1. **Determine whether the equation represents an exponential function.** The general form of an exponential function is: $$y = a \cdot b^x$$ where $a$ is a constant, $b$ is the base (a positive real number not equal to 1), and $x$ is the exponent. - For $y = 3(4)^x$: - The base is 4, which is positive. - Therefore, this is an exponential function. - For $y = x(-2)^x$: - The base is $-2$, which is negative. - Exponential functions require a positive base. - Therefore, this is not an exponential function. 2. **Graph the function $y = 3(2)^x$ and describe the domain and range.** Calculate $y$ for each given $x$: - For $x = -2$: $$y = 3 \cdot 2^{-2} = 3 \cdot \frac{1}{2^2} = 3 \cdot \frac{1}{4} = \frac{3}{4} = 0.75$$ - For $x = -1$: $$y = 3 \cdot 2^{-1} = 3 \cdot \frac{1}{2} = 1.5$$ - For $x = 0$: $$y = 3 \cdot 2^{0} = 3 \cdot 1 = 3$$ - For $x = 1$: $$y = 3 \cdot 2^{1} = 3 \cdot 2 = 6$$ - For $x = 2$: $$y = 3 \cdot 2^{2} = 3 \cdot 4 = 12$$ **Domain:** All real numbers, since $x$ can be any real number. **Range:** All positive real numbers $(0, \infty)$ because $3(2)^x$ is always positive. 3. **Exponential decay problem:** Given: - Initial population $a = 67000$ - Decrease rate $r = 0.06$ - Time $t$ in years **a. Write the function:** Since the population decreases by 6% each year, the function is exponential decay: $$P(t) = 67000 \cdot (1 - 0.06)^t = 67000 \cdot (0.94)^t$$ **b. Find the population after 8 years:** Calculate: $$P(8) = 67000 \cdot (0.94)^8$$ First calculate $(0.94)^8$: $$0.94^8 \approx 0.5948$$ Then: $$P(8) = 67000 \cdot 0.5948 = 39801.6$$ Rounded to the nearest hundredth: $$39801.60$$ **Final answers:** 1. Yes, $y = 3(4)^x$ is an exponential function. 2. No, $y = x(-2)^x$ is not an exponential function. 3. Table of values for $y = 3(2)^x$: | x | y | |---|---| | -2 | 0.75 | | -1 | 1.5 | | 0 | 3 | | 1 | 6 | | 2 | 12 | Domain: $(-\infty, \infty)$ Range: $(0, \infty)$ 4a. Population function: $$P(t) = 67000 \cdot (0.94)^t$$ 4b. Population after 8 years: $$P(8) \approx 39801.60$$