The given function is:
[tex]f(x)=\frac{5}{6}|x+3|-1[/tex]
The modulus function is a combination of a positive and a negative so let y=f(x) and get two equations as follows:
[tex]\begin{gathered} y=\frac{5}{6}|x+3|-1 \\ y=\frac{5}{6}(x+3)-1;y=\frac{-5}{6}(x+3)-1 \end{gathered}[/tex]
Find the point of intersection of these two lines to get the vertex of the modulus function.
Subtract the second equation from the first to get:
[tex]\begin{gathered} \frac{5}{6}(x+3)-1+\frac{5}{6}(x+3)+1=0 \\ x+3=0 \\ x=-3 \end{gathered}[/tex]
Substitute x=-3 to get:
[tex]y=-1[/tex]
Hence (-3,-1) is the vertex of the function.
Graph the function as shown below:
Solve for y=0 as follows:
[tex]\begin{gathered} \frac{5}{6}|x+3|-1=0 \\ \frac{5}{6}|x+3|=1 \\ |x+3|=\frac{6}{5} \\ x+3=\frac{6}{5},x+3=-\frac{6}{5} \\ x=\frac{6}{5}-3,x=-\frac{6}{5}-3 \\ x=1.2-3,x=-1.2-3 \\ x=-1.8,x=-4.2 \end{gathered}[/tex]
Hence the intercepts are (-1.8,0) and (-4.2,0).
