The maximum value of a quadratic function in vertex form is found directly from the vertex coordinates. Let's explore how.
Understanding Vertex Form
A quadratic function in vertex form is expressed as:
f(x) = a(x - h)² + k
Where:
- (h, k) represents the vertex of the parabola.
- a determines the direction and "width" of the parabola.
Determining Maximum Value
The vertex of the parabola represents either the maximum or minimum point of the quadratic function.
- If a < 0: The parabola opens downwards, and the vertex (h, k) represents the maximum point. The maximum value of the function is k.
- If a > 0: The parabola opens upwards, and the vertex (h, k) represents the minimum point. The function does not have a maximum value in this case; it extends infinitely upwards.
Steps to Find the Maximum Value
- Identify the vertex form: Ensure the quadratic function is in the form f(x) = a(x - h)² + k.
- Determine the sign of a: If a is negative, a maximum exists.
- Extract the k value: The k value of the vertex (h, k) is the maximum value of the function.
Example
Consider the function:
f(x) = -2(x - 3)² + 5
- a = -2 (negative, so a maximum exists)
- h = 3
- k = 5
Therefore, the maximum value of this quadratic function is 5, and it occurs at x = 3. The vertex is (3, 5).
Alternative Method for Standard Form
If your quadratic is in standard form (y = ax² + bx + c), you can find the x-coordinate of the vertex using the formula x = -b / (2a), as mentioned in the reference. Then, substitute this x value back into the equation to find the y-coordinate, which represents the maximum (or minimum) value. According to the reference, completing the square transforms a quadratic equation into vertex form so you can directly extract the vertex's coordinates, with the y-value representing the maximum (or minimum) value.
