Question

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

$$\begin{cases} -4x - 22 & \\text{for} & x < -4 \\\\ 4 & \\text{for} & x = -4 \\\\ \\frac{1}{2}x - 4 & \\text{for} & -4 < x < 0 \\end{cases}$$

$
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: Analyze \( f(x) = -4x - 22 \) ( \( x < -4 \))

Find two points. Let \( x = -5 \): \( f(-5)=-4(-5)-22 = 20 - 22=-2 \). Let \( x=-6 \): \( f(-6)=-4(-6)-22 = 24 - 22 = 2 \). The line has a slope of \(-4\), and at \( x=-4 \), \( f(-4)=-4(-4)-22=16 - 22=-6 \), but since \( x < -4 \), the endpoint at \( x=-4 \) is an open circle (value \(-6\)).

Step2: Analyze \( f(x) = 4 \) ( \( x = -4 \))

This is a closed circle at \( (-4, 4) \) since \( x=-4 \) is included.

Step3: Analyze \( f(x)=\frac{1}{2}x - 4 \) ( \( -4 < x < 0 \))

Find two points. Let \( x=-4 \): \( f(-4)=\frac{1}{2}(-4)-4=-2 - 4=-6 \), but since \( x > -4 \), the endpoint at \( x=-4 \) is an open circle (value \(-6\)). Let \( x = 0 \): \( f(0)=\frac{1}{2}(0)-4=-4 \), open circle at \( (0, -4) \). Another point: \( x=-2 \), \( f(-2)=\frac{1}{2}(-2)-4=-1 - 4=-5 \). The line has a slope of \(\frac{1}{2}\).

To graph:

• For \( x < -4 \): Draw the line through \( (-5, -2) \), \( (-6, 2) \), with an open circle at \( (-4, -6) \).
• At \( x=-4 \): Plot a closed circle at \( (-4, 4) \).
• For \( -4 < x < 0 \): Draw the line through \( (-2, -5) \), with open circles at \( (-4, -6) \) and \( (0, -4) \).

(Note: Since the actual graphing is interactive, the key is to identify the pieces, their endpoints (open/closed), and slopes. The given partial graph shows a point at \( x=-7 \) (maybe from the first piece) and an open circle near \( x=-4 \), \( y=4 \) (but the closed circle for \( x=-4 \) is at \( y=4 \), open for the first and third pieces at \( x=-4 \) is \( y=-6 \)).)

Answer:

To graph the piecewise function:

1. For \( \boldsymbol{x < -4} \) (function \( -4x - 22 \)):

• Slope \( -4 \), open circle at \( (-4, -6) \) (since \( x < -4 \)), pass through points like \( (-5, -2) \), \( (-6, 2) \).

1. For \( \boldsymbol{x = -4} \) (function \( 4 \)):

• Closed circle at \( (-4, 4) \).

1. For \( \boldsymbol{-4 < x < 0} \) (function \( \frac{1}{2}x - 4 \)):

• Slope \( \frac{1}{2} \), open circles at \( (-4, -6) \) (since \( x > -4 \)) and \( (0, -4) \) (since \( x < 0 \)), pass through points like \( (-2, -5) \).

(Interactive graphing: Use the line tool for each piece, set endpoints to open/closed as needed.)