Set a(x)=(1/2)^x; then,
[tex]\Rightarrow a(x-2)+1\to\text{two units to the right and one unit up}[/tex]
Furthermore,
[tex]a(x-2)+1=(\frac{1}{2})^{x-2}+1[/tex]
Then, the parent function of function 1) is (1/2)^x.
On the other hand, if b(x)=3^x; then,
[tex]\begin{gathered} \Rightarrow2b(x)=2\cdot3^x \\ \text{and} \\ 2b(x)\to\text{dilation of b(x) by a factor equal to 2} \end{gathered}[/tex]
Therefore, the parent function of 2*3^x is 3^x.
Now, we need to find 9 points on each parent function, as shown below
[tex]\begin{gathered} a(-4)=16 \\ a(-3)=8 \\ a(-2)=4 \\ a(-1)=2 \\ a(0)=1 \\ a(1)=\frac{1}{2} \\ a(2)=\frac{1}{4} \\ a(3)=\frac{1}{8} \\ a(4)=\frac{1}{16} \end{gathered}[/tex]
And
[tex]\begin{gathered} b(-4)=\frac{1}{81} \\ b(-3)=\frac{1}{27} \\ b(-2)=\frac{1}{9} \\ b(-1)=\frac{1}{3} \\ b(0)=1 \\ b(1)=3 \\ b(2)=9 \\ b(3)=27 \\ b(4)=81 \end{gathered}[/tex]
After graphing the points, we get
Parent function of function 1)
Parent function of function 2)
Now, translating the parent function in green two units to the right and 1 unit up, we obtain
In both images, the green curve is 1/2^(x)->parent function of (1/2)^(x-2)+1
the yellow curve is (1/2)^(x-2)+1
The red curve is 3^x->parent function of 2*3^x
and the blue curve is 2*3^x
