Question

the equation of the exponential function represented by the following table:| $x$ | $y$ ||-----|-----|| 0 | 2 || 1 | 6 || 2 | 18 || 3 | 54 |

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: Find $a$ using $x=0$

When $x=0$, $y=2$. Substitute into the formula:
$2 = ab^0$
Since $b^0=1$, we get $a=2$.

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

Substitute $a=2$, $x=1$, $y=6$ into $y=ab^x$:
$6 = 2b^1$
Solve for $b$: $b = \frac{6}{2}=3$

Step4: Verify with other points

Check $x=2$: $y=2(3)^2=2\times9=18$, which matches the table.
Check $x=3$: $y=2(3)^3=2\times27=54$, which matches the table.

Answer:

$y = 2(3)^x$