1. **State the problem:** We are given several exponential functions and need to find for each: i. The equation of the horizontal asymptote. ii. The y-intercept. Also, state the domain and range for given functions. 2. **Recall the general form and rules:** An exponential function is generally of the form $$y = a b^x + q$$ where: - $$a$$ is the vertical stretch/compression and reflection factor. - $$b$$ is the base of the exponential (positive, not equal to 1). - $$q$$ is the vertical shift. **Horizontal asymptote:** The line $$y = q$$ is the horizontal asymptote. **Y-intercept:** Set $$x=0$$, then $$y = a b^0 + q = a \cdot 1 + q = a + q$$. **Domain:** For all exponential functions, domain is $$(-\infty, \infty)$$. **Range:** Depends on $$a$$ and $$q$$: - If $$a > 0$$, range is $$(q, \infty)$$. - If $$a < 0$$, range is $$(-\infty, q)$$. --- 3. **Find asymptotes and y-intercepts for given functions:** **a) $$y = 2^x + 1$$** - Asymptote: $$y = 1$$ - Y-intercept: $$y = 2^0 + 1 = 1 + 1 = 2$$ **b) $$y = 3(2)^x - 2$$** - Asymptote: $$y = -2$$ - Y-intercept: $$y = 3 \cdot 2^0 - 2 = 3 - 2 = 1$$ **c) $$y = -2(2)^x + 3$$** - Asymptote: $$y = 3$$ - Y-intercept: $$y = -2 \cdot 1 + 3 = 1$$ **e) $$y = 3^x - 1$$** - Asymptote: $$y = -1$$ - Y-intercept: $$y = 1 - 1 = 0$$ **f) $$y = -3^x - 1$$** - Asymptote: $$y = -1$$ - Y-intercept: $$y = -1 - 1 = -2$$ **g) $$f(x) = 2(3)^x + 1$$** - Asymptote: $$y = 1$$ - Y-intercept: $$y = 2 \cdot 1 + 1 = 3$$ **i) $$y = -2(4)^x + 3$$** - Asymptote: $$y = 3$$ - Y-intercept: $$y = -2 \cdot 1 + 3 = 1$$ **j) $$f(x) = 5^x$$** - Asymptote: $$y = 0$$ - Y-intercept: $$y = 1$$ **k) $$y = 0.5(2)^x + 1$$** - Asymptote: $$y = 1$$ - Y-intercept: $$y = 0.5 \cdot 1 + 1 = 1.5$$ --- 4. **Domain and range for given functions:** **a) $$y = 4^x$$** - Domain: $$(-\infty, \infty)$$ - Range: $$(0, \infty)$$ (since $$a=1>0$$ and $$q=0$$) **b) $$y = 2^x - 3$$** - Domain: $$(-\infty, \infty)$$ - Range: $$(-3, \infty)$$ **c) $$y = 3(\frac{1}{4})^x + 4$$** - Domain: $$(-\infty, \infty)$$ - Range: $$(4, \infty)$$ (since $$a=3>0$$ and $$q=4$$) --- **Summary:** - Horizontal asymptote is always $$y = q$$. - Y-intercept is $$a + q$$. - Domain is all real numbers. - Range depends on sign of $$a$$ and vertical shift $$q$$. This completes the solution for the first question about asymptotes, y-intercepts, domain, and range.