The maximum of a quadratic function can be found by understanding the properties of its graph and using specific formulas. Quadratic equations are often expressed in the form ax² + bx + c. The graph of a quadratic is a parabola. Depending on the sign of 'a', the parabola will either open upwards or downwards. If 'a' is negative, the parabola opens downwards, which means it will have a maximum point. This maximum point is the vertex of the parabola.
Methods to Find the Maximum
1. Vertex Formula
The x-coordinate of the vertex can be found using the formula:
- x = -b / 2a
Once you have the x-coordinate, you can substitute it back into the quadratic equation to find the y-coordinate, which is the maximum value of the quadratic. This method directly gives you the coordinates of the vertex (h, k), where 'h' is the x-coordinate and 'k' is the maximum value.
2. Formula from the reference
As stated in the reference, the maximum value (when the parabola opens downwards) can be found using the following formula:
max= c- (b²/4a)
This formula directly calculates the y-coordinate of the vertex, which represents the maximum value.
3. Completing the Square
Another method is to complete the square. By rewriting the quadratic equation in the form a(x - h)² + k, the vertex can easily be identified as (h, k). 'k' is the maximum value when 'a' is negative.
Example:
Let's take an example quadratic equation: f(x) = -2x² + 8x - 3
Here, a = -2, b = 8, and c = -3.
Using the reference formula to find the maximum value:
max = c - (b²/4a)
max = -3 - (8² / 4 * -2)
max = -3 - (64 / -8)
max = -3 - (-8)
max = -3 + 8
max = 5
Therefore, the maximum value of this quadratic function is 5.
Summary
| Method | Formula/Process | When to Use |
|---|---|---|
| Vertex Formula | x = -b / 2a , then substitute x into the equation | Useful for finding the x-coordinate of the vertex. |
| Direct Maximum Formula (from reference) | max = c - (b²/4a) | When you need to find the maximum value directly without finding x value. |
| Completing the Square | Rewrite the equation in the form a(x-h)² + k | Helps in visually understanding the transformations of the quadratic. |
Conclusion
Finding the maximum value of a quadratic function involves analyzing the quadratic equation and applying appropriate formulas or techniques. Whether you use the vertex formula, the direct maximum formula, or completing the square, the goal is to find the y-coordinate of the vertex, which represents the maximum value when the parabola opens downwards (a < 0).
