1. **Problem 3: Analyze and graph** $f(x) = 3^{x-5} + 1$ 2. **Transformations:** - Horizontal shift right by 5 units (due to $x-5$ inside the exponent). - Vertical shift up by 1 unit (due to $+1$ outside the exponential). 3. **Domain:** The domain of an exponential function is all real numbers, so $\text{Domain} = (-\infty, \infty)$. 4. **Range:** The base function $3^x$ has range $(0, \infty)$. After shifting up by 1, range becomes $(1, \infty)$. 5. **End behavior:** - As $x \to \infty$, $3^{x-5} \to \infty$, so $f(x) \to \infty$. - As $x \to -\infty$, $3^{x-5} \to 0$, so $f(x) \to 1$. 6. **y-intercept:** Set $x=0$: $$f(0) = 3^{0-5} + 1 = 3^{-5} + 1 = \frac{1}{3^5} + 1 = \frac{1}{243} + 1 = \frac{244}{243} \approx 1.0041$$ 7. **Asymptote:** Horizontal asymptote at $y=1$. --- 1. **Problem 4: Analyze and graph** $f(x) = -\left(\frac{1}{3}\right)^{x+4}$ 2. **Transformations:** - Reflection over x-axis (due to negative sign). - Horizontal shift left by 4 units (due to $x+4$ inside the exponent). 3. **Domain:** All real numbers, $(-\infty, \infty)$. 4. **Range:** Base function $\left(\frac{1}{3}\right)^x$ has range $(0, \infty)$. Reflection flips it to $(-\infty, 0)$. 5. **End behavior:** - As $x \to \infty$, $\left(\frac{1}{3}\right)^{x+4} \to 0$, so $f(x) \to 0$ from below. - As $x \to -\infty$, $\left(\frac{1}{3}\right)^{x+4} \to \infty$, so $f(x) \to -\infty$. 6. **y-intercept:** Set $x=0$: $$f(0) = -\left(\frac{1}{3}\right)^{0+4} = -\left(\frac{1}{3}\right)^4 = -\frac{1}{81} \approx -0.0123$$ 7. **Asymptote:** Horizontal asymptote at $y=0$. --- **Summary:** - Problem 3: $f(x) = 3^{x-5} + 1$, domain $(-\infty, \infty)$, range $(1, \infty)$, asymptote $y=1$. - Problem 4: $f(x) = -\left(\frac{1}{3}\right)^{x+4}$, domain $(-\infty, \infty)$, range $(-\infty, 0)$, asymptote $y=0$.