Question

name: ____ date: __ period: ____
graphing piecewise function
sketch the graph of each function

1. $f(x)=\

$$\begin{cases}-4; & x \\leq 1 \\\\ x+2, & x > 1\\end{cases}$$

$

1. $g(x)=\

$$\begin{cases}-5, & x < 4 \\\\ 3, & x \\geq 4\\end{cases}$$

$

1. $g(x)=\

$$\begin{cases}x+1, & x < -4 \\\\ -x-4, & x \\geq -4\\end{cases}$$

$

1. $f(x)=\

$$\begin{cases}-6, & x \\leq -4 \\\\ 0, & -4 < x < 1 \\\\ -x-1, & x \\geq 1\\end{cases}$$

$

Explanation:

Step1: Analyze $f(x)$ first piece

For $x \leq 1$, $f(x) = -4$:
This is a horizontal line at $y=-4$, with a closed dot at $(1, -4)$ (since $x=1$ is included), extending left infinitely.

Step2: Analyze $f(x)$ second piece

For $x > 1$, $f(x) = x+2$:
This is a line with slope $1$ and y-intercept $2$. Calculate the point at $x=1$: $f(1)=1+2=3$, use an open dot at $(1, 3)$ (since $x=1$ is not included), then plot points like $(2,4), (3,5)$ and extend right.

Step3: Analyze $g(x)$ (problem 2) first piece

For $x < 4$, $g(x) = -5$:
Horizontal line at $y=-5$, open dot at $(4, -5)$ (since $x=4$ is excluded), extend left infinitely.

Step4: Analyze $g(x)$ (problem 2) second piece

For $x \geq 4$, $g(x) = 3$:
Horizontal line at $y=3$, closed dot at $(4, 3)$ (since $x=4$ is included), extend right infinitely.

Step5: Analyze $g(x)$ (problem3) first piece

For $x < -4$, $g(x) = x+1$:
Line with slope $1$, y-intercept $1$. Calculate $x=-4$: $g(-4)=-4+1=-3$, open dot at $(-4, -3)$, plot points like $(-5,-4), (-6,-5)$ and extend left.

Step6: Analyze $g(x)$ (problem3) second piece

For $x \geq -4$, $g(x) = -x-4$:
Line with slope $-1$, y-intercept $-4$. Calculate $x=-4$: $g(-4)=-(-4)-4=0$, closed dot at $(-4, 0)$, plot points like $(-3,-1), (0,-4)$ and extend right.

Step7: Analyze $f(x)$ (problem4) first piece

For $x \leq -4$, $f(x) = -6$:
Horizontal line at $y=-6$, closed dot at $(-4, -6)$, extend left infinitely.

Step8: Analyze $f(x)$ (problem4) second piece

For $-4 < x < 1$, $f(x) = 0$:
Horizontal line at $y=0$, open dot at $(-4, 0)$ and $(1, 0)$, between $x=-4$ and $x=1$.

Step9: Analyze $f(x)$ (problem4) third piece

For $x \geq 1$, $f(x) = -x-1$:
Line with slope $-1$, y-intercept $-1$. Calculate $x=1$: $f(1)=-1-1=-2$, closed dot at $(1, -2)$, plot points like $(2,-3), (0,-1)$ and extend right.

Answer:

1. For $f(x)=

$$\begin{cases}-4, & x \leq 1 \\ x+2, & x > 1\end{cases}$$

$:

• Horizontal line $y=-4$ (left from closed dot $(1,-4)$)
• Line $y=x+2$ (right from open dot $(1,3)$)

1. For $g(x)=

$$\begin{cases}-5, & x < 4 \\ 3, & x \geq 4\end{cases}$$

$:

• Horizontal line $y=-5$ (left from open dot $(4,-5)$)
• Horizontal line $y=3$ (right from closed dot $(4,3)$)

1. For $g(x)=

$$\begin{cases}x+1, & x < -4 \\ -x-4, & x \geq -4\end{cases}$$

$:

• Line $y=x+1$ (left from open dot $(-4,-3)$)
• Line $y=-x-4$ (right from closed dot $(-4,0)$)

1. For $f(x)=

$$\begin{cases}-6, & x \leq -4 \\ 0, & -4 < x < 1 \\ -x-1, & x \geq 1\end{cases}$$

$:

• Horizontal line $y=-6$ (left from closed dot $(-4,-6)$)
• Horizontal line $y=0$ (between open dots $(-4,0)$ and $(1,0)$)
• Line $y=-x-1$ (right from closed dot $(1,-2)$)

When plotting, use the provided grids to draw each segment with correct open/closed dots as specified.