The extreme value of a quadratic function is found at its vertex, which represents either the maximum or minimum point of the parabola.
Here's a breakdown of how to determine and find the extreme value:
Understanding Quadratic Functions
A quadratic function is generally represented as:
- f(x) = ax² + bx + c
Where:
- 'a', 'b', and 'c' are constants.
- 'a' determines whether the parabola opens upwards or downwards.
Determining Minimum or Maximum Value
The sign of the coefficient 'a' dictates whether the quadratic function has a minimum or maximum value:
- If a > 0: The parabola opens upwards, meaning the vertex represents the minimum value of the function.
- If a < 0: The parabola opens downwards, meaning the vertex represents the maximum value of the function.
This is important to identify the nature of the extreme value prior to calculation.
Finding the x-coordinate of the Vertex
The x-coordinate of the vertex is given by the formula:
- x = -b / 2a
Calculating the Extreme Value
Once you have the x-coordinate of the vertex, substitute this value back into the original quadratic function, f(x) = ax² + bx + c, to find the y-coordinate, which is the extreme value (maximum or minimum):
- f(-b / 2a)
Practical Steps
Here's a step-by-step guide:
- Identify 'a', 'b', and 'c': From the quadratic function f(x) = ax² + bx + c, note down the coefficients 'a', 'b', and 'c'.
- Determine if it is a minimum or maximum: Check if a > 0 or a < 0. If a > 0, you are calculating a minimum. If a < 0 you are calculating a maximum.
- Calculate the x-coordinate of the vertex: Use the formula x = -b / 2a.
- Substitute the x-coordinate of the vertex back into the quadratic function f(x) to find the corresponding y-coordinate which is the extreme value.
Example
Let's consider f(x) = 2x² - 8x + 6
- Identify: a = 2, b = -8, and c = 6.
- Determine: Since a = 2, which is greater than zero, there will be a minimum value at the vertex.
- Calculate x-coordinate of vertex: x = -(-8) / (2 * 2) = 8 / 4 = 2
- Calculate the extreme value: f(2) = 2(2)² - 8(2) + 6 = 8 - 16 + 6 = -2.
Thus, the minimum value for this quadratic function is -2.
Summary
| Step | Description | Formula/Action |
|---|---|---|
| 1. Identify Coefficients | Identify 'a', 'b', and 'c' from the function f(x) = ax² + bx + c | |
| 2. Determine Extreme Type | Check if 'a' > 0 (minimum) or 'a' < 0 (maximum). | |
| 3. Vertex x-coordinate | Calculate the x-coordinate of the vertex | x = -b / 2a |
| 4. Extreme Value | Substitute the x-coordinate back into f(x) to find the extreme (minimum or maximum) value or y-coordinate of vertex. | f(-b/2a) |
According to the reference, "a quadratic function f(x)=ax²+bx+c has an extreme value at its vertex, so if a > 0, then f(−b/2a) is the minimum, and if a < 0, then f(−b/2a) is the maximum."
