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 three segments: 1) A ray with slope 1, y-intercept 1, ending at the open point (0,1) and extending left; 2) A line segment with slope -1, from closed point (0,1) to closed point (2,-1); 3) A ray with slope 1, starting at the open point (2,1) and extending right.

Answer

Step 1: Analyze the first piece $y = x + 1$ for $x < 0$. Calculate key points: when $x=0$, $y=1$ (open dot); when $x=-2$, $y=-1$. Plot these points and draw a line extending left from the open dot at (0,1). Step 2: Analyze the second piece $y = -x + 1$ for $0 \leq x \leq 2$. Calculate endpoints: when $x=0$, $y=1$ (closed dot); when $x=2$, $y=-1$ (closed dot). Plot these points and draw a straight line segment connecting them. Step 3: Analyze the third piece $y = x - 1$ for $x > 2$. Calculate key points: when $x=2$, $y=1$ (open dot); when $x=4$, $y=3$. Plot these points and draw a line extending right from the open dot at (2,1). Step 4: Combine all three segments on the same coordinate plane to form the complete graph.