A model rocket is launched from the top of a building 160 feet from the ground. The relationship between the time in seconds, x, and the vertical height in feet, y, is represented by this equation.
y=-40x^2 + 120x+ 160

A. Rewrite the equation in a form that shows the time it takes the rocket to land on the ground. Use the equation to find the time. Explain your work.

B. Rewrite the original equation, y= -40x^2 + 120x +160, in a form that shows the maximum height of the rocket. Use the equation to determine the maximum height. Explain your work.

C. Write an equation that increases the maximum height reached by the rocket but doesn't change the time it takes for the rocket to land on the ground. Explain your answer.

Step-by-step explanation:

y = -40x² + 120x + 160

a) When the rocket lands, y = 0.  So we need to write the equation in factored form.

y = -40 (x² − 3x − 4)

y = -40 (x + 1) (x − 4)

Setting y equal to 0:

0 = -40 (x + 1) (x − 4)

x = -1 or x = 4

x can't be negative, so the rocket lands after 4 seconds.

b) The rocket reaches its maximum height at the vertex of the parabola.  So we need to write the equation in vertex form.

y = -40 (x² − 3x) + 160

y = -40 (x² − 3x + 9/4) + 40(9/4) + 160

y = -40 (x − 3/2)² + 250

The vertex is (3/2, 250).  So the rocket reaches a maximum height of 250 feet after 1.5 seconds.

c) There are many possible answers.  As long as one root is 4 and the other root is nonpositive, we can write an equation such that the rocket reaches a new maximum height without changing the time that it lands.

To do this, we must change either the acceleration (-40×2) or the initial height (160) or both.

If we change the acceleration, the new equation is:

y = (x − 4) (-ax − 40)

If we change the initial height, the new equation is:

y = (x − 4) (-40x − b)

In each case, a and b are any number greater than 40.

Graph: desmos.com/calculator/zxnjo0rm2r

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step-by-step explanation:

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