The graph of a quadratic polynomial is a parabola.
A quadratic polynomial is an expression of the form ax² + bx + c, where a, b, and c are constants, and a ≠ 0. The graph of this polynomial when plotted on a coordinate plane always results in a characteristic U-shaped curve known as a parabola.
Understanding Parabolas
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Definition: A parabola is a symmetrical curve formed by the intersection of a cone with a plane parallel to its side.
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Equation: The standard form of a quadratic equation is y = ax² + bx + c.
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Orientation: According to the provided reference, depending on the values of the coefficients in the expression and the discriminant, we get either a parabola opening upwards or downwards when x is the independent variable.
- If 'a' is positive (a > 0), the parabola opens upwards. The vertex represents the minimum point of the graph.
- If 'a' is negative (a < 0), the parabola opens downwards. The vertex represents the maximum point of the graph.
Key Features of a Parabola
| Feature | Description |
|---|---|
| Vertex | The highest or lowest point on the parabola. |
| Axis of Symmetry | A vertical line that passes through the vertex, dividing the parabola into two symmetrical halves. |
| Roots/Zeros/x-intercepts | The points where the parabola intersects the x-axis (where y = 0). These are the solutions to the quadratic equation. |
| y-intercept | The point where the parabola intersects the y-axis (where x = 0). |
Examples
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y = x²: This is the simplest parabola, opening upwards with its vertex at the origin (0,0).
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y = -x² + 4: This parabola opens downwards with its vertex at (0,4).
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y = (x - 2)² + 1: This parabola opens upwards with its vertex at (2,1).
Conclusion
The parabolic shape derived from graphing quadratic polynomials is a fundamental concept in algebra with significant applications in physics, engineering, and other fields. The orientation and specific features of the parabola are determined by the coefficients in the quadratic equation.
