Respuesta :
Given:
In a table,
x values are -2, -1, 0, 1, 2.
y vales are 25, 5, 1, 1/5, 1/25
The objective is to prove whether the table values represents an exponential function or not. And also to find the rule for the function.
Consider the x values in the given table. The difference between each value will be,
[tex]\begin{gathered} x_2-x_1=-1-(-2)_{} \\ =1 \end{gathered}[/tex]Similarly,
[tex]\begin{gathered} x_3-x_2=0-(-1) \\ =1 \end{gathered}[/tex]Thus, the values of x are constantly increasing with positive 1 unit.
Consider the y values in the given table.
Te ratio between each term will be,
[tex]\begin{gathered} \frac{y_2}{y_1}=\frac{5}{25} \\ =\frac{1}{5} \end{gathered}[/tex]Similarly,
[tex]\frac{y_3}{y_2}=\frac{1}{5}[/tex]Thus, there is a common ratio between each values of y in the table.
So, it is clear with the common ratio, that the given table value is an exponential functiton.
Now, the rule for the function can be written as,
[tex]y=a\cdot b^x[/tex]Consider the coordinate, (x,y) = (0,1) and substitute in the above formula.
[tex]\begin{gathered} 1=a\cdot b^0 \\ 1=a\cdot1 \\ a=1 \end{gathered}[/tex]Consider another coordinate (1, 1/5) and the value of a. Substitute the obtained values in the general formula.
[tex]\begin{gathered} \frac{1}{5}=1\cdot b^1 \\ b=\frac{1}{5} \end{gathered}[/tex]Now, substitue only the value of a and b in the general formula.
[tex]\begin{gathered} y=1\cdot(\frac{1}{5})^x^{} \\ y=(\frac{1}{5})^x \end{gathered}[/tex]Hence, the rule for the given exponential function is y=(1/5)^x.
