To convert a quadratic function from standard form to vertex form, you need to complete the square. Here's a breakdown of the process:
1. Understand the Forms
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Standard Form: A quadratic function in standard form is written as
y = ax² + bx + c, where a, b, and c are constants. -
Vertex Form: A quadratic function in vertex form is written as
y = a(x - h)² + k, where (h, k) represents the vertex of the parabola. The 'a' value remains the same in both forms.
2. Completing the Square: The Steps
Here's how to transform y = ax² + bx + c into y = a(x - h)² + k:
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Step 1: Factor out 'a' from the x² and x terms.
y = a(x² + (b/a)x) + c -
Step 2: Complete the square inside the parentheses.
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Take half of the coefficient of the x term inside the parentheses: (b/a) / 2 = b/(2a)
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Square the result: (b/(2a))² = b²/(4a²)
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Add and subtract this value inside the parentheses:
y = a(x² + (b/a)x + b²/(4a²) - b²/(4a²)) + c
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Step 3: Rewrite the expression inside the parentheses as a squared term.
The expression
x² + (b/a)x + b²/(4a²)is a perfect square trinomial and can be rewritten as(x + b/(2a))².y = a((x + b/(2a))² - b²/(4a²)) + c -
Step 4: Distribute 'a' to the term being subtracted.
y = a(x + b/(2a))² - a(b²/(4a²)) + c -
Step 5: Simplify and combine constants.
y = a(x + b/(2a))² - b²/(4a) + c
y = a(x + b/(2a))² + (4ac - b²)/(4a)
3. Identify the Vertex
Now the equation is in vertex form: y = a(x - h)² + k
h = -b/(2a)k = (4ac - b²)/(4a)or, more simply, k is the value of y when x = h.
Therefore, the vertex of the parabola is the point (-b/(2a), (4ac - b²)/(4a)).
Example:
Convert y = 2x² + 8x + 5 to vertex form.
- Factor out 2:
y = 2(x² + 4x) + 5 - Complete the square: Half of 4 is 2, and 2 squared is 4.
y = 2(x² + 4x + 4 - 4) + 5 - Rewrite as a square:
y = 2((x + 2)² - 4) + 5 - Distribute:
y = 2(x + 2)² - 8 + 5 - Simplify:
y = 2(x + 2)² - 3
The vertex form is y = 2(x + 2)² - 3, and the vertex is (-2, -3).
