72/5

The average rate of change is then calculated as:

average rate of change = (y2-y1)/(x2 –x1) = (72/5 – 45/4) / (12-15) = -1.05

 

Therefore this means that the height is decreasing at 1.05 ft per feet in horizontal distance from x=12 to x=15.

Suredude why not right

Where’s the stuff st? ? lol sorry