Question

$y = 96x - 18x^{2}$
complete the parts below.
(a) graph the parabola $y = 96x - 18x^{2}$. to do so, plot five points on the parabola: the vertex, two points to the left of the vertex, and two points to the right of the vertex. then click on the graph-a-function button.
(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:
the function $y = 96x - 18x^{2}$ is increasing over these values of $x$
the function $y = 96x - 18x^{2}$ is decreasing over these values of $x$
(c) find the maximum value of $y$ for the function $y = 96x - 18x^{2}$. then choose the best description of the maximum value of $y$
maximum value of $y$: $\square$
description:
the highest elevation, or apex of the cannonballs flight
the time in seconds when the cannonball hits the ground
the height in feet when the cannonball leaves the ground

Explanation:

Step1: Find vertex of parabola

The function is $y = 96x - 16x^2$, rewrite as $y = -16x^2 + 96x$. For $ax^2+bx+c$, vertex $x$-value is $-\frac{b}{2a}$.
$x = -\frac{96}{2(-16)} = \frac{96}{32} = 3$
Substitute $x=3$: $y = 96(3) -16(3)^2 = 288 - 144 = 144$. Vertex: $(3, 144)$

Step2: Find 4 other points

Left of vertex: $x=1$: $y=96(1)-16(1)^2=80$ → $(1,80)$; $x=2$: $y=96(2)-16(4)=192-64=128$ → $(2,128)$
Right of vertex: $x=4$: $y=96(4)-16(16)=384-256=128$ → $(4,128)$; $x=5$: $y=96(5)-16(25)=480-400=80$ → $(5,80)$

Step3: Find increasing interval

Downward-opening parabola increases left of vertex.
Inequality: $0 < x < 3$

Step4: Find maximum value

Vertex is the maximum point, so $y=144$.

Answer:

(i) Plot the points $(3,144)$, $(1,80)$, $(2,128)$, $(4,128)$, $(5,80)$ and draw a smooth parabola through them.
(ii)
Inequality: $0 < x < 3$
Description: The function $y = 96x - 16x^2$ is increasing over these values of $x$
(iii)
Maximum value of $y$: $144$
Description: This is the maximum value the cannonball reaches (the vertex of the parabola)