The vertex form gives the maximum at x = -b/(2a) = 4/2 = 2, f(2) = -2^2 + 4*2 = 4.
Questions & Step-by-step Solutions
1 item
Q
Q: What is the maximum value of f(x) = -x^2 + 4x?
Solution: The vertex form gives the maximum at x = -b/(2a) = 4/2 = 2, f(2) = -2^2 + 4*2 = 4.
Steps: 8
Step 1: Identify the function we are working with, which is f(x) = -x^2 + 4x.
Step 2: Recognize that this is a quadratic function in the form of ax^2 + bx + c, where a = -1, b = 4, and c = 0.
Step 3: Understand that the maximum value of a quadratic function occurs at the vertex. The x-coordinate of the vertex can be found using the formula x = -b/(2a).
Step 4: Substitute the values of a and b into the formula: x = -4/(2 * -1).
Step 5: Calculate the value: x = -4 / -2 = 2.
Step 6: Now, substitute x = 2 back into the function to find the maximum value: f(2) = -2^2 + 4*2.
Step 7: Calculate f(2): f(2) = -4 + 8 = 4.
Step 8: Conclude that the maximum value of f(x) is 4.
