Sketch the graph of the function. \[ g(x)=\left\{\begin{array}{ll}x+2, & x \leq-4 \\ \frac{1}{2} x-6, & x>-4\end{array}\right. \]

To sketch the graph of the piecewise function: \[ g(x) = \begin{cases} x + 2, & \text{if } x \leq -4 \\ \frac{1}{2}x - 6, & \text{if } x > -4 \end{cases} \] follow these steps: ### 1. **Identify the Pieces of the Function** - **First Piece:** \( g(x) = x + 2 \) for \( x \leq -4 \) - **Slope:** 1 - **Y-intercept:** 2 - **Second Piece:** \( g(x) = \frac{1}{2}x - 6 \) for \( x > -4 \) - **Slope:** \( \frac{1}{2} \) - **Y-intercept:** -6 ### 2. **Determine Key Points and Continuity at \( x = -4 \)** - **At \( x = -4 \):** - Use the first piece since \( x \leq -4 \). - \( g(-4) = (-4) + 2 = -2 \) - This point is included in the first piece: **\((-4, -2)\)** (closed circle). - **Limit from the Right as \( x \) Approaches -4:** - Use the second piece: \( \lim_{x \to -4^+} g(x) = \frac{1}{2}(-4) - 6 = -2 - 6 = -8 \) - This indicates a **jump discontinuity** at \( x = -4 \). - Represented by an open circle at \((-4, -8)\). ### 3. **Plot Each Piece** - **First Piece (\( g(x) = x + 2 \) for \( x \leq -4 \)):** - Plot the point \((-4, -2)\) with a closed circle. - Since the slope is 1, for every 1 unit increase in \( x \), \( y \) increases by 1. - Draw a straight line extending to the left from \((-4, -2)\). - **Second Piece (\( g(x) = \frac{1}{2}x - 6 \) for \( x > -4 \)):** - Start just to the right of \( x = -4 \) with an open circle at \((-4, -8)\). - Since the slope is \( \frac{1}{2} \), for every 2 units increase in \( x \), \( y \) increases by 1. - Draw a straight line extending to the right from the open circle. ### 4. **Visual Representation** Here's a rough sketch of the graph: ``` y | | Second Piece: y = (1/2)x - 6 | / | / | / | / |__________●________/__________ x (-4, -2) Open Circle at (-4, -8) | | | First Piece: y = x + 2 | | | | ``` ### 5. **Key Characteristics** - **Slope:** The first piece has a steeper slope (1) compared to the second piece (\( \frac{1}{2} \)). - **Y-Intercepts:** - First piece intersects the y-axis at (0, 2). - Second piece intersects the y-axis at (0, -6). - **Discontinuity:** There is a jump at \( x = -4 \): - The function value jumps from \(-2\) (from the first piece) to approaching \(-8\) (from the second piece) as \( x \) crosses \(-4\) from left to right. ### 6. **Summary** - For \( x \leq -4 \), the graph is a straight line with slope 1 passing through \((-4, -2)\). - For \( x > -4 \), the graph is a different straight line with slope \( \frac{1}{2} \) starting just above \((-4, -8)\). - There's a visible jump at \( x = -4 \) indicating the discontinuity between the two pieces. By following these steps, you can accurately sketch the graph of the given piecewise function.