Question

Given y=x²−4x 8, determine which is the equivalent equation in vertex form:

a) y=(x−2)²+4
b) y=(x+2)²−4
c) y=(x−2)² −4
d) y=(x+2)²+4

Answer 1

To convert the quadratic equation y = x² - 4x + 8 to vertex form, we complete the square by adding and subtracting 4, then grouping and factoring to get y = (x - 2)² + 4, which is option (a).

Step-by-step explanation:

The quadratic equation provided, y = x² - 4x + 8, needs to be converted into vertex form, which is y = a(x - h)² + k, where (h,k) is the vertex of the parabola. To convert the standard form to vertex form, we need to complete the square.

First, factor out the coefficient of x², which is 1 in this case (so we don't need to actually factor anything out since it won't change the expression).

Next, we take half of the coefficient of x (which is -4), square it, and add and subtract this number in the equation. Half of -4 is -2, and -2 squared is 4.

Add and subtract 4 within the equation: y = x² - 4x + 4 + 8 - 4.

Group the first three terms and the last two terms separately: y = (x² - 4x + 4) + 8 - 4.

Now factor the first three terms, which is now a perfect square: y = (x - 2)² + 4.

Therefore, the equivalent equation in vertex form is y = (x - 2)² + 4, which corresponds to option (a).