y=a(x-h)^2+k

Dou Vamel yi

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15-01-2025

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The equation y = a(x-h)² + k represents a parabola in its vertex form. Understanding this form is crucial for quickly graphing parabolas and extracting key information about their shape and position. This article will break down this equation, explaining each component and demonstrating its practical applications.

Understanding the Components of y = a(x-h)² + k

The beauty of the vertex form lies in its explicit revelation of the parabola's key features:

• (h, k): The Vertex: This ordered pair represents the coordinates of the parabola's vertex – the point where the parabola reaches its minimum (if a > 0) or maximum (if a < 0) value. Note that 'h' is subtracted within the parentheses, so the x-coordinate of the vertex is actually 'h', not '-h'.

• a: The Scaling Factor: This value determines the parabola's vertical stretch or compression, and its direction (opening upwards or downwards).

  • |a| > 1: The parabola is vertically stretched (narrower).
  • 0 < |a| < 1: The parabola is vertically compressed (wider).
  • a > 0: The parabola opens upwards (U-shaped).
  • a < 0: The parabola opens downwards (inverted U-shaped).

Let's illustrate with an example: Consider the equation y = 2(x - 3)² + 1. Here, the vertex is (3, 1), the parabola opens upwards (a = 2 > 0), and it's vertically stretched (|a| = 2 > 1).

Graphing Parabolas Using the Vertex Form

Graphing a parabola from its vertex form is straightforward:

1. Identify the vertex (h, k): This gives you the starting point for your graph.

2. Determine the direction and stretch/compression (a): This tells you the parabola's shape and whether it opens upwards or downwards.

3. Plot additional points: Choose a few x-values on either side of the vertex, substitute them into the equation to find their corresponding y-values, and plot these points. The parabola's symmetry helps; points equidistant from the vertex will have the same y-value (except for the sign if the parabola opens downwards).

Finding the Equation from the Vertex and a Point

If you know the vertex (h, k) and another point (x, y) on the parabola, you can determine the value of 'a' and thus the complete equation:

1. Substitute the vertex (h, k) and the point (x, y) into the equation: This will give you one equation with one unknown (a).

2. Solve for 'a': Isolate 'a' to find its value.

3. Substitute 'a', 'h', and 'k' back into the vertex form: This gives you the complete equation of the parabola.

Example: Let's say the vertex is (2, -1) and the point (4, 3) lies on the parabola.

Substitute: 3 = a(4 - 2)² - 1

Solve for a: 4 = 4a => a = 1

Therefore, the equation is y = (x - 2)² - 1.

Applications of the Vertex Form

The vertex form has numerous applications in various fields:

• Physics: Modeling projectile motion (e.g., the trajectory of a ball).

• Engineering: Designing parabolic antennas or reflectors.

• Economics: Representing quadratic cost or revenue functions.

• Computer Graphics: Creating curved shapes and paths.

Frequently Asked Questions (FAQs)

Q: How do I convert the standard form (y = ax² + bx + c) to vertex form?

A: Complete the square for the x terms. This involves manipulating the equation to obtain the (x-h)² form.

Q: What if the vertex is at the origin (0, 0)?

A: The equation simplifies to y = ax².

Q: Can a parabola have multiple vertices?

A: No, a parabola has only one vertex, which is its highest or lowest point.

This comprehensive guide should equip you with a strong understanding of the vertex form of a parabola (y = a(x-h)² + k). Remember to practice applying these concepts to various examples to solidify your understanding. Mastering this form will greatly enhance your ability to analyze and work with parabolic functions.