1. **Stating the problem:** We are given a function $y=f(x)$ with points and secant lines labeled A, B, C, D, E on its graph. We want to understand the differences in function values and slopes between points $x=0$, $x=2$, and $x=4$. 2. **Given values:** - $f(2)-f(0) = 2-0 = 2$ - $f(4)-f(2) = 4-2 = 2$ - $f(4)-f(0) = 4-0 = 4$ 3. **Formula for average rate of change (slope of secant line):** $$\text{slope} = \frac{f(b)-f(a)}{b-a}$$ This formula calculates the average rate of change of $f$ between $x=a$ and $x=b$. 4. **Calculate slopes for each interval:** - Between $0$ and $2$: $$\frac{f(2)-f(0)}{2-0} = \frac{2}{2} = 1$$ - Between $2$ and $4$: $$\frac{f(4)-f(2)}{4-2} = \frac{2}{2} = 1$$ - Between $0$ and $4$: $$\frac{f(4)-f(0)}{4-0} = \frac{4}{4} = 1$$ 5. **Interpretation:** The average rate of change (slope of secant lines) between these points is consistently $1$, indicating a linear average increase over these intervals. 6. **Additional notes:** The graph is concave down with a peak near $x=0$, so the instantaneous rate of change (derivative) varies, but the average slopes between these points are equal. **Final answer:** The average rate of change of $f$ between $x=0$ and $x=2$, between $x=2$ and $x=4$, and between $x=0$ and $x=4$ is $1$ in each case.