We have to graph the function:
[tex]f(x)=-|2x-6|+1[/tex]To do that we can start by looking at which point the absolute function changes direction.
This happens when 2x-6=0.
[tex]\begin{gathered} 2x-6=0 \\ 2x=6 \\ x=\frac{6}{2} \\ x=3 \end{gathered}[/tex]We have one line for x<3 and other line, with different slope, for x>3.
Then we define the function as a piecewise function:
For x<3, we have:
[tex]\begin{gathered} x<3\longrightarrow2x-6<0\longrightarrow|2x-6|=-(2x-6) \\ f(x)=-(-(2x-6))+1 \\ f(x)=2x-6+1 \\ f(x)=2x-5 \end{gathered}[/tex]And for x>3:
[tex]\begin{gathered} x>3\longrightarrow2x-6>0\longrightarrow|2x-6|=2x-6 \\ f(x)=-(2x-6)+1 \\ f(x)=-2x+6+1 \\ f(x)=-2x+7 \end{gathered}[/tex]We can resume the function f(x) as:
[tex]f(x)=\begin{cases}2x-5;x<3 \\ -2x+7\colon x\ge3\end{cases}[/tex]For x<3, we have a positive-slope line (m=2), with y-intercept b=-5.
For x>3, we have a negative-slope line (m=-2), with y-intercept b=7 (but as x>3, the y-axis is not intercepted by this line). We also can use the reference that, when x=3, y=1.
We can graph this as:
NOTE: I draw them with different colours for each part of the function, but it is the same function and should be drawn with the same colour.
