Please help!!
A ball is thrown straight up from a height of 3 ft with a speed of 32 ft/s. Its height above the ground after x seconds is given by the quadratic function y = -16x^2 + 32x + 3. Explain the steps you would use to determine the path of the ball in terms of a transformation of the graph of y = x^2.

Complete the square to get the equation in vertex form with
a = -16, h = 1, and k = 19. The path is a reflection over the x-axis and narrower. It is also translated right 1 unit and up 19 units.

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MrRoyal MrRoyal · Answer 2 · 2021-10-08T05:17:56+02:00

Transformation involves changing the position and size of a function

The steps to determine the path of the ball involves changing the equation to vertex form and then interpret the parts of the function.

The parameters are given as:

[tex]\mathbf{y = x^2}[/tex] --- the parent function

[tex]\mathbf{y = -16x^2 + 32x + 3}[/tex] --- the new function

First, we write the new function in vertex form

We have:

[tex]\mathbf{y = -16x^2 + 32x + 3}[/tex]

Factor out -16

[tex]\mathbf{y = -16(x^2 - 2x) + 3}[/tex]

--------------------------------------------------------------

Take the coefficient of x

[tex]\mathbf{k = -2}[/tex]

Divide by 2

[tex]\mathbf{\frac k2 = -1}[/tex]

Take square of both sides

[tex]\mathbf{(\frac k2)^2 = (-1)^2}[/tex]

[tex]\mathbf{(\frac k2)^2 = 1}[/tex]

--------------------------------------------------------------

So, the equation becomes

[tex]\mathbf{y = -16(x^2 - 2x) + 3}[/tex]

[tex]\mathbf{y = -16(x^2 - 2x + 1 - 1) + 3}[/tex]

Factor out -1

[tex]\mathbf{y = -16(x^2 - 2x + 1 ) + 16 + 3}[/tex]

[tex]\mathbf{y = -16(x^2 - 2x + 1 ) +13}[/tex]

Express the bracket as perfect squares

[tex]\mathbf{y = -16(x - 1 )^2 +19}[/tex]

Next, we compare the above function to the parent function [tex]\mathbf{y = x^2}[/tex]

By comparison, we have the following observations

-16 means that the path of the ball is reflected over the x-axis, and it follows a narrower path compared to [tex]\mathbf{y = x^2}[/tex]
-1 means that the function is translated right by 1 unit
19 means that the function is translated up by 19 units

See attachment for the graph of both functions