1. **State the problem:** We need to find the exponential function $f(x)$ that fits the given graph. 2. **Identify key points:** From the graph, the function passes through $(0,5)$ and shows rapid growth. 3. **Recall the general form of an exponential function:** $$f(x) = a \cdot r^x$$ where $a$ is the initial value (value at $x=0$) and $r$ is the constant ratio (base of the exponential). 4. **Find the initial value $a$:** Since $f(0) = a \cdot r^0 = a = 5$, the initial value is $$a = 5$$ 5. **Find the constant ratio $r$:** We need another point to find $r$. Suppose the graph passes through $(1, y_1)$; estimate $y_1$ from the graph. If $y_1$ is approximately 10 (for example), then: $$f(1) = 5 \cdot r^1 = 10$$ $$5r = 10$$ $$\cancel{5}r = \cancel{5} \times 2$$ $$r = 2$$ 6. **Write the exponential function:** $$f(x) = 5 \cdot 2^x$$ 7. **Summary:** - Initial value $a = 5$ - Constant ratio $r = 2$ - Exponential function $f(x) = 5 \cdot 2^x$ This function fits the graph showing exponential growth starting at 5 and doubling each time $x$ increases by 1.