Question

sketch a graph of $f(x)=\

$$\begin{cases}-1.5x - 5&\\text{if }x\\leq - 2\\\\1&\\text{if }-2 < x\\leq1\\\\x - 4&\\text{if }x > 1\\end{cases}$$

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

Explanation:

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

Find endpoint at $x=-2$:
$f(-2) = -1.5(-2) -5 = 3 -5 = -2$ (closed dot at $(-2, -2)$).
Find a second point: let $x=-4$, $f(-4) = -1.5(-4)-5=6-5=1$ (point $(-4,1)$).
Draw line through these, leftward from $(-2,-2)$.

Step2: Analyze second piece ($-2 < x \leq 1$)

Horizontal line $y=1$.
Open dot at $x=-2$: $(-2,1)$, closed dot at $x=1$: $(1,1)$.
Draw horizontal segment between these points.

Step3: Analyze third piece ($x > 1$)

Find endpoint at $x=1$:
$f(1)=1-4=-3$ (open dot at $(1,-3)$).
Find a second point: let $x=4$, $f(4)=4-4=0$ (point $(4,0)$).
Draw line through these, rightward from $(1,-3)$.

Answer:

The graph consists of:

1. A line segment/ray for $y=-1.5x-5$ with a closed dot at $(-2, -2)$, passing through $(-4, 1)$, extending left.
2. A horizontal segment for $y=1$ with an open dot at $(-2, 1)$ and a closed dot at $(1, 1)$.
3. A line segment/ray for $y=x-4$ with an open dot at $(1, -3)$, passing through $(4, 0)$, extending right.