1. **State the problem:** We are given a table of 8 ordered pairs $(x, y)$ and asked to find the average rate of change (AROC) between consecutive points and determine when it becomes constant or zero. 2. **Recall the formula for average rate of change:** $$\text{AROC} = \frac{y_2 - y_1}{x_2 - x_1}$$ This measures the change in $y$ per unit change in $x$ between two points. 3. **Calculate AROC between each consecutive pair:** - Between $x=0$ and $x=0.4$: $$\frac{10.6 - 5}{0.4 - 0} = \frac{5.6}{0.4} = 14$$ - Between $x=0.4$ and $x=0.9$: $$\frac{15.4 - 10.6}{0.9 - 0.4} = \frac{4.8}{0.5} = 9.6$$ - Between $x=0.9$ and $x=1.2$: $$\frac{17.1 - 15.4}{1.2 - 0.9} = \frac{1.7}{0.3} \approx 5.67$$ - Between $x=1.2$ and $x=1.7$: $$\frac{18.0 - 17.1}{1.7 - 1.2} = \frac{0.9}{0.5} = 1.8$$ - Between $x=1.7$ and $x=2.2$: $$\frac{16.5 - 18.0}{2.2 - 1.7} = \frac{-1.5}{0.5} = -3$$ - Between $x=2.2$ and $x=2.9$: $$\frac{10.2 - 16.5}{2.9 - 2.2} = \frac{-6.3}{0.7} = -9$$ - Between $x=2.9$ and $x=3.4$: $$\frac{2.8 - 10.2}{3.4 - 2.9} = \frac{-7.4}{0.5} = -14.8$$ 4. **Analyze the AROC values:** The AROC starts positive and decreases steadily from 14 to -14.8, passing through zero between $x=1.7$ and $x=2.2$ where the rate changes from positive to negative. 5. **When does AROC become constant or zero?** - The AROC is never constant; it changes at every interval. - The AROC becomes zero approximately between $x=1.7$ and $x=2.2$ because it changes sign from positive to negative, indicating a maximum point. **Final answer:** The average rate of change is not constant; it decreases from 14 to -14.8 and crosses zero between $x=1.7$ and $x=2.2$.