1. **Problem:** Graph and compare the functions to $f(x) = 2^x$ for the following: (a) $f(x) = 2^x - \frac{1}{2}$ (b) $f(x) = 2^x + 3$ (c) $f(x) = 2^{x-2}$ (d) $f(x) = 2^{-x}$ 2. **Problem:** Graph and compare the functions to $f(x) = 3^x$ for the following: (a) $f(x) = 3^x - 2$ (b) $f(x) = 3^x + 4$ (c) $f(x) = 3^{x-1}$ (d) $f(x) = 3^{x+4}$ 3. **Problem:** Graph $f(x) = 3^x$ 4. **Problem:** Graph $f(x) = (0.5)^x$ 5. **Problem:** Graph $f(x) = \left(\frac{2}{3}\right)^x$ 6. **Problem:** Graph $f(x) = 4^{-x}$ 7. **Problem:** Graph $f(x) = e^x$ 8. **Problem:** Graph $f(x) = e^{1-x}$ 9. **Problem:** Graph $f(x) = 4^x$ 10. **Problem:** Graph $f(x) = 9^x$ --- **Step 1:** Understand the base function $f(x) = a^x$ where $a > 0$ and $a \neq 1$. This is an exponential function. **Step 2:** Vertical shifts add or subtract a constant outside the exponential, moving the graph up or down. **Step 3:** Horizontal shifts change the input $x$ to $x - h$ or $x + h$, moving the graph left or right. **Step 4:** Negative exponents reflect the graph about the y-axis. --- ### Detailed explanations and transformations: 1(a) $f(x) = 2^x - \frac{1}{2}$ - This is a vertical shift down by $\frac{1}{2}$. 1(b) $f(x) = 2^x + 3$ - Vertical shift up by 3. 1(c) $f(x) = 2^{x-2}$ - Horizontal shift right by 2. 1(d) $f(x) = 2^{-x}$ - Reflection about the y-axis. 2(a) $f(x) = 3^x - 2$ - Vertical shift down by 2. 2(b) $f(x) = 3^x + 4$ - Vertical shift up by 4. 2(c) $f(x) = 3^{x-1}$ - Horizontal shift right by 1. 2(d) $f(x) = 3^{x+4}$ - Horizontal shift left by 4. 3. $f(x) = 3^x$ - Base exponential function with base 3. 4. $f(x) = (0.5)^x$ - Exponential decay since base $0.5 < 1$. 5. $f(x) = \left(\frac{2}{3}\right)^x$ - Exponential decay since base $\frac{2}{3} < 1$. 6. $f(x) = 4^{-x}$ - Reflection of $4^x$ about y-axis. 7. $f(x) = e^x$ - Natural exponential function. 8. $f(x) = e^{1-x}$ - Rewrite as $e^{1} \cdot e^{-x} = e \cdot e^{-x}$, reflection and vertical stretch. 9. $f(x) = 4^x$ - Exponential growth with base 4. 10. $f(x) = 9^x$ - Exponential growth with base 9. --- **Summary:** - Vertical shifts: add/subtract outside the exponent. - Horizontal shifts: add/subtract inside the exponent. - Negative exponent: reflection about y-axis. - Base $>1$: growth, base $<1$: decay.