Find the maximum value of the function f(x) = -2x^2 + 8x - 3.

Find the maximum value of the function f(x) = -2x^2 + 8x - 3.

Correct Answer: 8

Step 1: Identify the function we are working with, which is f(x) = -2x^2 + 8x - 3.
Step 2: Recognize that this function is a quadratic function and its graph is a parabola.
Step 3: Determine the direction of the parabola. Since the coefficient of x^2 is negative (-2), the parabola opens downwards.
Step 4: Find the x-coordinate of the vertex of the parabola, which gives the maximum value. Use the formula x = -b/(2a). Here, a = -2 and b = 8.
Step 5: Substitute the values into the formula: x = -8/(2 * -2) = 8/4 = 2.
Step 6: Now, calculate the maximum value of the function by substituting x = 2 back into the function f(x).
Step 7: Calculate f(2): f(2) = -2(2^2) + 8(2) - 3.
Step 8: Simplify the calculation: f(2) = -2(4) + 16 - 3 = -8 + 16 - 3 = 8 - 3 = 5.
Step 9: Therefore, the maximum value of the function is 5.

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