Let's find out how many values of $p$ make the difference quotient on $[2, p]$ equal to $0.5$! 1. Imagine you have points on the graph at $x=2$ and $x=p$. We want the average rate of change (difference quotient) between these points to be $0.5$. 2. The difference quotient formula is: $$\frac{y(p) - y(2)}{p - 2} = 0.5$$ 3. From the graph, $y(2) = 15$. 4. So, we have: $$\frac{y(p) - 15}{p - 2} = 0.5$$ 5. Multiply both sides by $(p - 2)$: $$y(p) - 15 = 0.5(p - 2)$$ 6. Add $15$ to both sides: $$y(p) = 0.5(p - 2) + 15$$ 7. Now, let's check the points on the graph to see when $y(p)$ equals this value: - At $p=4$, $y(4) = 10$ but $0.5(4-2)+15 = 0.5(2)+15=16$ (not equal) - At $p=6$, $y(6) = 18$ and $0.5(6-2)+15=0.5(4)+15=17$ (close but not equal) - At $p=8$, $y(8)=20$ and $0.5(8-2)+15=0.5(6)+15=18$ (not equal) - At $p=10$, $y(10)=15$ and $0.5(10-2)+15=0.5(8)+15=19$ (not equal) - At $p=12$, $y(12)=0$ and $0.5(12-2)+15=0.5(10)+15=20$ (not equal) 8. The difference quotient equals $0.5$ only if $y(p)$ matches $0.5(p-2)+15$. The graph crosses this line only once around $p=6.5$ (between $6$ and $8$). **Answer:** There is exactly 1 value of $p$ where the difference quotient equals $0.5$. Great job exploring the graph! 🎉