[tex]\begin{gathered} \text{the parent function is f(x)=2(x-4)}^2+3 \\ 1stcase\colon\text{f(x-1) means that you must substitute x by x-1 in the parent funcion, i.e.,} \\ \text{f(x-1)=2((x-1)-4)}^2+3 \\ \text{f(x-1) =2(x-5)}^2+3 \end{gathered}[/tex][tex]\begin{gathered} \text{therefore, f(x-1)-4 is equal to:} \\ f(x-1)-4=(2(x-5)^2+3)-4 \\ f(x-1)-4=2(x-5)^2+3-4 \\ f(x-1)-4=2(x-5)^2-1 \end{gathered}[/tex][tex]\begin{gathered} 2ndcase\colon f(x+3)\text{ means that you must substitute x-3 in x, i.e.} \\ f(x+3)=2((x+3)-4)^2+3 \\ f(x+3)=2(x-1)^2+3 \\ \end{gathered}[/tex][tex]\begin{gathered} \text{Now, f(x+3)-5 =(}2(x-1)^2+3)-5 \\ \text{f(x+3)-5 =}2(x-1)^2+3-5 \\ \text{f(x+3)-5 =}2(x-1)^2-2 \end{gathered}[/tex][tex]\begin{gathered} 3rdcase\colon-f(x)\text{ =-(}2(x-4)^2+3) \\ -f(x)\text{ =}-2(x-4)^2-3 \end{gathered}[/tex][tex]\begin{gathered} 4th\text{ case: f(-x) means that you must substitute -x in x, i.e.,} \\ f(-x)\text{ =}2((-x)-4)^2+3 \\ f(-x)\text{ =}2(-x-4)^2+3 \\ f(-x)\text{ =}2((-1)(x+4)^2+3 \\ f(-x)\text{ =}2(x+4)^2+3 \end{gathered}[/tex]
