Question

The functions f(x) = (x² - 2) and g(x) = -(x-2)² - 1 have been re-written using the completing the square method. What are the new expressions for f(x) and g(x)?

1) (x-1)² - 3 and -(x-2)² - 1
2) (x+1)² - 3 and -(x+2)² - 1
3) (x-1)² + 3 and -(x-2)² + 1
4) (x+1)² + 3 and -(x+2)² + 1

Answer 1

The new expressions for the functions f(x) and g(x) after completing the square are (x-1)² - 3 and -(x-2)² - 1, respectively, corresponding to option 1.

Step-by-step explanation:

The function f(x) = (x² - 2) can be rewritten by completing the square as follows:

1. Identify the coefficient of the x² term (which is 1 in this case) and the constant term (-2).
2. Find the value that completes the square. This is the value that, when added and subtracted inside the parenthesis, creates a perfect square trinomial. The number we need to add and subtract is (½ * coefficient of x)².
3. In this case, that value is (½ * 1)² = (0.5)² = 0.25. So, we add and subtract 0.25 inside the parenthesis.
4. Rewrite the expression: x² - 2 + 0.25 - 0.25 = (x ²-0.5²) - 1.75. This simplifies further to (x - 1)² - 3.

The function g(x) = -(x-2)² - 1 is already in completed square form, so it does not need to be rewritten.

Therefore, the new expressions for f(x) and g(x) are (x-1)² - 3 and -(x-2)² - 1, respectively, which corresponds to option 1.