Question

this is the graph of an exponential function.
write an equation for the function in the form ( f(x) = a(b)^x ).
use whole numbers, decimals, or simplified fractions for the values of a and b.
( f(x) = )

Explanation:

Step1: Find the value of \( a \)

The general form of an exponential function is \( f(x) = a(b)^x \). When \( x = 0 \), \( f(0)=a(b)^0=a(1)=a \). From the graph, when \( x = 0 \), \( y = 6 \). So, \( a = 6 \).

Step2: Find the value of \( b \)

We know \( a = 6 \), so the function is \( f(x)=6(b)^x \). Now, we need another point to find \( b \). Let's take \( x = 1 \), from the graph, when \( x = 1 \), \( y = 12 \) (since the graph passes through (1,12)). Substitute \( x = 1 \), \( f(1)=12 \) into \( f(x)=6(b)^x \):
\( 12 = 6(b)^1 \)
Divide both sides by 6: \( \frac{12}{6}=b \)
So, \( b = 2 \).

Step3: Write the function

Now that we have \( a = 6 \) and \( b = 2 \), the exponential function is \( f(x)=6(2)^x \).

Answer:

\( 6(2)^x \)