Find the maximum value of the function f(x) = -2x^2 + 8x - 5.

Find the maximum value of the function f(x) = -2x^2 + 8x - 5.

Step 1: Identify the function we are working with, which is f(x) = -2x^2 + 8x - 5.
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: Calculate -b/(2a): Substitute b = 8 and a = -2 into the formula: x = -8/(2 * -2) = 8/4 = 2.
Step 6: Now, substitute x = 2 back into the function to find the maximum value: f(2) = -2(2^2) + 8(2) - 5.
Step 7: Calculate f(2): f(2) = -2(4) + 16 - 5 = -8 + 16 - 5 = 3.
Step 8: The maximum value of the function f(x) is 3.

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.
Function Evaluation – Evaluating the function at the vertex to find the maximum or minimum value.