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

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

Step 1: Identify the function you want to analyze, which is f(x) = -x^2 + 4x + 1.
Step 2: Recognize that this is a quadratic function in the form of f(x) = ax^2 + bx + c, where a = -1, 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 = -1.
Step 4: Calculate x = -4/(2 * -1) = -4 / -2 = 2.
Step 5: Now, substitute x = 2 back into the function to find the maximum value: f(2) = -2^2 + 4(2) + 1.
Step 6: Calculate f(2): f(2) = -4 + 8 + 1 = 5.
Step 7: Conclude that the maximum value of the function f(x) is 5.

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 determine the maximum or minimum value.