Question

graph the following function on the axes provided.
$f(x) = \

$$\begin{cases} -x + 1 & \\text{for} & x \\leq -3 \\\\ -3x + 2 & \\text{for} & x > 0 \\end{cases}$$

$
line closed circle open circle
click and drag to make a line. click the line to delete it.
click on an endpoint of a line to change it.

Explanation:

Step1: Graph \( y = -x + 1 \) for \( x \leq -3 \)

First, find the endpoint at \( x = -3 \). Substitute \( x = -3 \) into \( y = -x + 1 \):
\( y = -(-3) + 1 = 3 + 1 = 4 \). So the point is \( (-3, 4) \), and since \( x \leq -3 \), this is a closed circle.
Next, find another point for \( x < -3 \), e.g., \( x = -4 \):
\( y = -(-4) + 1 = 4 + 1 = 5 \), so the point is \( (-4, 5) \). Draw a line through \( (-3, 4) \) (closed circle) and \( (-4, 5) \), extending left.

Step2: Graph \( y = -3x + 2 \) for \( x > 0 \)

Find the endpoint at \( x = 0 \) (but \( x > 0 \), so open circle). Substitute \( x = 0 \) into \( y = -3x + 2 \):
\( y = -3(0) + 2 = 2 \), so the open circle is at \( (0, 2) \).
Find another point for \( x > 0 \), e.g., \( x = 1 \):
\( y = -3(1) + 2 = -1 \), so the point is \( (1, -1) \). Draw a line through \( (0, 2) \) (open circle) and \( (1, -1) \), extending right.

(Note: Since this is a graphing task, the final answer is the visual graph as described. For the purpose of this format, the key steps to construct the graph are outlined above.)

Answer:

Step1: Graph \( y = -x + 1 \) for \( x \leq -3 \)

First, find the endpoint at \( x = -3 \). Substitute \( x = -3 \) into \( y = -x + 1 \):
\( y = -(-3) + 1 = 3 + 1 = 4 \). So the point is \( (-3, 4) \), and since \( x \leq -3 \), this is a closed circle.
Next, find another point for \( x < -3 \), e.g., \( x = -4 \):
\( y = -(-4) + 1 = 4 + 1 = 5 \), so the point is \( (-4, 5) \). Draw a line through \( (-3, 4) \) (closed circle) and \( (-4, 5) \), extending left.

Step2: Graph \( y = -3x + 2 \) for \( x > 0 \)

Find the endpoint at \( x = 0 \) (but \( x > 0 \), so open circle). Substitute \( x = 0 \) into \( y = -3x + 2 \):
\( y = -3(0) + 2 = 2 \), so the open circle is at \( (0, 2) \).
Find another point for \( x > 0 \), e.g., \( x = 1 \):
\( y = -3(1) + 2 = -1 \), so the point is \( (1, -1) \). Draw a line through \( (0, 2) \) (open circle) and \( (1, -1) \), extending right.

(Note: Since this is a graphing task, the final answer is the visual graph as described. For the purpose of this format, the key steps to construct the graph are outlined above.)