Question

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

$x$

$y$

---

---

0

0.2

1

0.08

2

0.032

3

0.0128

Explanation:

Step1: Recall exponential form

The general exponential function is $y = ab^x$, where $a$ is the initial value (when $x=0$), and $b$ is the growth/decay factor.

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

When $x=0$, $y=0.2$. Substitute into the formula:
$0.2 = ab^0$
Since $b^0=1$, we get $a=0.2$.

Step3: Find $b$ using $x=1$

Substitute $x=1$, $y=0.08$, and $a=0.2$ into $y=ab^x$:
$0.08 = 0.2b^1$
Solve for $b$:
$b = \frac{0.08}{0.2} = 0.4$

Step4: Verify $b$ with other points

Check $x=2$: $y=0.2\times(0.4)^2 = 0.2\times0.16 = 0.032$, which matches the table.
Check $x=3$: $y=0.2\times(0.4)^3 = 0.2\times0.064 = 0.0128$, which matches the table.

Step5: Write the final function

Substitute $a=0.2$ and $b=0.4$ into $y=ab^x$.

Answer:

$y = 0.2(0.4)^x$