Question

sketch a graph of $f(x)=\

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

$
note: be sure to include closed or open dots, but only at breaks in the graph

Explanation:

Step1: Plot first piece ($x\leq-2$)

Draw a horizontal line $y=-5$ from $x=-\infty$ to $x=-2$, with a closed dot at $(-2, -5)$ (since $x=-2$ is included in this interval).

Step2: Plot second piece ($-2

First, find endpoints:

• At $x=-2$ (not included): $y=2(-2)-1=-5$, so open dot at $(-2, -5)$.
• At $x=2$ (included): $y=2(2)-1=3$, so closed dot at $(2, 3)$.

Draw a line connecting these two points, following $y=2x-1$.

Step3: Plot third piece ($x>2$)

Draw a horizontal line $y=1$ from $x=2$ to $x=+\infty$, with an open dot at $(2, 1)$ (since $x=2$ is not included in this interval).

Answer:

The graph consists of:

1. A horizontal line at $y=-5$ for all $x\leq-2$, ending with a closed dot at $(-2, -5)$.
2. A line segment of $y=2x-1$ from $(-2, -5)$ (open dot) to $(2, 3)$ (closed dot).
3. A horizontal line at $y=1$ for all $x>2$, starting with an open dot at $(2, 1)$.