Question

this table shows several values of an exponential function.
x | f(x)
1 | -9
2 | -27
3 | -81
4 | -243
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: Recall exponential function form

The general form of an exponential function is \( f(x) = a(b)^x \). We need to find the values of \( a \) and \( b \).

Step2: Find the common ratio \( b \)

To find \( b \), we can divide consecutive \( f(x) \) values. Let's take \( x = 2 \) and \( x = 1 \): \( \frac{f(2)}{f(1)}=\frac{-27}{-9} = 3 \). Check with \( x = 3 \) and \( x = 2 \): \( \frac{-81}{-27}=3 \), and \( x = 4 \) and \( x = 3 \): \( \frac{-243}{-81}=3 \). So \( b = 3 \).

Step3: Find \( a \) using a point

Use the point \( (1, -9) \). Substitute \( x = 1 \), \( f(x) = -9 \), and \( b = 3 \) into \( f(x) = a(b)^x \):
\( -9 = a(3)^1 \)
Solve for \( a \): \( a=\frac{-9}{3}=-3 \).

Step4: Write the function

Substitute \( a = -3 \) and \( b = 3 \) into \( f(x) = a(b)^x \):
\( f(x)=-3(3)^x \)

Answer:

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