Graphing the Three-Part Piecewise Function $f(x)=\begin{cases} x+1, x 2 \end{cases}$

Question

Graph the piecewise function: $f(x)=\begin{cases} x+1, \text{ if } x < 0 \\ -x+1, \text{ if } 0 \leq x \leq 2 \\ x-1, \text{ if } x > 2 \end{cases}$

Accepted Answer

A piecewise graph with: 1) A ray for $y=x+1$ ending at open point $(0,1)$ and extending left; 2) A line segment for $y=-x+1$ connecting closed points $(0,1)$ and $(2,-1)$; 3) A ray for $y=x-1$ starting at open point $(2,1)$ and extending right.

Answer

Step 1: Analyze the first piece $y=x+1$ for $x < 0$: This is a linear function with slope 1 and y-intercept 1. Plot the endpoint at $x=0$ as an open dot at $(0,1)$ (since $x<0$ excludes 0). Plot an additional point (e.g., $x=-2$, $f(-2)=-1$) and draw a ray leftward from the open dot through this point. Step 2: Analyze the second piece $y=-x+1$ for $0 \leq x \leq 2$: This is a linear function with slope -1 and y-intercept 1. Plot the left endpoint at $x=0$: $f(0)=1$, a closed dot at $(0,1)$. Plot the right endpoint at $x=2$: $f(2)=-2+1=-1$, a closed dot at $(2,-1)$. Draw a straight line segment connecting these two closed dots. Step 3: Analyze the third piece $y=x-1$ for $x > 2$: This is a linear function with slope 1 and y-intercept -1. Plot the endpoint at $x=2$ as an open dot at $(2,1)$ (since $x>2$ excludes 2). Plot an additional point (e.g., $x=4$, $f(4)=3$) and draw a ray rightward from the open dot through this point. Step 4: Combine all three parts on the same coordinate grid to form the complete graph.