Graph a Piecewise-Defined Function Part A Use the tools or pencil and paper to graph \( f(x)=\left\{\begin{array}{rr}x+3 & \text { if } x \leq 1 \\ -2 x & \text { if } x>1\end{array}\right. \)

To graph the piecewise-defined function: \[ f(x) = \begin{cases} x + 3 & \text{if } x \leq 1 \\ -2x & \text{if } x > 1 \end{cases} \] follow these steps: ### 1. Graph the First Piece: \( y = x + 3 \) for \( x \leq 1 \) - **Equation:** \( y = x + 3 \) - **Domain:** \( x \leq 1 \) - **Type of Function:** Linear **Key Points:** - **When \( x = 0 \):** \( y = 0 + 3 = 3 \) → Point \( (0, 3) \) - **When \( x = 1 \):** \( y = 1 + 3 = 4 \) → Point \( (1, 4) \) **Plotting:** - Draw a straight line with a slope of 1 (rising to the right) starting from \( x = -\infty \) up to \( x = 1 \). - **Solid Dot at \( (1, 4) \):** Since the condition includes \( x = 1 \) (\( x \leq 1 \)), use a solid dot to indicate that the point is part of this piece. ### 2. Graph the Second Piece: \( y = -2x \) for \( x > 1 \) - **Equation:** \( y = -2x \) - **Domain:** \( x > 1 \) - **Type of Function:** Linear **Key Points:** - **When \( x = 1 \):** \( y = -2(1) = -2 \) → Point \( (1, -2) \) (Note: This point is **not** included in this segment) - **When \( x = 2 \):** \( y = -2(2) = -4 \) → Point \( (2, -4) \) **Plotting:** - Draw a straight line with a slope of -2 (falling to the right) starting just after \( x = 1 \) and extending to \( x = +\infty \). - **Open Circle at \( (1, -2) \):** Since the condition is \( x > 1 \), use an open circle to indicate that the point \( (1, -2) \) is **not** part of this piece. ### 3. Combine Both Pieces - **Intersection at \( x = 1 \):** - **First Piece:** \( (1, 4) \) with a solid dot. - **Second Piece:** Starts just after \( x = 1 \) with an open circle at \( (1, -2) \). This results in a graph composed of two distinct linear segments: - A line with a positive slope (\( y = x + 3 \)) extending to the left of \( x = 1 \), ending at \( (1, 4) \). - A line with a negative slope (\( y = -2x \)) starting just after \( x = 1 \), headed downward as \( x \) increases. ### Visual Representation Below is a rough sketch of how the graph looks: ``` y | 5 | 4 |● / 3 | \ / 2 | \ / 1 | \ / 0 |----•---------•----> x -2 -1 0 1 2 ``` - The solid dot **●** at \( (1, 4) \) represents the endpoint of the first piece. - The open circle **•** at \( (1, -2) \) marks the starting point of the second piece, which continues downward. ### Summary - **For \( x \leq 1 \):** The graph is the line \( y = x + 3 \) with a solid dot at \( (1, 4) \). - **For \( x > 1 \):** The graph is the line \( y = -2x \) starting just after \( x = 1 \) with an open circle at \( (1, -2) \). This piecewise function has a clear break at \( x = 1 \), where it switches from increasing to decreasing.