To write a quadratic function in standard form from vertex form, you expand the vertex form equation by squaring the binomial term, distributing any leading coefficient, and then combining like terms.
Here's a step-by-step breakdown:
1. Understand the Forms:
- Vertex Form: f(x) = a(x - h)² + k, where (h, k) is the vertex of the parabola, and 'a' determines the direction and width.
- Standard Form: f(x) = ax² + bx + c, where 'a', 'b', and 'c' are constants.
2. The Conversion Process:
Let's say you have the quadratic function in vertex form: f(x) = a(x - h)² + k
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Step 1: Expand the squared binomial.
(x - h)² = (x - h)(x - h) = x² - 2hx + h² -
Step 2: Substitute the expanded form back into the equation.
f(x) = a(x² - 2hx + h²) + k -
Step 3: Distribute the leading coefficient 'a'.
f(x) = ax² - 2ahx + ah² + k -
Step 4: Combine the constant terms. Let c = ah² + k
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Step 5: Write the equation in standard form.
f(x) = ax² + (-2ah)x + (ah² + k) which simplifies to f(x) = ax² + bx + c
3. Example:
Convert f(x) = 2(x - 5)² + 3 to standard form.
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Step 1: Expand (x - 5)².
(x - 5)² = (x - 5)(x - 5) = x² - 10x + 25 -
Step 2: Substitute.
f(x) = 2(x² - 10x + 25) + 3 -
Step 3: Distribute the '2'.
f(x) = 2x² - 20x + 50 + 3 -
Step 4: Combine constant terms.
50 + 3 = 53 -
Step 5: Standard Form.
f(x) = 2x² - 20x + 53
Therefore, the quadratic function in standard form is f(x) = 2x² - 20x + 53.
