The task requires matching graphs of exponential functions to their corresponding equations. Here’s a detailed solution for the problem:

Equations Given:

1. A: $y = 2 \cdot 2^x$
2. B: $y = 2 \cdot \left( \frac{1}{2} \right)^x$
3. C: $y = \frac{1}{2} \cdot 2^x$
4. D: $y = 2^{-x}$

Matching Process:

1. Graph a): This graph has a decreasing trend (exponential decay). The base is less than 1 since the curve approaches zero as $x$ increases. This corresponds to B: $y = 2 \cdot \left( \frac{1}{2} \right)^x$.

2. Graph b): This graph shows an increasing exponential trend. The slope grows rapidly, suggesting it has a base greater than 1. This corresponds to A: $y = 2 \cdot 2^x$.

3. Graph c): This graph also shows a decreasing trend (exponential decay), but its shape matches when the equation is $y = 2^{-x}$, indicating a specific decay rate. This corresponds to D: $y = 2^{-x}$.

4. Graph d): This graph exhibits a slower increasing trend compared to graph b). The coefficient $\frac{1}{2}$ in the equation reduces the growth rate. This corresponds to C: $y = \frac{1}{2} \cdot 2^x$.

Final Matches:

• a) $y = 2 \cdot \left( \frac{1}{2} \right)^x$ (B)
• b) $y = 2 \cdot 2^x$ (A)
• c) $y = 2^{-x}$ (D)
• d) $y = \frac{1}{2} \cdot 2^x$ (C)

Let me know if you need further clarification or details!

5 Related Questions:

1. What are the key differences between exponential growth and decay functions?
2. How does the base of an exponential function affect its graph?
3. What happens to an exponential graph when the coefficient changes?
4. How can you find the equation of an exponential function from its graph?
5. What are real-world applications of exponential growth and decay?

Tip:

To identify exponential growth or decay, check whether the base $b > 1$ (growth) or $0 < b < 1$ (decay).