Answer:
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:
⇒ (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).