Question

a teacher displays the graph of a function, as shown.\
\
which function correctly produces the graph displayed?\
\
\\( \circ \\ d(x) = \

$$\begin{cases} -x + 1, & x \\leq 0 \\\\ 2x - 4, & x \\geq 1 \\end{cases}$$

\\)\
\\( \circ \\ d(x) = \

$$\begin{cases} -x - 2, & x < 1 \\\\ 2x - 4, & x \\geq 1 \\end{cases}$$

\\)\
\\( \circ \\ d(x) = \

$$\begin{cases} 2x + 4, & x \\leq 0 \\\\ -x + 1, & x \\geq 1 \\end{cases}$$

\\)\
\\( \circ \\ d(x) = \

$$\begin{cases} 2x - 4, & x < 1 \\\\ -x + 2, & x \\geq 1 \\end{cases}$$

\\)

Explanation:

Step1: Find left segment equation

First, identify two points on the left segment: $(-2, 0)$ and $(0, 4)$. Calculate the slope:
$$m=\frac{4-0}{0-(-2)}=\frac{4}{2}=2$$
Use slope-intercept form $y=mx+b$, with $b=4$ (y-intercept). The equation is $y=2x+4$, valid for $x<1$ (since the segment ends at an open point at $(1,6)$).

Step2: Find right segment equation

Identify two points on the right segment: $(1,1)$ and $(2,0)$. Calculate the slope:
$$m=\frac{0-1}{2-1}=-1$$
Use point-slope form $y-y_1=m(x-x_1)$ with $(x_1,y_1)=(1,1)$:
$$y-1=-1(x-1) \implies y=-x+2$$
This is valid for $x\geq1$ (since the segment starts at a closed point at $(1,1)$).

Step3: Match to options

Compare the two equations to the given options. The piecewise function matches the fourth option.

Answer:

$\boldsymbol{d(x) =

$$\begin{cases} 2x + 4 & x < 1 \\ -x + 2 & x \geq 1 \end{cases}$$

}$ (the fourth option)