Question

How does the graph of g(x) = \cos(x) - 12 compare with the parent function g(x) = \cos x?
A. The graph is shifted 12 units up.
B. The graph is shifted 12 units left.
C. The graph is shifted 12 units right.
D. The graph is shifted 12 units down.
Answer
Are you sure
Answered by GPT-4o mini
The function \( g(x) = \cos(x) - 12 \) is a transformation of the parent function \( g(x) = \cos x \).

When you have a function of the form \( f(x) - k \), where \( k \) is a constant, it represents a vertical shift of the graph downward by \( k \) units. In this case, since \( k = 12 \), the graph of \( g(x) = \cos(x) - 12 \) is shifted 12 units down from the parent function \( g(x) = \cos x \).

Therefore, the correct answer is:

**D. The graph is shifted 12 units down.**
Answered by GPT-4o mini
Yes, I am sure.

In the function \( g(x) = \cos(x) - 12 \), the term \(-12\) indicates a vertical shift of the graph. Specifically, subtracting 12 from the cosine function shifts the entire graph down by 12 units.

To clarify further:

- The parent function \( g(x) = \cos(x) \) oscillates between 1 and -1.
- The modified function \( g(x) = \cos(x) - 12 \) oscillates between -11 and -13.

This results in a downward shift, confirming that the correct response is:

**D. The graph is shifted 12 units down.**