Question

graph this function.
$f(x) = \

$$\begin{cases} -6 & \\text{if } -7 < x < 0 \\\\ -\\frac{1}{4}x & \\text{if } 0 \\leq x \\leq 8 \\end{cases}$$

$
select points on the graph to plot them. select \point fill\ to change a point from closed to open.
graph with x-axis from -8 to 8 and y-axis from -5 to 8, grid lines, and axes labeled x and y

Explanation:

Step1: Analyze the first piece

The first piece is \( f(x) = -6 \) for \( -7 < x < 0 \). This is a horizontal line. We need to plot open circles at \( x = -7 \) and \( x = 0 \) (since the inequality is strict) with \( y = -6 \). So points like \( (-6, -6) \), \( (-3, -6) \) etc. lie on this segment.

Step2: Analyze the second piece

The second piece is \( f(x) = -\frac{1}{4}x \) for \( 0 \leq x \leq 8 \). This is a linear function. At \( x = 0 \), \( f(0) = 0 \) (closed circle, since \( x = 0 \) is included). At \( x = 8 \), \( f(8)=-\frac{1}{4}(8)= - 2 \) (closed circle). We can also find other points: at \( x = 4 \), \( f(4)=-\frac{1}{4}(4)= - 1 \). So we plot the line segment from \( (0, 0) \) to \( (8, -2) \) with closed circles at the endpoints.

Step3: Plot the points

For the first segment, mark open circles at \( (-7, -6) \) and \( (0, -6) \), and draw a horizontal line between them (through points like \( (-5, -6) \)). For the second segment, mark closed circles at \( (0, 0) \) and \( (8, -2) \), and draw the line through \( (4, -1) \) etc.

Answer:

To graph the piece - wise function:

1. For \( -7 < x < 0 \), draw a horizontal line \( y=-6 \) with open circles at \( (-7, -6) \) and \( (0, -6) \).
2. For \( 0\leq x\leq8 \), draw the line \( y =-\frac{1}{4}x \) with closed circles at \( (0, 0) \) and \( (8, -2) \), and plot additional points like \( (4, -1) \) on this line segment.