Question

mitchell knows that $f(x) = x^2$ is the parent graph of a quadratic. what steps should he take if he wants to transform the parent graph to obtain the graph of $f(x) = (x + 6)^2 - 3$?
a. he should shift it 6 units to the right and 3 units up.
b. he should shift it 6 units to the left and 3 units up.
c. he should shift it 6 units to the left and 3 units down.
d. he should shift it 6 units to the right and 3 units down.

Explanation:

Step1: Recall Transformations

For a quadratic function \( y = (x - h)^2 + k \), horizontal shift: \( h \) units (right if \( h>0 \), left if \( h<0 \)), vertical shift: \( k \) units (up if \( k>0 \), down if \( k<0 \)).

Step2: Analyze \( f(x)=(x + 6)^2 - 3 \)

Compare with \( y = (x - h)^2 + k \). Here, \( h=-6 \) (so shift left 6 units) and \( k = -3 \) (shift down 3 units).

Answer:

C. He should shift it 6 units to the left and 3 units down.