From the image, we see four exponential functions and corresponding graphs. The task is to match each function to its respective graph. The functions are:

1. $m(x) = 2(4)^x$
2. $p(x) = 2\left(\frac{1}{4}\right)^x$
3. $n(x) = 4(2)^x$
4. $q(x) = 4\left(\frac{1}{2}\right)^x$

Key observations for matching:

• Exponential growth occurs when the base is greater than 1.
• Exponential decay occurs when the base is a fraction between 0 and 1.

Analysis of the functions:

1. $m(x) = 2(4)^x$:

   • Growth (base $4 > 1$).
   • Starts small and grows rapidly as $x$ increases.

2. $p(x) = 2\left(\frac{1}{4}\right)^x$:

   • Decay (base $\frac{1}{4} < 1$).
   • Starts large and decreases as $x$ increases.

3. $n(x) = 4(2)^x$:

   • Growth (base $2 > 1$).
   • Starts with a higher initial value and grows rapidly.

4. $q(x) = 4\left(\frac{1}{2}\right)^x$:

   • Decay (base $\frac{1}{2} < 1$).
   • Starts with a higher initial value and decreases.

Matching:

• The top-left graph shows decay, starting from a high value and decreasing (matches $q(x)$).
• The top-right graph shows growth, starting from a low value and increasing rapidly (matches $m(x)$).
• The bottom-left graph shows growth with a higher starting value (matches $n(x)$).
• The bottom-right graph shows decay with a slower decrease (matches $p(x)$).

Final Pairing:

1. $m(x) = 2(4)^x$ → Top-right graph.
2. $p(x) = 2\left(\frac{1}{4}\right)^x$ → Bottom-right graph.
3. $n(x) = 4(2)^x$ → Bottom-left graph.
4. $q(x) = 4\left(\frac{1}{2}\right)^x$ → Top-left graph.

Would you like further explanation on exponential functions? Here are five related questions to expand understanding:

1. How do we determine the growth/decay rate of an exponential function?
2. What role does the coefficient (e.g., 2 or 4) play in shaping the graph?
3. How do transformations (e.g., $f(x) = a(b)^{x-c}$) affect the graph's shape?
4. Why does the base value determine whether the function grows or decays?
5. How are exponential graphs used in real-world applications?

Tip: Always check the base of the exponent to determine whether the function represents growth or decay.