Question

graph the following function on the axes provided.

( f(x) = \begin{cases} 4x + 17 & \text{for } x leq -3 \\ x + 3 & \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.

graph with x - axis from -10 to 10, y - axis from -3 to 10, grid lines

Explanation:

Step1: Graph \( y = 4x + 17 \) for \( x \leq -3 \)

First, find the endpoint at \( x = -3 \). Substitute \( x = -3 \) into \( y = 4x + 17 \):
\( y = 4(-3) + 17 = -12 + 17 = 5 \). So the point is \( (-3, 5) \), and since \( x \leq -3 \), this is a closed circle.
Next, find another point on this line. Let's choose \( x = -4 \):
\( y = 4(-4) + 17 = -16 + 17 = 1 \). So the point is \( (-4, 1) \). Draw a line through \( (-3, 5) \) (closed circle) and \( (-4, 1) \), extending left (since \( x \leq -3 \)).

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

Find the endpoint at \( x = 0 \). Substitute \( x = 0 \) into \( y = x + 3 \):
\( y = 0 + 3 = 3 \). Since \( x > 0 \), this is an open circle at \( (0, 3) \).
Find another point on this line. Let's choose \( x = 1 \):
\( y = 1 + 3 = 4 \). So the point is \( (1, 4) \). Draw a line through \( (0, 3) \) (open circle) and \( (1, 4) \), extending right (since \( x > 0 \)).

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

Answer:

The graph consists of two lines:

• For \( x \leq -3 \), a line with a closed circle at \( (-3, 5) \) and passing through \( (-4, 1) \) (extending left).
• For \( x > 0 \), a line with an open circle at \( (0, 3) \) and passing through \( (1, 4) \) (extending right).