To convert a quadratic equation from vertex form to standard form, you need to expand and simplify the equation. Here's a breakdown of the process:
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Understand the Forms:
- Vertex Form: y = a(x - h)² + k, where (h, k) is the vertex of the parabola and 'a' determines the direction and stretch.
- Standard Form: y = ax² + bx + c, where 'a', 'b', and 'c' are constants.
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Expand the Squared Term: The key step is to expand the squared term (x - h)². Remember that (x - h)² means (x - h)(x - h). Use the FOIL (First, Outer, Inner, Last) method or the distributive property to expand this.
- (x - h)(x - h) = x² - hx - hx + h² = x² - 2hx + h²
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Substitute and Simplify: Substitute the expanded form back into the vertex form equation:
- y = a(x² - 2hx + h²) + k
Distribute the 'a' across the terms inside the parentheses:
- y = ax² - 2ahx + ah² + k
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Combine Constants: Combine the constant terms (ah² and k) to get the 'c' term in the standard form.
- y = ax² - 2ahx + (ah² + k)
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Identify a, b, and c: Now the equation is in standard form, y = ax² + bx + c. You can identify the coefficients:
- The 'a' value remains the same as in vertex form.
- b = -2ah
- c = ah² + k
Example:
Convert y = 2(x - 5)² + 3 from vertex form to standard form.
- Expand: (x - 5)² = (x - 5)(x - 5) = x² - 10x + 25
- Substitute: y = 2(x² - 10x + 25) + 3
- Distribute: y = 2x² - 20x + 50 + 3
- Combine: y = 2x² - 20x + 53
Therefore, the standard form is y = 2x² - 20x + 53.
