Let's analyze the graph and match the functions to their respective formulas:

Given functions:

1. $f(x) = 2^x$ (exponential growth)
2. $f(x) = \left(\frac{1}{3}\right)^x$ (exponential decay)
3. $f(x) = -\left(\frac{1}{3}\right)^x$ (reflected exponential decay)
4. $f(x) = -2^x$ (reflected exponential growth)

Key observations for matching:

• Graph 1: The curve grows steeply as $x \to +\infty$ and approaches $y = 0$ as $x \to -\infty$. This is the classic exponential growth. Match: $f(x) = 2^x$.

• Graph 2: The curve decreases as $x \to +\infty$, approaching $y = 0$, and grows in the negative direction for $x \to -\infty$. This represents exponential decay. Match: $f(x) = \left(\frac{1}{3}\right)^x$.

• Graph 3: The curve looks similar to exponential decay but is flipped below the x-axis (negative values). This is a reflection of exponential decay. Match: $f(x) = -\left(\frac{1}{3}\right)^x$.

• Graph 4: The curve is a reflection of exponential growth, below the x-axis (negative values). Match: $f(x) = -2^x$.

Final Matches:

1. Graph 1 → $f(x) = 2^x$ (a)
2. Graph 2 → $f(x) = \left(\frac{1}{3}\right)^x$ (b)
3. Graph 3 → $f(x) = -\left(\frac{1}{3}\right)^x$ (c)
4. Graph 4 → $f(x) = -2^x$ (d)

Would you like further details on why these graphs were matched this way?

Related Questions:

1. What happens to exponential graphs when the base is between 0 and 1?
2. How do reflections across the x-axis affect the graph of exponential functions?
3. How does changing the base in $f(x) = a^x$ impact the graph?
4. What is the domain and range of exponential functions?
5. Can exponential graphs have horizontal asymptotes other than $y = 0$?

Tip:

To determine whether an exponential graph represents growth or decay, check the base:

• $a > 1$: Growth
• $0 < a < 1$: Decay