Question

use your graphing calculator to sketch the graph of the function, and then determine the horizontal interval over which the function is decreasing.
$y = -x^2 + 96$
sketch the graph of the function in the viewing window $-20,20\times-100,100$. choose the correct graph below.
\\(\bigcirc\\) a. \\(\bigcirc\\) b. \\(\bigcirc\\) c. \\(\bigcirc\\) d.
look at the graph of the function. find the horizontal interval, expressed as a inequality, over which the function is decreasing.
\\(\square\\) (type an inequality.)

Explanation:

Step1: Analyze the function type

The function $y = -x^2 + 96$ is a quadratic function. The general form of a quadratic function is $y = ax^2 + bx + c$. Here, $a=-1$, $b = 0$, $c = 96$. Since $a=-1<0$, the parabola opens downward.

Step2: Find the vertex of the parabola

For a quadratic function $y = ax^2+bx + c$, the x - coordinate of the vertex is given by $x=-\frac{b}{2a}$. Substituting $a=-1$ and $b = 0$ into the formula, we get $x =-\frac{0}{2\times(-1)}=0$. The vertex of the parabola $y=-x^2 + 96$ is at $(0,96)$.

Step3: Determine the interval where the function is decreasing

For a parabola that opens downward (because $a<0$), the function is decreasing to the right of the vertex. The vertex is at $x = 0$, so the function $y=-x^2 + 96$ is decreasing for $x>0$.

Answer:

$x > 0$