Leaving the calculations of the table of values to get full marks aside as a clear replacement for hard but obvious work a student must do himself, here is an explanation of the graph transformation
from #f(x)=3^x# to #g(x)=3^-(x+1)-2#.
First of all, let's recall the definition of a graph of a function #y=F(x)#: it's a set of all points on a Cartesian plane with coordinates #(a,b)# such that #b=F(a)#.
We will transform a graph of #f(x)# to a graph of #g(x)# in the following steps:
Step 1: from #f(x)=3^x# to #f_1(x)=3^-x#.
Step 2: from #f_1(x)=3^-x# to #f_2(x)=3^-(x+1)#.
Step 3: from #f_2(x)=3^-(x+1)# to #g(x)=3^-(x+1)-2#.
Here are the transformations in steps.
Step 1: from #f(x)=3^x# to #f_1(x)=3^-x#.
Consider general transformation
from #y=F(x)# to #y=F(-x)#
If a point #(a,b)# belongs to a graph of #y=F(x)#, that is if #b=F(a)#, then a point #(-a,b)# belongs to a graph of #y=F(-x)# because #F(-(-a))=F(a)=b#
Therefore, each point #(a,b)# of a graph of #y=F(x)# corresponds to a point #(-a,b)# of a graph of #y=F(-x)#. So, the whole graph of #y=F(-x)# is symmetrically reflected to a graph of #y=F(x)# relatively to the Y-axis.
In our case #F(x)=3^x#, so the transformation looks like:
#f(x)=3^x#:
graph{3^x [-10, 10, -5, 5]}
#f_1(x)=3^-x#:
graph{3^-x [-10, 10, -5, 5]}
Step 2: from #f_1(x)=3^-x# to #f_2(x)=3^-(x+1)#.
Consider general transformation
from #y=F(x)# to #y=F(x+epsilon)#
If a point #(a,b)# belongs to a graph of #y=F(x)#, that is if #b=F(a)#, then a point #(a-epsilon,b)# belongs to a graph of #y=F(x+epsilon)# because #F(a-epsilon+epsilon)=F(a)=b#
Therefore, each point #(a,b)# of a graph of #y=F(x)# corresponds to a point #(a-epsilon,b)# of a graph of #y=F(x+epsilon)#. So, the whole graph shifts to the left by the value #epsilon#.
In our case #F(x)=3^-x# and #epsilon=1# to transform a function to #3^-(x+1)#. So, the graph of #3^-x# is shifted to the left by #epsilon=1#.
#f_1(x)=3^-x#:
graph{3^-x [-10, 10, -5, 5]}
#f_2(x)=3^-(x+1)#:
graph{3^-(x+1) [-10, 10, -5, 5]}
Step 3: from #f_2(x)=3^-(x+1)# to #g(x)=3^-(x+1)-2#.
Consider general transformation
from #y=F(x)# to #y=F(x)+delta#
If a point #(a,b)# belongs to a graph of #y=F(x)#, that is if #b=F(a)#, then a point #(a,b+delta)# belongs to a graph of #y=F(x)+delta# because #F(a)+delta=b+delta#
Therefore, each point #(a,b)# of a graph of #y=F(x)# corresponds to a point #(a,b+delta)# of a graph of #y=F(x)+delta#. So, the whole graph shifts vertically by #delta# (up for positive #delta# and down for negative).
In our case #F(x)=3^-(x+1)# and #delta=-2# to transform a function to #3^-(x+1)-2#. So, the graph of #3^-(x+1)# is shifted down by #delta=-2#.
#f_2(x)=3^-(x+1)#:
graph{3^-(x+1) [-10, 10, -5, 5]}
#g(x)=3^-(x+1)-2#:
graph{3^-(x+1)-2 [-10, 10, -5, 5]}
This completes the transformation from #f(x)=3^x# to #g(x)=3^-(x+1)-2#.
