Step 1: Identify the function we are working with, 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: Since the coefficient of x^2 (a) is negative, we know 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 = 4 and a = -1.
Step 5: Calculate x = -4/(2 * -1) = -4/-2 = 2.
Step 6: Now, substitute x = 2 back into the function to find the maximum value: f(2) = -2^2 + 4(2) + 1.
Step 7: Calculate f(2): f(2) = -4 + 8 + 1 = 5.
Step 8: Therefore, the maximum value of the function is 5.
Quadratic Functions – Understanding the properties of quadratic functions, including how to find their maximum or minimum values using the vertex formula.
Vertex of a Parabola – Identifying the vertex of a parabola represented by a quadratic function, which gives the maximum or minimum value.
Function Evaluation – Calculating the value of a function at a specific point to determine the maximum value.
