Part A
h(d) = (-1/5)d^2 + 2d is the same as h(d) = -0.2d^2 + 2d because -1/5 = -0.2
I'll use x in place of d, and y in place of h(d) to get this equivalent equation: y = -0.2x^2 + 2x
Let's find the vertex of y = -0.2x^2 + 2x
Note how y = -0.2x^2 + 2x is in the form y = ax^2+bx+c with
a = -0.2
b = 2
c = 0
So the x coordinate of the vertex is
h = -b/(2a)
h = -2/(2(-0.2))
h = 5
Plug this value into the h(d) function to compute the corresponding y coordinate
h(d) = -0.2d^2 + 2d
h(5) = -0.2(5)^2 + 2(5)
h(5) = 5
Coincidentally, the x and y coordinates of the vertex are both the same. This wont be the case in general.
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Answers:
The vertex is located at (5,5)
The interpretation is that the highest the dolphin can get is 5 feet off the surface of the water and this occurs 5 horizontal feet away from the starting point. Imagine a parabola that opens downward to have a highest point at the vertex mentioned.
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Part B
From part A above, we found x = 5 to be the x coordinate of the vertex. Simply double this value to get 10 as the answer. The reason why this works is because x = 0 is the starting x intercept, or the starting point which the dolphin jumps from. Also, its because of the hint given how the vertex is between each x intercept.
If the dolphin started from any other point on the x axis, then this trick of "double the result from part A" wouldn't apply.
Part C below shows that x = 0 and x = 10 are the two x intercepts. This helps confirm the proper range.
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Answer: 10 feet
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Part C
h(d) = y = height of dolphin
we want to find the two x values that make y = 0, which help us figure out when the dolphin jumps out of the water and when it lands back in the water
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Plug y = 0 into the equation y = -0.2x^2 + 2x and solve for x
y = -0.2x^2 + 2x
-0.2x^2 + 2x = y
-0.2x^2 + 2x = 0
x(-0.2x + 2) = 0 .... factor out x
x = 0 or -0.2x + 2 = 0 .... zero product property
x = 0 or -0.2x = -2
x = 0 or x = -2/(-0.2)
x = 0 or x = 10
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Answer:
The two x intercepts are x = 0 and x = 10
This means the parabola crosses the x axis at the locations (0,0) and (10,0).
