Question

sketch a graph of $f(x)=\

$$\begin{cases}5&\\text{if }x\\leq -2\\\\-2x + 1&\\text{if }-2 < x\\leq 1\\\\1&\\text{if }x > 1\\end{cases}$$

$
note: be sure to include closed or open dots, but only at breaks in the graph.
question help: video 1 video 2 post to forum

Explanation:

Step1: Plot first piece: $f(x)=5$

This is a horizontal line at $y=5$ for all $x \leq -2$. Place a closed dot at $(-2, 5)$ (since $x=-2$ is included) and extend the line leftwards to $x=-5$ (the edge of the grid).

Step2: Plot second piece: $f(x)=-2x+1$

First, find endpoints:

• At $x=-2$ (not included here, since $x>-2$ for this piece): $f(-2)=-2(-2)+1=5$, so place an open dot at $(-2, 5)$.
• At $x=1$ (included): $f(1)=-2(1)+1=-1$, so place a closed dot at $(1, -1)$.

Draw a straight line connecting these two points.

Step3: Plot third piece: $f(x)=1$

This is a horizontal line at $y=1$ for all $x > 1$. Place an open dot at $(1, 1)$ (since $x=1$ is not included) and extend the line rightwards to $x=5$ (the edge of the grid).

Answer:

The graph consists of:

1. A horizontal line at $y=5$ from $x=-5$ to $x=-2$ (closed dot at $(-2,5)$).
2. A line segment from open dot $(-2,5)$ to closed dot $(1,-1)$ (following $y=-2x+1$).
3. A horizontal line at $y=1$ from open dot $(1,1)$ to $x=5$.