1. **State the problem:** We are given a piecewise function: $$f(x) = \begin{cases} x + 1, & x \leq -2 \\ -6, & -2 < x \leq 2 \\ 5, & x > 2 \end{cases}$$ We want to understand and analyze this function, including its graph segments. 2. **Explain the formula and rules:** - For $x \leq -2$, the function is linear: $f(x) = x + 1$. - For $-2 < x \leq 2$, the function is constant: $f(x) = -6$. - For $x > 2$, the function is constant: $f(x) = 5$. 3. **Analyze each piece:** - For $x \leq -2$, the line $y = x + 1$ starts from $x = -6$ (given in the graph description) and goes up to $x = -2$. At $x = -2$, $f(-2) = -2 + 1 = -1$. - For $-2 < x \leq 2$, $f(x) = -6$ is a horizontal line segment from just right of $x = -2$ to $x = 2$. - For $x > 2$, $f(x) = 5$ is a constant horizontal line starting just after $x = 2$. 4. **Check continuity at boundaries:** - At $x = -2$, from the left $f(-2) = -1$, from the right $f(-2) = -6$ (since $x > -2$), so there is a jump discontinuity. - At $x = 2$, from the left $f(2) = -6$, from the right $f(2) = 5$, another jump discontinuity. 5. **Summary:** The graph consists of three segments: - A line segment from $x = -6$ to $x = -2$ with $f(x) = x + 1$. - A horizontal segment at $y = -6$ for $-2 < x \leq 2$. - A horizontal segment at $y = 5$ for $x > 2$. This matches the description of the first graph (bottom-center). **Final answer:** The piecewise function is correctly described and graphed as above.