To convert a quadratic equation from standard form (y = ax² + bx + c) to vertex form (y = a(x - h)² + k), you need to complete the square. Here's a step-by-step guide:
1. Understand the Forms
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Standard Form: y = ax² + bx + c
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Vertex Form: y = a(x - h)² + k
- Where (h, k) represents the vertex of the parabola.
2. Completing the Square
Let's work with the standard form: y = ax² + bx + c
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Step 1: Factor out 'a' from the x² and x terms.
y = a(x² + (b/a)x) + c
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Step 2: Complete the square inside the parentheses.
- Take half of the coefficient of the x term (b/a), square it ((b/2a)²), and add it inside the parentheses. Remember to also subtract a * (b/2a)² outside the parentheses to maintain the equation's balance.
y = a(x² + (b/a)x + (b/2a)²) + c - a(b/2a)²
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Step 3: Rewrite the expression inside the parentheses as a squared term.
y = a(x + b/2a)² + c - a(b/2a)²
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Step 4: Simplify the constant term.
y = a(x + b/2a)² + c - b²/4a
3. Identify h and k
Now that the equation is in vertex form, y = a(x - h)² + k, you can identify the vertex (h, k):
- h = -b/2a (Note the sign change!)
- k = c - b²/4a
Example:
Convert y = 2x² + 8x + 5 to vertex form.
- Step 1: y = 2(x² + 4x) + 5
- Step 2: y = 2(x² + 4x + 4) + 5 - 2(4) (Half of 4 is 2, 2² is 4)
- Step 3: y = 2(x + 2)² + 5 - 8
- Step 4: y = 2(x + 2)² - 3
Therefore, the vertex form is y = 2(x + 2)² - 3, and the vertex is (-2, -3).
In Summary: Converting to vertex form involves completing the square, allowing you to easily identify the vertex (h, k) of the parabola represented by the quadratic equation. The 'a' value remains the same in both standard and vertex forms and determines the direction and "width" of the parabola.
