convert to vertex form, write mapping notation 𝑦 =1/4𝑥^2 − 4𝑥 + 5
To convert the given quadratic equation to vertex form, we complete the square. Here's the step-by-step process:
1. Start with the equation: 𝑦 = (1/4)𝑥^2 − 4𝑥 + 5.
2. Factor out the common factor (1/4) from the first two terms: 𝑦 = (1/4)(𝑥^2 − 16𝑥) + 5.
3. Now, to complete the square, take half of the coefficient of 𝑥 (-16/2 = -8) and square it (64).
4. Add this value inside the parentheses and subtract it outside the parentheses to maintain the balance of the equation:
𝑦 = (1/4)(𝑥^2 − 16𝑥 + 64 - 64) + 5.
Simplifying the equation inside the parentheses and combining like terms, we get:
𝑦 = (1/4)((𝑥 - 8)^2 - 64) + 5.
5. Distribute (1/4) to both terms inside the parentheses:
𝑦 = (1/4)(𝑥 - 8)^2 - 16 + 5.
Simplify further:
𝑦 = (1/4)(𝑥 - 8)^2 - 11.
So, the given equation 𝑦 = (1/4)𝑥^2 − 4𝑥 + 5 can be written in vertex form as 𝑦 = (1/4)(𝑥 - 8)^2 - 11.
In mapping notation, this equation can be stated as follows:
(𝑥, 𝑦) → (𝑥 - 8, (1/4)(𝑥 - 8)^2 - 11).