Find the maximum value of the function f(x) = -x^2 + 6x - 8.

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Question: Find the maximum value of the function f(x) = -x^2 + 6x - 8.

Options:

Correct Answer: 8

Solution:

The vertex occurs at x = 3. f(3) = -3^2 + 6(3) - 8 = 6.

Find the maximum value of the function f(x) = -x^2 + 6x - 8.

Find the maximum value of the function f(x) = -x^2 + 6x - 8.

Correct Answer: 6

Step 1: Identify the function you need to analyze, which is f(x) = -x^2 + 6x - 8.
Step 2: Recognize that this is a quadratic function in the form of f(x) = ax^2 + bx + c, where a = -1, b = 6, and c = -8.
Step 3: Find the x-coordinate of the vertex using the formula x = -b/(2a). Here, b = 6 and a = -1.
Step 4: Calculate x = -6/(2 * -1) = -6/-2 = 3.
Step 5: Now, substitute x = 3 back into the function to find the maximum value: f(3) = -3^2 + 6(3) - 8.
Step 6: Calculate f(3): f(3) = -9 + 18 - 8 = 1.
Step 7: The maximum value of the function is 1.

Quadratic Functions – Understanding the properties of quadratic functions, including their vertex and maximum/minimum values.
Vertex Formula – Using the vertex formula x = -b/(2a) to find the x-coordinate of the vertex for a quadratic function.
Function Evaluation – Evaluating the function at the vertex to find the maximum or minimum value.