a The function can be written on the form y=ax(x-250). To find the value of a, we need to know a point on the function that is not an x-intercept.
B
b The function can be written on the form y=ax(x-x_1). To find the value of a and x_1, we need to know two points on the function that are not an x-intercept.
A
a y=- 3/3125x(x-250)
B
bHigher shot: Dwayne's
Further shot: David's
Practice makes perfect
a If we let the origin represent the start of the golf ball's flight path, we can model it with the following parabola.
Since the graph has intercepts at x=0 and x=250, we can write the graph's function in the following format.
y=ax(x-250)
In order for us to complete the equation we have to find the value of a. To do that, we need a third point on the graph. Because of the parabolas symmetry, we can determine the vertex's x-coordinate by calculating the average of two points on the graph that lie on the same y-coordinate. The x-intercepts are two such points, as they are both on y=0.
x_(sym)=0+250/2= 125
The line of symmetry is x=125, which means the vertex has the coordinates (125,15). By substituting this point into the equation, we can solve for a.
Now that we know the value of a, we can complete the function.
y=- 3/3125x(x-250)
b We want to compare the hits of the two brothers. From Part A we have the full information about David's hit — we know that we can model it using the following function.
y_(Da)=- 3/3125x(x-250)
First, we want to find similar model for Dwayne's hit.
Finding the Function
This time we can use the given table to find the function of the other hit.
Horizontal Distance (yd)
Height (yd)
0
0
20
5.5
60
13.5
180
13.5
220
5.5
To get an idea how the function should look like, let's plot these points and connect them with a smooth curve.
This function looks like a parabola with x-intercepts at x = 0 and at x = 240. If we mark point (240, 0) on our graph, we can notice that the six points form a set which is symmetric across the x = 120 line. This makes sense, since all parabolas have an axis of symmetry.
This suggests that we should look for the function in the following format.
y_(Dwayne) = ax(x - 240)
To find the value of a, we should use one of the other points that we were given. Let's choose point ( 180, 13.5). Hence, let's substitute 180 for x and 13.5 in the above fomula.
We can use the above information to complete our function.
y_(Dw) = ax(x - 240)
⇓
y_(Dw) = -13.5/10800x(x - 240)
Having found the function, we can move to finding which brother threw the ball higher, and which brother through the ball further.
Higher Shot
From Part A of the exercise we know that the ball hit by David reached the height of 15 yards. To find the maximal height of Dwayne's hit, notice that the highest point of our function is at the axis of symmetry. Hence, let's substitute 120 for x in our function to find the height.
We see that Dwayne's hit reached the height of about 16.66 yards. Hence, it was higher David's 15 yards.
Further Shot
In both cases we can determine the length of the shot to be the x-coordinate of the second x-intercept of the functions we found. Let's recall both of these functions, along with the mentioned coordinate.
Brother
Function
Length
David
y=- 3/3125x(x- 250)
250
Dwayne
y=-13.5/10800x(x - 240)
240
We see that David's shot was 250 yards long, while Dwayne's was 240 yards. Hence, it was David who hit the ball further.
