Question

m(x) = 3(2)^x + 4, g(x) = 2(3)^x + 4, h(x) = 3(3)^x + 2, k(x) = 4(3)^x + 2 (and four graphs of exponential functions)

Explanation:

Step1: Find horizontal asymptote

For exponential functions of the form $f(x)=a(b)^x + c$, the horizontal asymptote is $y=c$.

• $m(x)$: $y=4$
• $g(x)$: $y=4$
• $h(x)$: $y=2$
• $k(x)$: $y=2$

So top two graphs (asymptote $y=4$) match $m(x),g(x)$; bottom two (asymptote $y=2$) match $h(x),k(x)$.

Step2: Calculate y-intercept ($x=0$)

• $m(0)=3(2)^0 + 4=3+4=7$
• $g(0)=2(3)^0 + 4=2+4=6$
• $h(0)=3(3)^0 + 2=3+2=5$
• $k(0)=4(3)^0 + 2=4+2=6$

Step3: Match graphs to functions

1. Top-left graph: asymptote $y=4$, y-intercept $7$ → $m(x)=3(2)^x + 4$
2. Top-right graph: asymptote $y=4$, y-intercept $6$ → $g(x)=2(3)^x + 4$
3. Bottom-left graph: asymptote $y=2$, y-intercept $5$ → $h(x)=3(3)^x + 2$
4. Bottom-right graph: asymptote $y=2$, y-intercept $6$ → $k(x)=4(3)^x + 2$

Answer:

Top-left graph: $\boldsymbol{m(x)=3(2)^x + 4}$
Top-right graph: $\boldsymbol{g(x)=2(3)^x + 4}$
Bottom-left graph: $\boldsymbol{h(x)=3(3)^x + 2}$
Bottom-right graph: $\boldsymbol{k(x)=4(3)^x + 2}$