Question

PLEASE I BEG ANYONE WHO SEES THIS QUESTION TO PLEASE HELP ME TO PLEASE GIVING A CORRECT ANSWERS OR CHOICES PLEASE I'LL GIVE 50 POINTS!!!!!!!!!

Hi I really need help with this problem please.

Correct answers only

the choices are:

equal to

1 greater than

1 less than

2 greater than

2 less than

3 greater than

3 less than

Answer 1

The maximum value of g(x) is 3 greater than the maximum value of f(x).

Explanation:

Quadratic Function:

Definitions:

The function modeled by the equation, f(x) = - (x - 3)² + 9, represents a quadratic function in vertex form.

The vertex form is given by: f(x) = a(x - h)² + k, where:

⇒ The value of a is the vertical compression or stretch factor, which essentially determines the width of a parabola. The value of a also determines where the graph of a parabola opens:

• a > 1: graph opens up;

• a < 1, graph opens down.

⇒ (h, k) = Vertex of the parabola.

Given these definitions, we can assume the following information regarding the quadratic function, f(x) = - (x - 3)² + 9:

• The vertex occurs at point, (3, 9).
• The graph opens down because a = -1.
• Therefore, the vertex is the maximum point of the graph.

Absolute Value Function:

Definitions:

The vertex form of an absolute value function is given by: g(x) = a|x - h| + k, where:

⇒ The value of a determines the following:

• |a| > 1 represents the vertical stretch of the graph.
• 0 < |a| < 1 represents the vertical compression or shrink of the graph.
• a < 0 represents the reflection across the x-axis.

⇒ (h, k) = Vertex of the absolute value function.

⇒ x = h represents the axis of symmetry.

The axis of symmetry is the imaginary point on the graph that intersects through the h-coordinate of the vertex, thereby dividing the graph into two symmetrical parts.

Observing the given table of values, especially how the y-values increasingly peak towards g(x) = 12, then decreases back down starting at g(x) = 8. It is important to take notice of the y-values, as it represents the resulting output provided by the given input (x-value).

The increasing, followed by the decreasing y-values imply that the graph must be facing downward. Since we specified that the maximum point provided by g(x) = 12, then it means that it is the vertex occurs at point, (1, 12).

Compare the vertices of f(x) and g(x):

• The vertex of the quadratic function, f(x) = (3, 9).

• The vertex of the absolute value function, g(x) = (1, 12).

• The basis of comparing the minimum or maximum value of a graph is the output, or the y-value.

• The difference between the maximum values of g(x) and f(x) = 3.

Therefore, the maximum value of g(x) is 3 greater than the maximum value of f(x).