1. **State the problem:** We are given a graph with two lines intersecting at the origin. One line passes through points $(-6,-4)$ and $(6,5)$, and the other is the x-axis line $y=0$. We need to find which given function or table has the same rate of change (slope) as the slanting line. 2. **Find the slope of the slanting line:** The slope formula is $$m=\frac{y_2 - y_1}{x_2 - x_1}$$ where $(x_1,y_1)=(-6,-4)$ and $(x_2,y_2)=(6,5)$. 3. Calculate the slope: $$m=\frac{5 - (-4)}{6 - (-6)}=\frac{5 + 4}{6 + 6}=\frac{9}{12}=\frac{3}{4}$$ 4. **Check each option's rate of change:** - A: $y=\frac{1}{2}x - 3$ has slope $\frac{1}{2}$. - B: $y=x+5$ has slope $1$. - C: Table values: - From $x=4$ to $x=8$, $y$ changes from $4$ to $5$, slope $=\frac{5-4}{8-4}=\frac{1}{4}$. - From $x=8$ to $x=12$, slope $=\frac{6-5}{12-8}=\frac{1}{4}$. - Consistent slope $\frac{1}{4}$. - D: Table values: - From $x=4$ to $x=8$, $y$ changes from $-1$ to $-6$, slope $=\frac{-6 - (-1)}{8-4}=\frac{-5}{4}=-\frac{5}{4}$. - From $x=8$ to $x=12$, slope $=\frac{-11 - (-6)}{12-8}=\frac{-5}{4}=-\frac{5}{4}$. - Consistent slope $-\frac{5}{4}$. 5. **Compare slopes:** The slanting line has slope $\frac{3}{4}$. None of the options have exactly $\frac{3}{4}$, but option A has slope $\frac{1}{2}$, B has slope $1$, C has $\frac{1}{4}$, and D has $-\frac{5}{4}$. 6. **Conclusion:** None match exactly, but the closest positive slope to $\frac{3}{4}$ is option A with $\frac{1}{2}$. Since the question asks for the same rate of change, none are exact, but option A is the closest linear function. **Final answer:** Option A: $y=\frac{1}{2}x - 3$ has the closest rate of change to the given line.