To determine which of the given forms correctly represents an exponential model, let's analyze each option and see which one fits the characteristics of exponential growth or decay.
1. [tex]\( y = A \cdot B^x \)[/tex]
This form suggests that [tex]\( y \)[/tex] is based on an initial value [tex]\( A \)[/tex] (the y-intercept) and an exponential base [tex]\( B \)[/tex] raised to the power of [tex]\( x \)[/tex]. This is a classic representation of exponential growth or decay. For example, if [tex]\( B > 1 \)[/tex], the function depicts exponential growth, and if [tex]\( 0 < B < 1 \)[/tex], it depicts exponential decay.
2. [tex]\( y = A \cdot x^B \)[/tex]
This form indicates a power model, not an exponential model. Here, [tex]\( y \)[/tex] depends on [tex]\( x \)[/tex] raised to the power [tex]\( B \)[/tex], and is then scaled by [tex]\( A \)[/tex]. This is typically used to describe polynomial-like behavior rather than exponential growth or decay.
3. [tex]\( y = A \cdot x + B \)[/tex]
This represents a linear model, not an exponential model. In this form, [tex]\( y \)[/tex] changes linearly with [tex]\( x \)[/tex], scaled by [tex]\( A \)[/tex] and with an additional constant [tex]\( B \)[/tex] (the y-intercept).
Given the criteria for an exponential model, the correct form is:
[tex]\[ y = A \cdot B^x \][/tex]
