1. **Problem Statement:** We are given discrete points with $x$ values from $-5$ to $5$ and corresponding $y$ values as $-2,-3,-4,-5,-6,-7,-8,-7,-6,-5,-4$. We want to analyze the function represented by these points and find a formula or pattern. 2. **Observing the Data:** The points are: $$ \begin{aligned} &(-5,-2), (-4,-3), (-3,-4), (-2,-5), (-1,-6), (0,-7), (1,-8), (2,-7), (3,-6), (4,-5), (5,-4) \end{aligned} $$ 3. **Looking for a Pattern:** Notice that from $x=-5$ to $x=1$, $y$ decreases by 1 as $x$ increases by 1. Then from $x=1$ to $x=5$, $y$ increases by 1 as $x$ increases by 1. This suggests a V-shaped pattern with a minimum at $x=1$. 4. **Hypothesis:** The function looks like an absolute value function shifted horizontally and vertically. The general form is: $$ y = a|x - h| + k $$ where $(h,k)$ is the vertex. 5. **Finding the Vertex:** The minimum $y$ value is $-8$ at $x=1$, so $h=1$ and $k=-8$. 6. **Finding $a$:** Use a point to find $a$. For example, at $x=0$, $y=-7$: $$ -7 = a|0 - 1| - 8 \Rightarrow -7 = a(1) - 8 \Rightarrow a = 1 $$ 7. **Final Formula:** $$ y = |x - 1| - 8 $$ 8. **Verification:** Check at $x=-5$: $$ y = |-5 - 1| - 8 = | -6 | - 8 = 6 - 8 = -2 $$ which matches the given data. **Answer:** The function that fits the data is $$ y = |x - 1| - 8 $$