The vertex form of a quadratic equation is an expression that allows for easy identification of the vertex of the parabola represented by the equation. This form is particularly useful in graphing and analyzing the properties of the parabola, such as its maximum or minimum value and direction of opening.
What is the Vertex Form?
The vertex form of a quadratic equation is given by: y = a(x – h)² + k, where:
- ‘a’ determines the width and direction of the parabola (upwards if ‘a’ is positive, downwards if ‘a’ is negative).
- ‘(h, k)’ is the vertex of the parabola. ‘h’ affects the horizontal position, and ‘k’ affects the vertical position.
Converting to Vertex Form
To convert a standard quadratic equation (y = ax² + bx + c) to vertex form, you can complete the square, which involves algebraic manipulation to restructure the equation.
Examples of Vertex Form
Example 1: The vertex form of y = 2(x – 3)² + 4 shows a parabola with a vertex at (3, 4) that opens upwards (since ‘a’ is positive).
Example 2: For y = -(x + 2)² – 5, the vertex form indicates a parabola with a vertex at (-2, -5) that opens downwards (as ‘a’ is negative).
Importance of Vertex Form
The vertex form is particularly valuable in graphing quadratic functions, solving optimization problems, and understanding the motion of objects in physics, as it clearly identifies the vertex and the direction of the parabola.
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