Question

use your graphing calculator to sketch the graph of the function, and then determine the horizontal interval over which the function is increasing.
$y = x^2 - 96$
sketch the graph of the function in the viewing window $-20,20×-100,100$. choose the correct graph below.
a. b. c. d.
(images of graphs for a, b, c, d are shown with zoom and other icons)
find the horizontal interval, expressed as an inequality, over which the function is increasing
(type an inequality.)

Explanation:

Step1: Identify the function type

The function \( y = x^2 - 96 \) is a quadratic function in the form \( y = ax^2 + bx + c \), where \( a = 1 \), \( b = 0 \), and \( c = -96 \). Since \( a>0 \), the parabola opens upward.

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 \( x =-\frac{0}{2\times1}=0 \). The vertex of the parabola \( y=x^2 - 96 \) is at \( (0,- 96) \).

Step3: Determine the interval where the function is increasing

For a parabola that opens upward ( \( a>0 \) ), the function is increasing to the right of the vertex. Since the vertex has an x - coordinate of \( x = 0 \), the function \( y=x^2-96 \) is increasing for all \( x \) values greater than 0. So the interval where the function is increasing is \( x>0 \).

Answer:

\( x > 0 \)