Answer:
The parent function of the function represented in the table is exponential
If function f is vertically compressed by a factor of 1/4, the f(x) - values will be divided by 4.
A point in the table for the transformed function is (1, 6)
Explanation:
The x values are increasing by 1 unit at a time, but the f(x) is always half the value before, for example
96/2 = 48
48/2 = 24
24/2 = 12
12/2 = 6
This is the behavior of an exponential function and the equation that describes this table is:
[tex]f(x)=96(\frac{1}{2})^{x+1}[/tex]Because
[tex]\begin{gathered} f(-1)=96(\frac{1}{2})^{-1+1}=96(\frac{1}{2})^0=96 \\ f(0)=96(\frac{1}{2})^{0+1}=96(\frac{1}{2})^1=96(\frac{1}{2})=48 \\ f(1)=96(\frac{1}{2})^{1+1}=96(\frac{1}{2})^2=96(\frac{1}{4})=24 \\ f(2)=96(\frac{1}{2})^{2+1}=96(\frac{1}{2})^3=96(\frac{1}{8})=12 \\ f(3)=96(\frac{1}{2})^{3+1}=96(\frac{1}{2})^4=96(\frac{1}{16})=6 \end{gathered}[/tex]Therefore, the parent function represented in the table is Exponential.
Then, to make a vertical compression by a factor of 1/4, we need to multiply the values of f(x) by 1/4. It is the same to divide the values of f(x) by 4.
So, if function f is vertically compressed by a factor of 1/4, the f(x) - values will be divided by 4.
Therefore, the equation for the vertical compressed function is:
[tex]f(x)=\frac{1}{4}(96)(\frac{1}{2})^{x+1}=24(\frac{1}{2})^{x+1}[/tex]So, we can verify that (1, 6) is a point in the transformed function by replacing x by 1 on the equation above.
[tex]f(1)=24(\frac{1}{2}_{})^{1+1}=24(\frac{1}{2})^2=24(\frac{1}{4})=6[/tex]Then, a point in the table for the transformed function is (1, 6)
