Question

At a diving competition, Holly jumps from a springboard that is 3 meters above the surface of the water at time t=0 seconds. She reaches a maximum height of 4.5 meters above the surface of the water after 0.5 seconds and enters the water 1.5 seconds after jumping. This situation may be modeled by a quadratic function. Part A: What is the vertex of this function? Interpret its meaning. Part B: What is the y-intercept of this function? How do you know?

Answer 1

Answer:

Part A:  Vertex of function is at (0.5, 4.5)

h(t) = -6*(t - 0.5)^2  + 4.5

Part B:  We want the y-intercept to be (0, 3)  because  Holly starts at 3 meters above the water surface and at time t =0 seconds.

Solving for when she enters the water, we do not need to use t = 1.5 seconds, because it is unclear if h< 0 at that time.

h(t ) =0  =  -6*(t - 0.5)^2 + 4.5

-4.5 = -6*(t - 0.5)^2

4.5/6 = (t - 0.5)^2

0.866025  = t - 0.5

t = 0.5 + 0.866025 =  1.366025  seconds

So at 1.366 seconds, Holly is at the surface of the water.

Step-by-step explanation:

We know that for a quadratic function that models a trajectory path, that the vertex is the maximum point.

With a maximum at h = 4.5 meters at time t = 0.5 seconds

Vertex = (0.5, 4.5)

x-intercept at   (1.5, 0)

Parabolic equation:   h(t) = a *(t - h)^2  + k

h(t) = a*(t - 0.5)^2 + 4.5

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We need  h(t) = 3  at t = 0

3 =  a*(0 - 0.5)^2  + 4.5

-1.5 = a * (-0.5)^2

-1.5 = a*0.25

a = -1.5/0.25

a = -6

so:   h(t) = -6*(t - 0.5)^2  + 4.5

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With  t = 1.5 at  h < 0

0 = a*(1.5 - 0.5)^2  + 4.5

0 = a*1 + 4.5

a = -4.5 ?   this part we really do not know if t = 1.5 seconds is when h = 0

it says she enters the water, it could mean h(t) < 0 here.