# Applications of Quadratic Functions

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## Applications of Quadratic Functions Answer Key

1.

A small model rocket is launched straight up into the air. After rising to its highest point, its parachute fails to open and the rocket falls straight back to the ground. Let [math]h(t) = -4.9t^2 + 30t[/math] represent the height of the rocket in meters, dependent on the time, [math]t[/math], in seconds. The function is only valid for times [math]t[/math] such that [math]t[/math] is greater or equal to the start time and less than or equal to when it hits the ground.

A.

Complete the square so that the function [math]h(t)[/math] is in vertex form. Which of the following is correct? Round computed coefficients and terms to one decimal place, if necessary.

- [math]h(t) = -4.9(t - 3.1)^2 - 9.4[/math]
- [math]h(t) = -4.9(t - 3.1)^2 + 45.9[/math]
- [math]h(t) = (-4.9t + 15)^2 - 225[/math]
- [math]h(t) = -4.9(t + 3.1)^2 + 9.4[/math]

B.

What facts about the rocket can be seen directly from the vertex form of the function? Choose all that apply.

- The time when the rocket reached its maximum height.
- The total duration of the rocket's flight.
- The initial velocity of the rocket.
- The rocket's maximum height.

C.

Which of the following is the correct fully factored form of the function [math]h(t) ?[/math]

- [math]h(t) = (2.2t - 5.5)(2.2 + 5.5)[/math]
- [math]h(t) = 4.9(t^2 + 6.1t)[/math]
- [math]h(t) = -4.9t (t - 6.1)[/math]
- [math]h(t) = (-4.9t - 3)(t + 10)[/math]

D.

What information about the rocket can be directly seen from the factored form of [math]h(t) ?[/math] Choose all that apply.

- The initial height height.
- The total duration of the flight.
- The velocity when the rocket reaches its maximum height.
- How far the rocket traveled.

E.

Sketch the function [math]h(t)[/math], and fully label the graph. Indicate the maximum and intercepts, and state what these represent in terms of the rocket's flight.

- Graph should be parabolic. Intercepts: (0, 0) and (6.1, 0). Maximum at approximately (3.1, 45.9). The intercepts represent the starting time and end time (when the rocket crashes to the ground). The maximum represents the max height the rocket reached and the time at which this height was reached.

2.

The trajectory of a golf ball hit over relatively flat terrain can be modeled by the function [math]h(d) = -0.0013d^2 + 0.64d + 0.13[/math], where [math]h[/math] is the height of the golf ball in feet and [math]d[/math] is the horizontal distance it has traveled from its original spot in feet. The function is only valid for values [math]d[/math] such that [math]d[/math] is greater than or equal to the ball's initial position and less than or equal to when it first makes contact with the ground after being hit.

A.

Use the method of completing the square to find the vertex form of the function [math]h(d)[/math]. Round computed coefficients and terms to one decimal place, if necessary.

- [math]h(d) = -0.0013(d + 246.2)^2 + 0.2[/math]
- [math]h(d) = (-0.0013d - 0.6)^2 + 0.2[/math]
- [math]h(d) = -0.0013 ( d^2 -246.2) - 60","594.6[/math]
- [math]h(d) = -0.0013(d - 246.2)^2 + 78.9[/math]

B.

What facts about the golf ball's trajectory can be determined directly from the vertex form? Choose all that apply.

- The amount of time it was in the air for.
- Its maximum height.
- At what distance the maximum height occurred.
- At what time the maximum height occurred.

C.

Which of the following is the fully factored form of [math]h(d) ? [/math] While performing calculations, keep at least 4 decimal places.

- [math]h(d) = -0.0013 ( d + 0.2)(d - 492.5)[/math]
- [math]h(d) = -0.0013d (d - 492.3) + 0.13[/math]
- [math]h(d) = ( d - 0.2)(d + 492.5)[/math]
- [math]h(d) = -0.0013 (d + 3.7)(d - 134.4)[/math]

D.

What information can be determined directly from the factored form of the function [math]h(d) ?[/math] Choose all correct answers.

- The ball's initial height.
- The total horizontal distance it traveled.
- The angle the ball was hit at.
- The total time the ball was in the air for.

E.

Graph the function [math]h(d)[/math], being sure to indicate its maximum and intercepts. Also, indicate what these values represent in terms of the golf ball's trajectory.

- Graph is a parabola, downwards facing. Intercepts are (492.5, 0), which is the total horizontal distance the ball traveled, and (0, 0.13), which is the ball's initial height before being hit (the height of the tee). The value (-0.2, 0) isn't included, because not in the restricted domain (and has no meaning to the trajectory). Maximum is 78.9, which is the max height the ball reaches, in feet, at about 246.2 (feet).

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