Question

find the equation of the exponential function represented by the table below: x | y 0 | 0.2 1 | 0.6 2 | 1.8 3 | 5.4

Explanation:

Step1: Recall exponential form

The general exponential function is $y = ab^x$, where $a$ is the initial value, $b$ is the growth factor, $x$ is the input, and $y$ is the output.

Step2: Solve for $a$ 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: Solve for $b$ using $x=1$

Substitute $a=0.2$, $x=1$, $y=0.6$ into $y=ab^x$:
$0.6 = 0.2b^1$
Rearrange to solve for $b$:
$b = \frac{0.6}{0.2} = 3$

Step4: Verify $b$ with other points

Check $x=2$: $y = 0.2(3)^2 = 0.2(9) = 1.8$, which matches the table.
Check $x=3$: $y = 0.2(3)^3 = 0.2(27) = 5.4$, which matches the table.

Answer:

$y = 0.2(3)^x$