Question

(b) fill in the blanks below to write an inequality for all the values of x for which the cannonball is gaining height (going up). then choose the best description of this inequality.inequality: $square < x < square$description:$\bigcirc$ the function $y = -16x^{2}+32x$ is increasing over these values of $x$$\bigcirc$ the function $y = -16x^{2}+32x$ is decreasing over these values of $x$(c) find the maximum value of $y$ for the function $y = -16x^{2}+32x$. then choose the best description of the maximum value of $y$maximum value of $y$: $square$description:$\bigcirc$ the time in seconds when the cannonball hits the ground$\bigcirc$ the height in feet above the ground the cannonball starts at$\bigcirc$ the highest elevation in feet the cannonball reaches

Explanation:

Step1: Find vertex x-coordinate

For $y=ax^2+bx$, vertex $x=-\frac{b}{2a}$.
Here $a=-16, b=32$, so $x=-\frac{32}{2(-16)} = 1$.

Step2: Identify increasing interval

Downward parabola increases left of vertex.
Inequality: $0 < x < 1$.

Step3: Match increasing description

Function increases on this interval.

Step4: Calculate max y-value

Substitute $x=1$ into $y=-16x^2+32x$.
$y=-16(1)^2 + 32(1) = 16$.

Step5: Match max value description

This is the highest height reached.

Answer:

Part (b)

Inequality: $0 < x < 1$
Description: $\bigcirc$ The function $y = -16x^2 + 32x$ is increasing over these values of $x$

Part (c)

Maximum value of $y$: $16$
Description: $\bigcirc$ The highest elevation in feet the cannonball reaches