Exponential Decay E54A96

1. **State the problem:** Graph the function $$f(x) = \frac{3}{4} \left(\frac{3}{7}\right)^x$$ and answer questions about its behavior. 2. **Identify the type of function:** This is an exponential function of the form $$f(x) = a b^x$$ where $$a = \frac{3}{4}$$ and $$b = \frac{3}{7}$$. 3. **Determine if the function is increasing or decreasing:** Since the base $$b = \frac{3}{7}$$ is less than 1, the function is **decreasing**. 4. **Domain and range:** - The domain of any exponential function is all real numbers: $$(-\infty, \infty)$$. - The range depends on $$a$$ and $$b$$. Since $$a = \frac{3}{4} > 0$$ and $$b^x > 0$$ for all $$x$$, the function is always positive. Thus, the range is $$\left(0, \infty\right)$$. 5. **Find the y-intercept:** The y-intercept occurs at $$x=0$$. Calculate: $$f(0) = \frac{3}{4} \left(\frac{3}{7}\right)^0 = \frac{3}{4} \times 1 = \frac{3}{4}$$ So the y-intercept is $$\left(0, \frac{3}{4}\right)$$. 6. **Find the horizontal asymptote:** As $$x \to \infty$$, since $$0 < b < 1$$, $$b^x \to 0$$. Therefore, $$\lim_{x \to \infty} f(x) = \frac{3}{4} \times 0 = 0$$ The horizontal asymptote is $$y = 0$$. **Final answers:** - The function is **decreasing**. - Domain: $$(-\infty, \infty)$$. - Range: $$\left(0, \infty\right)$$. - Y-intercept: $$\left(0, \frac{3}{4}\right)$$. - Horizontal asymptote: $$y = 0$$.