Determine the maximum value of f(x) = -x^2 + 6x - 8. (2022)

Determine the maximum value of f(x) = -x^2 + 6x - 8. (2022)

Step 1: Identify the function we are working with, which is f(x) = -x^2 + 6x - 8.
Step 2: Recognize that this is a quadratic function in the form of ax^2 + bx + c, where a = -1, b = 6, and c = -8.
Step 3: Since the coefficient of x^2 (a) is negative, the parabola opens downwards, meaning it has a maximum point.
Step 4: To find the x-coordinate of the maximum point, use the formula x = -b/(2a). Here, b = 6 and a = -1.
Step 5: Calculate x = -6/(2 * -1) = -6/-2 = 3.
Step 6: Now, substitute x = 3 back into the function to find the maximum value: f(3) = -3^2 + 6(3) - 8.
Step 7: Calculate f(3): f(3) = -9 + 18 - 8 = 1.
Step 8: Therefore, the maximum value of f(x) is 1, which occurs at x = 3.

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 maximum or minimum point.
Function Evaluation – Evaluating the function at specific points to find maximum or minimum values.