A graph of a quadratic function is a parabola, which is a U-shaped curve.
To determine which graph represents a quadratic function, you need to look for a U-shaped curve. Here’s a breakdown of why this is the case and what to look for:
Understanding Quadratic Functions
A quadratic function is defined by the general form:
f(x) = ax² + bx + c
Where:
- 'a', 'b', and 'c' are constants, and 'a' is not zero.
- The x² term is what makes the function quadratic.
Key Features of a Parabola
- U-Shape: The most distinctive characteristic of a parabola is its U-shape. It can open upwards or downwards.
- Vertex: The vertex is the lowest point (if the parabola opens upwards) or the highest point (if it opens downwards). It is also a point of symmetry.
- Symmetry: A parabola is symmetrical around a vertical line that passes through the vertex (called the axis of symmetry).
- No Straight Lines: Quadratic functions are smooth curves without straight line segments.
How to Identify a Parabola
- Look for the U-Shape: The most straightforward way is to look for the distinctive U-shaped curve. If a graph is a straight line or a V-shape, then it's definitely not a graph of a quadratic function.
- Smooth Curve: The curve should be smooth without any sharp corners or breaks.
- Vertical Axis of Symmetry: Check for the imaginary line about which the graph is symmetric.
Example
Let’s illustrate with a table:
| Graph Shape | Quadratic Function? | Reason |
|---|---|---|
| U-shape | Yes | The graph meets the description of a parabola. |
| Straight Line | No | Linear functions result in straight lines. |
| V-shape | No | Absolute value functions can look like V-shapes. |
| S-shape | No | Cubic or other functions might result in S-shapes. |
In summary, remember that the graph of a quadratic function is always a parabola, a U-shaped curve.
