Question

find the equation of the exponential function represented by the table below:

x	y
1	0.03
2	0.09
3	0.27

Explanation:

Step1: Recall exponential form

The general exponential function is $y = ab^x$, where $a$ is the y-intercept, $b$ is the growth factor.

Step2: Find $a$ using $x=0$

When $x=0$, $y=0.01$. Substitute into $y=ab^x$:
$0.01 = ab^0$
Since $b^0=1$, $a=0.01$.

Step3: Find $b$ using another point

Use $x=1, y=0.03$. Substitute $a=0.01$, $x=1$, $y=0.03$ into $y=ab^x$:
$0.03 = 0.01b^1$
Solve for $b$: $b = \frac{0.03}{0.01} = 3$

Step4: Verify $b$ with other points

Check $x=2$: $y=0.01(3)^2 = 0.01(9)=0.09$, which matches the table.
Check $x=3$: $y=0.01(3)^3 = 0.01(27)=0.27$, which matches the table.

Answer:

$y = 0.01(3)^x$