The vertex form of a quadratic equation is a specific way to express a quadratic equation that reveals the vertex of the parabola directly. It is defined as:
y = a(x - h)² + k
Here's a breakdown of each part:
- y: The y-coordinate of a point on the parabola.
- x: The x-coordinate of a point on the parabola.
- a: A constant that determines the direction the parabola opens (upwards if a > 0, downwards if a < 0) and the width of the parabola.
- h: The x-coordinate of the vertex of the parabola.
- k: The y-coordinate of the vertex of the parabola.
Key Characteristics of Vertex Form
The vertex form is particularly useful because:
- Directly Provides the Vertex: The vertex of the parabola is easily identified as the point (h, k).
- Ease of Graphing: Knowing the vertex, and the direction the parabola opens from a, makes it easier to quickly sketch the graph.
Practical Insights and Examples
Let's consider a few examples:
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Example 1: y = 2(x - 3)² + 4
- Here, a = 2, the vertex is (3, 4), and the parabola opens upwards.
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Example 2: y = -1(x + 2)² - 1
- Note that (x + 2) is the same as (x - (-2)). Here a = -1, the vertex is (-2, -1), and the parabola opens downwards.
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Example 3: y = 0.5(x - 0)² + 5 which can be simplified to y = 0.5x² + 5
- Here, a = 0.5, the vertex is (0, 5), and the parabola opens upwards.
How to Use Vertex Form
- Identify 'a', 'h', and 'k': Once a quadratic equation is in vertex form, identify the values for a, h, and k.
- Find the Vertex: The vertex coordinates are (h, k).
- Determine the Parabola's Direction: If a > 0, the parabola opens upwards; if a < 0, it opens downwards.
Summary
The vertex form of a quadratic equation, y = a ( x − h ) 2 + k, is a powerful tool for identifying the vertex and easily visualizing the graph of a parabola. The coordinates of the vertex are directly available as (h, k), which facilitates both sketching and analysis of quadratic functions. As emphasized by the reference, it is designed to easily identify the vertex of the parabola.
