Question

how does the graph of (g(x) = 3^x - 2) compare to the graph of (f(x) = 3^x)?

the graph of (g(x)) is a translation of (f(x)) 2 units left.
the graph of (g(x)) is a translation of (f(x)) 2 units right.
the graph of (g(x)) is a translation of (f(x)) 2 units up.
the graph of (g(x)) is a translation of (f(x)) 2 units down.

Explanation:

Identify the relationship between the functions

$$ g(x) = f(x) - 2 $$

Determine the geometric transformation

$$ y = f(x) - c \implies \text{Vertical translation down by } c \text{ units} $$

Apply to the given functions

$$ g(x) = 3^x - 2 \implies \text{Translation of } f(x) = 3^x \text{ down by } 2 \text{ units} $$

Answer:

• The graph of \(g(x)\) is a translation of \(f(x)\) 2 units left.
• The graph of \(g(x)\) is a translation of \(f(x)\) 2 units right.
• The graph of \(g(x)\) is a translation of \(f(x)\) 2 units up.
• The graph of \(g(x)\) is a translation of \(f(x)\) 2 units down. (Correct answer)