Question

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

x	y
0	4
1	12
2	36
3	108

Explanation:

Step1: Recall exponential function form

The general form of an exponential function is \( y = ab^x \), where \( a \) is the initial value (when \( x = 0 \)) and \( b \) is the common ratio.

Step2: Find the value of \( a \)

When \( x = 0 \), \( y = 4 \). Substitute into \( y = ab^x \):
\( 4 = ab^0 \)
Since \( b^0 = 1 \), we get \( a = 4 \).

Step3: Find the value of \( b \)

Use another point, say \( x = 1 \), \( y = 12 \). Substitute \( a = 4 \), \( x = 1 \), \( y = 12 \) into \( y = ab^x \):
\( 12 = 4b^1 \)
Divide both sides by 4: \( b = \frac{12}{4} = 3 \).

Step4: Write the equation

Substitute \( a = 4 \) and \( b = 3 \) into \( y = ab^x \):
\( y = 4 \times 3^x \)

Answer:

\( y = 4 \cdot 3^x \)