Question

find the equation of the exponential function represented by the table below: (table with x and y values: x=0,y=2; x=1,y=4; x=2,y=8; x=3,y=16)

Explanation:

Step1: Recall exponential form

The general form of an exponential function is $y = ab^x$, where $a$ is the initial value, $b$ is the growth factor, $b>0, b
eq1$.

Step2: Solve for $a$ using $x=0$

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

Step3: Solve for $b$ using $x=1$

Substitute $a=2$, $x=1$, $y=4$ into $y=ab^x$:
$4 = 2b^1$
Solve for $b$: $b = \frac{4}{2}=2$

Step4: Verify with other points

Check $x=2$: $y=2\times2^2=2\times4=8$, which matches the table.
Check $x=3$: $y=2\times2^3=2\times8=16$, which matches the table.

Answer:

$y=2\times2^x$ (or simplified as $y=2^{x+1}$)