Question
Change the function to vertex form 𝑦 = 𝑥^2 − 8𝑥 + 11
Answers
Answered by
GPT 3.5
To convert the function 𝑦 = 𝑥^2 − 8𝑥 + 11 to vertex form, we need to complete the square.
First, let's factor out the coefficient of 𝑥^2, which is 1:
𝑦 = 𝑥^2 − 8𝑥 + 11
𝑦 = 1(𝑥^2 − 8𝑥) + 11
To complete the square, we take half of the coefficient of 𝑥 (-8/2 = -4), square it (-4^2 = 16), and add it inside the parentheses. However, we can't just add 16 inside the parentheses without changing the equation, so we must subtract 16 outside the parentheses to compensate for this addition:
𝑦 = 1(𝑥^2 - 8𝑥 + 16 - 16) + 11
Now, we can rewrite the equation and factor the square:
𝑦 = 1(𝑥^2 - 8𝑥 + 16) - 16 + 11
The quadratic inside the parentheses can be rewritten as a perfect square:
𝑦 = 1(𝑥 - 4)^2 - 5
So, the function 𝑦 = 𝑥^2 − 8𝑥 + 11 in vertex form is 𝑦 = (𝑥 - 4)^2 - 5.
First, let's factor out the coefficient of 𝑥^2, which is 1:
𝑦 = 𝑥^2 − 8𝑥 + 11
𝑦 = 1(𝑥^2 − 8𝑥) + 11
To complete the square, we take half of the coefficient of 𝑥 (-8/2 = -4), square it (-4^2 = 16), and add it inside the parentheses. However, we can't just add 16 inside the parentheses without changing the equation, so we must subtract 16 outside the parentheses to compensate for this addition:
𝑦 = 1(𝑥^2 - 8𝑥 + 16 - 16) + 11
Now, we can rewrite the equation and factor the square:
𝑦 = 1(𝑥^2 - 8𝑥 + 16) - 16 + 11
The quadratic inside the parentheses can be rewritten as a perfect square:
𝑦 = 1(𝑥 - 4)^2 - 5
So, the function 𝑦 = 𝑥^2 − 8𝑥 + 11 in vertex form is 𝑦 = (𝑥 - 4)^2 - 5.