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.
