The function is given to be:
[tex]y=\frac{1}{4}^x[/tex]
The graph of the function is shown below:
The graph is shifted up and to the left.
The graph is shifted up by 1 unit. This transformation rule is given to be:
[tex]f(x)\to f(x)+1[/tex]
Therefore, the function becomes:
[tex]y=\frac{1}{4}^x+1[/tex]
The graph is shifted to the left by 2 units. The transformation rule is given to be:
[tex]f(x)\to f(x+2)_{}[/tex]
Therefore, the function becomes:
[tex]y=\frac{1}{4}^{(x+2)}+1[/tex]
To check if the function is correct, we can use the points (-3, 5) if it gives a true statement:
[tex]\begin{gathered} (x,y)=(-3,5) \\ \therefore \\ 5=\frac{1}{4}^{(-3+2)}+1 \\ 5=\frac{1}{4}^{(-1)}+1 \\ 5=4+1 \\ 5=5(\text{True)} \end{gathered}[/tex]
Therefore, the function is:
[tex]y=\frac{1}{4}^{(x+2)}+1[/tex]
The coefficient is 1.
The exponent is x + 2.
The added constant is 1.
