Final answer:
To find the maximum value of a quadratic function, use the vertex form of the equation. The vertex form is f(x) = a(x - h)^2 + k, where the vertex is at (h, k). Determine the x-coordinate of the vertex using x = -b/(2a), then substitute it into the equation to find the y-coordinate which represents the maximum value.
Step-by-step explanation:
To find the maximum value of a quadratic function, we can use the vertex form of the quadratic equation: f(x) = a(x - h)^2 + k. In this form, the vertex of the parabola is at the point (h, k), and the coefficient 'a' determines whether the parabola opens upward or downward. If 'a' is positive, the parabola opens upward and the vertex represents the minimum value. If 'a' is negative, the parabola opens downward and the vertex represents the maximum value.
To find the maximum value, we need to determine the coordinates of the vertex. The x-coordinate of the vertex can be found using the formula x = -b/(2a), where 'a' and 'b' are coefficients of the quadratic function in the standard form f(x) = ax^2 + bx + c. Once we have the x-coordinate, we can substitute it into the quadratic function to find the corresponding y-coordinate, which represents the maximum value.
For example, let's consider the quadratic function f(x) = 2x^2 - 8x + 5. The coefficient 'a' is 2 and the coefficient 'b' is -8. Using the formula for the x-coordinate of the vertex, we have x = -(-8)/(2(2)) = 2. Substituting this value into the quadratic function, we get f(2) = 2(2)^2 - 8(2) + 5 = 5. Therefore, the maximum value of the quadratic function is 5.