To transform a quadratic equation into vertex form, you'll essentially complete the square. Vertex form is expressed as y = a(x - h)² + k, where (h, k) represents the vertex of the parabola. Here's a breakdown of the process:
Steps to Convert to Vertex Form
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Start with the standard form: Begin with your quadratic equation in the standard form: y = ax² + bx + c.
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Factor out 'a': Factor the coefficient a from the ax² and bx terms: y = a(x² + (b/a)x) + c.
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Complete the square: Take half of the coefficient of the x term (b/a), square it, and then add and subtract it inside the parentheses. This value is (b/2a)². Here's how it looks: y = a(x² + (b/a)x + (b/2a)² - (b/2a)²) + c.
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Rewrite as a perfect square trinomial: The first three terms inside the parentheses now form a perfect square trinomial, which can be rewritten as: y = a[(x + (b/2a))² - (b/2a)²] + c.
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Distribute and simplify: Distribute a back into the bracket and simplify to get: y = a(x + (b/2a))² - a(b/2a)² + c.
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Simplify the Constants: Combine the constant terms to achieve the vertex form. y = a(x - h)² + k. Here: h = -b/2a and k = c - a(b/2a)²
Example
Let's look at an example to illustrate the process: Suppose we have the equation y = 2x² + 8x + 5.
- Factor out 'a': y = 2(x² + 4x) + 5
- Complete the square: Take half of 4 (which is 2), square it (2²=4). Then, add and subtract 4 inside the parentheses: y = 2(x² + 4x + 4 - 4) + 5
- Rewrite as a perfect square: y = 2[(x + 2)² - 4] + 5
- Distribute: y = 2(x + 2)² - 8 + 5
- Simplify: y = 2(x + 2)² - 3
Now, the equation is in vertex form: y = 2(x + 2)² - 3. The vertex of the parabola is at (-2, -3).
Key Points
- Understanding 'a', 'h', and 'k': The 'a' value determines whether the parabola opens up or down, as well as its narrowness. The vertex is at point (h,k) which represents the parabola's highest or lowest point depending on whether the parabola is opening downwards or upwards respectively.
- h: Note that in the vertex form y = a(x - h)² + k, the x-coordinate of the vertex is 'h,' not '-h'. This is why the value of h is -b/2a as derived above.
- Reference Insight: The reference mentions that using the formula x = -b/2a to find the x-coordinate of the vertex can be quicker (19:14).
| Term | Description |
|---|---|
| Vertex Form | y = a(x - h)² + k |
| Standard Form | y = ax² + bx + c |
| h | x-coordinate of the vertex; -b/2a |
| k | y-coordinate of the vertex; c - a(b/2a)² |
| a | Coefficient of x² term, affecting parabola's shape |
By following these steps, you can effectively transform any quadratic equation from standard form to vertex form. This form is especially useful for quickly identifying the vertex of the parabola and understanding the function's transformations.
