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 \[ g(x) = \begin{cases} x + 2, & x \leq -4 \\ \frac{1}{2}x - 6, & x > -4 \end{cases} \] follow these steps: ### 1. **Understand Each Piece of the Function** - **For \( x \leq -4 \):** The function is \( g(x) = x + 2 \). - **Slope:** 1 - **Y-intercept:** (0, 2) - **For \( x > -4 \):** The function is \( g(x) = \frac{1}{2}x - 6 \). - **Slope:** 0.5 - **Y-intercept:** (0, -6) ### 2. **Determine Key Points and Continuity** - **At \( x = -4 \):** - **From the left (\( x \leq -4 \)):** \( g(-4) = (-4) + 2 = -2 \) So, the point \((-4, -2)\) is included in this part of the graph. - **From the right (\( x > -4 \)):** \( \lim_{x \to -4^+} g(x) = \frac{1}{2}(-4) - 6 = -2 - 6 = -8 \) Since \( x > -4 \) does not include \( x = -4 \), we represent this with an open circle at \((-4, -8)\). - **Continuity Check:** The left-hand limit at \( x = -4 \) is \(-2\), and the right-hand limit is \(-8\). Since \(-2 \neq -8\), there is a **jump discontinuity** at \( x = -4 \). ### 3. **Plotting the Graph** Here's how you can sketch the graph step-by-step: #### **a. Plotting \( g(x) = x + 2 \) for \( x \leq -4 \):** - **Choose points:** - \( x = -6 \): \( g(-6) = -6 + 2 = -4 \) → \((-6, -4)\) - \( x = -5 \): \( g(-5) = -5 + 2 = -3 \) → \((-5, -3)\) - \( x = -4 \): \( g(-4) = -4 + 2 = -2 \) → \((-4, -2)\) **(Closed circle)** - **Draw the line** passing through these points extending to the left. #### **b. Plotting \( g(x) = \frac{1}{2}x - 6 \) for \( x > -4 \):** - **Choose points:** - \( x = -2 \): \( g(-2) = \frac{1}{2}(-2) - 6 = -1 - 6 = -7 \) → \((-2, -7)\) - \( x = 0 \): \( g(0) = 0 - 6 = -6 \) → \((0, -6)\) - \( x = 2 \): \( g(2) = \frac{1}{2}(2) - 6 = 1 - 6 = -5 \) → \((2, -5)\) - **At \( x = -4 \):** Open circle at \((-4, -8)\) to indicate that this point is not included in this part of the graph. - **Draw the line** passing through these points extending to the right. ### 4. **Final Graph Overview** - **For \( x \leq -4 \):** A straight line with slope 1 passing through \((-4, -2)\). Include the point \((-4, -2)\) with a closed circle. - **For \( x > -4 \):** A straight line with slope 0.5 starting just after \( x = -4 \) with an open circle at \((-4, -8)\), passing through points like \((-2, -7)\), \((0, -6)\), and so on. - **Discontinuity:** Notice the jump from \(-2\) to \(-8\) at \( x = -4 \), indicating a jump discontinuity. ### 5. **Visual Representation** While I can't provide a visual sketch here, you can follow these instructions to draw the graph: 1. **Draw the Axes:** Draw horizontal (x-axis) and vertical (y-axis) lines intersecting at the origin. 2. **Plot the First Line (\( x \leq -4 \)):** - Plot points \((-6, -4)\), \((-5, -3)\), and \((-4, -2)\). - Connect these points with a straight line extending to the left. - Place a **closed circle** at \((-4, -2)\). 3. **Plot the Second Line (\( x > -4 \)):** - Plot points \((-2, -7)\), \((0, -6)\), and \((2, -5)\). - Connect these points with a straight line extending to the right. - Place an **open circle** at \((-4, -8)\) to indicate that this point is not included. 4. **Label Key Points:** Clearly label the points where the function changes its definition and the points of discontinuity. ### 6. **Summary** - **Lines:** Two distinct straight lines with different slopes. - **Continuity:** Jump discontinuity at \( x = -4 \). - **Behavior:** For \( x \leq -4 \), the function increases with a slope of 1. For \( x > -4 \), it increases more slowly with a slope of 0.5. By following these steps, you should be able to accurately sketch the graph of the piecewise function \( g(x) \).