Comparing quadratic functions involves analyzing their key features to understand their similarities and differences. The process primarily focuses on comparing their vertices and the direction they open (determined by the 'a' value). Here's a structured approach based on the reference provided and standard methods:
Understanding Quadratic Functions
A quadratic function is typically expressed in the form f(x) = ax² + bx + c, where a, b, and c are constants, and a ≠ 0. The graph of a quadratic function is a parabola, which can open upwards or downwards. The vertex is the highest or lowest point of the parabola.
Methods for Comparing Quadratic Functions
Here's how you can compare quadratic functions:
1. Identify the Vertex
- Definition: The vertex is the turning point of the parabola. It represents either the minimum (if the parabola opens upwards) or the maximum (if the parabola opens downwards) value of the function.
- Finding the Vertex:
- Using the Formula: For a quadratic in the form f(x) = ax² + bx + c, the x-coordinate of the vertex is given by x = -b / 2a. Substitute this value back into the function to find the corresponding y-coordinate.
- From a Table: As per the reference, if a table of values is provided, the vertex is identified by finding where the y-values start to reverse their trend (decreasing then increasing for upward-facing parabolas, or vice-versa). This point is the vertex (x,y).
2. Compare Vertex Y-Values
- Reference: As stated in step 3 of the reference: "Compare the y values of each point to determine which graph has a higher maximum or lower minimum."
- Interpretation:
- Minimum Value Comparison: For two upward-opening parabolas, the parabola with the lower y-coordinate at its vertex has the lower minimum value.
- Maximum Value Comparison: For two downward-opening parabolas, the parabola with the higher y-coordinate at its vertex has the higher maximum value.
3. Compare the 'a' Value
- Definition: The coefficient 'a' in f(x) = ax² + bx + c determines whether the parabola opens upwards (a > 0) or downwards (a < 0).
- Comparison:
- A positive 'a' indicates an upward opening parabola (minimum value at the vertex).
- A negative 'a' indicates a downward opening parabola (maximum value at the vertex).
- The absolute value of 'a' affects the "width" of the parabola. Larger |a| values make the parabola narrower, while smaller |a| values make it wider.
4. Analyzing the Parabola's Properties
- Direction of Opening: Compare if parabolas open upwards or downwards based on their 'a' values.
- Vertical Stretch/Compression: The absolute value of 'a' gives the parabola's stretch or compression relative to the standard parabola y = x².
- Axis of Symmetry: The vertical line that passes through the vertex which is useful in understanding the symmetry of the function.
Example
Let's consider two quadratic functions:
- f(x) = 2x² - 8x + 6
- g(x) = -x² + 4x + 1
Analysis:
- Vertices:
- For f(x): x = -(-8) / (2*2) = 2. f(2) = 2(2)² - 8(2) + 6 = -2. Vertex: (2, -2).
- For g(x): x = -4 / (2 * -1) = 2. g(2) = -(2)² + 4(2) + 1 = 5. Vertex: (2, 5).
- Comparison:
- f(x) opens upwards (a = 2 > 0), so its vertex is a minimum.
- g(x) opens downwards (a = -1 < 0), so its vertex is a maximum.
- The minimum of f(x) is -2.
- The maximum of g(x) is 5.
Conclusion: g(x) has a significantly higher maximum value than the minimum value of f(x).
Summary Table
| Feature | How to Compare |
|---|---|
| Vertex | Find the coordinates (x, y) of the vertex for each function. |
| Vertex Y-values | Compare the y-values; greater values for maximums, lower values for minimums. |
| 'a' value | Positive ‘a’ values open upwards; negative values open downwards. |
| Direction of Opening | Based on the sign of the 'a' value. |
