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: Identify the y-intercept (when x = 0)

For the exponential function \( f(x) = a(b)^x \), when \( x = 0 \), \( f(0) = a(b)^0 = a(1) = a \). From the graph, when \( x = 0 \), \( y = 3 \), so \( a = 3 \).

Step2: Find the value of b

Now we have \( f(x) = 3(b)^x \). Let's find another point on the graph. Looking at the graph, when \( x = 1 \), let's estimate the y-value. From the graph, when \( x = 1 \), the function passes through (1, 12) (since it goes from (0,3) up to around y=12 when x=1). Substitute \( x = 1 \) and \( f(1) = 12 \) into the equation:
\( 12 = 3(b)^1 \)
Divide both sides by 3: \( \frac{12}{3} = b \)
So \( b = 4 \).

Answer:

\( f(x) = 3(4)^x \)