Question

which equation is represented by the graph below?
answer
\\( y = 3^{x - 2} \\)
\\( y = 3^{x} + 2 \\)
\\( y = \left(\frac{1}{3}\
ight)^{x} + 2 \\)
\\( y = \left(\frac{1}{3}\
ight)^{x - 2} \\)

Explanation:

Step1: Check y-intercept (x=0)

Test x=0 in each option:

• Option1: $y=3^{0-2}=3^{-2}=\frac{1}{9}\approx0.11$
• Option2: $y=3^{0}+2=1+2=3$
• Option3: $y=(\frac{1}{3})^0+2=1+2=3$
• Option4: $y=(\frac{1}{3})^{0-2}=3^{2}=9$

The graph has y-intercept 3, so eliminate Option1 and Option4.

Step2: Check growth direction

The graph increases as x increases (exponential growth).

• Option2: $y=3^x+2$, base $3>1$, so it grows as x increases.
• Option3: $y=(\frac{1}{3})^x+2$, base $0<\frac{1}{3}<1$, so it decays as x increases.

This matches the graph's upward trend for positive x.

Answer:

$\boldsymbol{y = 3^x + 2}$ (Option B)