1. **State the problem:** We are given the exponential function $$k(x) = 6.7 (0.85)^{-x}$$ and asked to analyze its graph, including the y-intercept and horizontal asymptote. 2. **Recall the general form and properties:** An exponential function modeling growth or decay can be written as $$f(x) = a b^x$$ where: - $$a$$ is the initial value (y-intercept). - $$b$$ is the base; if $$0 < b < 1$$, the function models exponential decay, if $$b > 1$$, it models growth. 3. **Rewrite the function for clarity:** Since the exponent is negative, rewrite: $$k(x) = 6.7 (0.85)^{-x} = 6.7 \left(\frac{1}{0.85}\right)^x = 6.7 (1.1765)^x$$ This shows the function is actually exponential growth because the base $$1.1765 > 1$$. 4. **Find the y-intercept:** The y-intercept occurs at $$x=0$$: $$k(0) = 6.7 (0.85)^0 = 6.7 \times 1 = 6.7$$ So the y-intercept is $$(0, 6.7)$$. 5. **Determine the horizontal asymptote:** The problem states the asymptote is $$y=5$$. For exponential functions of the form $$k(x) = a b^x + c$$, the horizontal asymptote is $$y = c$$. 6. **Adjust the function to include the asymptote:** Since the given function does not explicitly show $$+5$$, the horizontal asymptote at $$y=5$$ suggests the function is: $$k(x) = 5 + 6.7 (0.85)^{-x}$$ 7. **Behavior of the graph:** - At $$x=0$$, $$k(0) = 5 + 6.7 = 11.7$$ - As $$x \to \infty$$, $$k(x) \to 5$$ from above. 8. **Summary:** The function models exponential growth shifted up by 5 units, with y-intercept at $$(0, 11.7)$$ and horizontal asymptote $$y=5$$. **Final answers:** - y-intercept: $$(0, 11.7)$$ - Horizontal asymptote: $$y=5$$