Determine the maximum value of f(x) = -2x^2 + 4x + 1. (2023)

Determine the maximum value of f(x) = -2x^2 + 4x + 1. (2023)

Step 1: Identify the function you want to analyze, which is f(x) = -2x^2 + 4x + 1.
Step 2: Recognize that this is a quadratic function in the form of ax^2 + bx + c, where a = -2, b = 4, and c = 1.
Step 3: Determine the x-coordinate of the vertex using the formula x = -b/(2a). Here, b = 4 and a = -2.
Step 4: Plug in the values: x = -4/(2 * -2) = -4/-4 = 1.
Step 5: Now, find the maximum value of the function by substituting x = 1 back into the function f(x).
Step 6: Calculate f(1): f(1) = -2(1)^2 + 4(1) + 1.
Step 7: Simplify the calculation: f(1) = -2(1) + 4 + 1 = -2 + 4 + 1 = 3.
Step 8: Conclude that the maximum value of the function f(x) is 3.

Quadratic Functions – Understanding the properties of quadratic functions, including how to find the vertex and 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 determine the maximum or minimum value.