1. **State the problem:** We are given two points on a graph, (2,2) and (6,1), and we want to find the exponential function of the form $$y = ab^x$$ that passes through these points. 2. **Formula and explanation:** The general form of an exponential function is $$y = ab^x$$ where: - $$a$$ is the initial value (when $$x=0$$), - $$b$$ is the base or growth/decay factor. 3. **Use the points to create equations:** From point (2,2): $$2 = ab^2$$ From point (6,1): $$1 = ab^6$$ 4. **Divide the two equations to eliminate $$a$$:** $$\frac{2}{1} = \frac{ab^2}{ab^6} = b^{2-6} = b^{-4}$$ So, $$2 = b^{-4}$$ 5. **Solve for $$b$$:** $$b^{-4} = 2 \implies b^4 = \frac{1}{2}$$ $$b = \sqrt[4]{\frac{1}{2}} = \frac{1}{\sqrt[4]{2}}$$ 6. **Find $$a$$ using one of the points, say (2,2):** $$2 = a \left(\frac{1}{\sqrt[4]{2}}\right)^2 = a \frac{1}{\sqrt{2}}$$ So, $$a = 2 \sqrt{2}$$ 7. **Write the final function:** $$y = 2 \sqrt{2} \left(\frac{1}{\sqrt[4]{2}}\right)^x$$ This function represents an exponential decay because $$b < 1$$, which matches the decreasing trend observed in the graph. **Final answer:** $$y = 2 \sqrt{2} \left(\frac{1}{\sqrt[4]{2}}\right)^x$$