Question

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

x	y
1	0.01
2	0.005
3	0.0025

Explanation:

Step1: Recall exponential form

The general exponential function is $y = ab^x$, where $a$ is the y-intercept (value of $y$ when $x=0$), and $b$ is the common ratio.

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

When $x=0$, $y=0.02$. Substitute into $y=ab^x$:
$0.02 = ab^0$
Since $b^0=1$, $a=0.02$.

Step3: Calculate common ratio $b$

Use $x=1$, $y=0.01$ and $a=0.02$:
$0.01 = 0.02b^1$
Solve for $b$: $b = \frac{0.01}{0.02} = 0.5$
Verify with $x=2$: $y=0.02(0.5)^2 = 0.02(0.25)=0.005$, which matches the table.

Step4: Write final equation

Substitute $a=0.02$ and $b=0.5$ into $y=ab^x$.

Answer:

$y = 0.02(0.5)^x$