1. **State the problem:** We need to find the equation of an exponential function in the form $$f(x) = a b^x$$ given points on the graph and its shape. 2. **Analyze the graph:** The graph is a decreasing exponential curve with horizontal asymptote $$y=0$$, passing near points $$(0, -\frac{1}{2}), (1, -1), (2, -2)$$. 3. **Recall the form:** The general form is $$f(x) = a b^x$$ where $$a$$ is the initial value $$f(0)$$ and $$b$$ is the base. 4. **Find $$a$$:** Since $$f(0) = a b^0 = a = -\frac{1}{2}$$, we have $$a = -\frac{1}{2}$$. 5. **Use another point to find $$b$$:** Using point $$(1, -1)$$: $$-1 = -\frac{1}{2} b^1$$ Divide both sides by $$-\frac{1}{2}$$: $$\frac{-1}{-\frac{1}{2}} = b$$ $$\cancel{-1} \times \frac{1}{\cancel{-\frac{1}{2}}} = b$$ Simplify: $$2 = b$$ 6. **Check with point $$(2, -2)$$:** $$f(2) = -\frac{1}{2} \times 2^2 = -\frac{1}{2} \times 4 = -2$$ which matches the point. 7. **Final equation:** $$f(x) = -\frac{1}{2} \times 2^x$$ This matches the decreasing exponential shape with horizontal asymptote at $$y=0$$ and the given points.