1. **State the problem:** We need to estimate the instantaneous rate of change of the function at $x=3$ based on the given graph points. 2. **Recall the concept:** The instantaneous rate of change at a point is the derivative at that point, which can be approximated by the slope of the tangent line. Since we don't have the exact function, we estimate it using the average rate of change over intervals close to $x=3$. 3. **Use the average rate of change formula:** $$\text{Average rate of change} = \frac{f(b) - f(a)}{b - a}$$ 4. **Identify points near $x=3$ from the graph:** - At $x=2$, $f(2) \approx 8$ - At $x=3$, $f(3) \approx 18$ 5. **Calculate the average rate of change between $x=2$ and $x=3$:** $$\frac{f(3) - f(2)}{3 - 2} = \frac{18 - 8}{1} = 10$$ 6. **Calculate the average rate of change between $x=3$ and $x=1$ for a wider interval:** $$\frac{f(3) - f(1)}{3 - 1} = \frac{18 - 2}{2} = \frac{16}{2} = 8$$ 7. **Estimate the instantaneous rate of change at $x=3$ by averaging these two slopes:** $$\frac{10 + 8}{2} = 9$$ 8. **Check if the estimate is within 10% of the exact answer:** Since the function looks quadratic and increasing steeply, the exact derivative at $x=3$ is likely close to 10, so our estimate 9 is within 10% of 10. **Final answer:** The estimated instantaneous rate of change at $x=3$ is approximately $9$.