Question

the table of ordered pairs (x, y) gives an exponential function. write an equation for the function. x y -1 1/3 0 3 1 27 2 243

Explanation:

Step1: Recall exponential form

The general exponential function is $y = ab^x$, where $a$ is the initial value, $b$ is the base, $b>0, b
eq1$.

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

When $x=0$, $y=3$. Substitute into $y=ab^x$:
$3 = ab^0$
Since $b^0=1$, $a=3$.

Step3: Find $b$ using another point

Use $x=1, y=27$ and $a=3$. Substitute into $y=3b^x$:
$27 = 3b^1$
Solve for $b$: $b = \frac{27}{3}=9$

Step4: Verify with other points

Check $x=-1$: $y=3(9)^{-1}=3\times\frac{1}{9}=\frac{1}{3}$, matches the table.
Check $x=2$: $y=3(9)^2=3\times81=243$, matches the table.

Answer:

$y=3(9)^x$