The image you uploaded contains text and a graph related to "Exponential Models." Let me extract the key information from the image and provide a detailed explanation.

Extracted Content:

1. Title: Exponential Models

2. Introduction:

   • Exponential functions have the form $y = ab^x$, where $a$ is a non-zero constant and $b$ is a positive constant other than 1.
   • The function models how a quantity $y$ grows or decays over time.
   • The value of $b$ determines the nature of the function: $b > 1$ indicates growth, while $0 < b < 1$ indicates decay.

3. Example Problem:

   • For the function $y = 2(3)^x$, let $x$ take on values 0, 1, 2, and 3.
   • The corresponding $y$-values are calculated and listed in a table.
   • A graph is shown with a curve that passes through the points generated by the table.

4. Table:

   • A table showing values of $x$ and $y$:
     • $x = 0$, $y = 2$
     • $x = 1$, $y = 6$
     • $x = 2$, $y = 18$
     • $x = 3$, $y = 54$

5. Graph:

   • A graph illustrating the exponential growth of the function $y = 2(3)^x$.
   • The curve is steep, showing rapid increase as $x$ increases.

Explanation:

Exponential functions are fundamental in modeling situations where quantities grow or decay at a constant relative rate. The general form of an exponential function is $y = ab^x$, where:

• $a$ represents the initial amount (when $x = 0$).
• $b$ is the growth factor (if $b > 1$) or decay factor (if $0 < b < 1$).
• $x$ is the independent variable, typically representing time.

In the provided example $y = 2(3)^x$, the function models exponential growth because the base $b = 3$ is greater than 1. For each unit increase in $x$, the value of $y$ is multiplied by 3. The table provided shows the computed values of $y$ for different values of $x$, and the corresponding graph depicts how rapidly $y$ increases as $x$ increases.

Questions and Further Exploration:

1. What happens to the graph of $y = ab^x$ if the value of $b$ is between 0 and 1?
2. How does the value of $a$ affect the starting point of the graph?
3. Can you identify real-world phenomena that are modeled by exponential growth or decay?
4. How would the graph change if $y = 2(3)^{-x}$ was plotted instead?
5. What are the characteristics of a logarithmic function, and how does it relate to exponential functions?

Tip:

When plotting exponential functions, carefully choose a range of $x$-values that showcases the behavior of the function, especially around $x = 0$, to accurately represent its growth or decay.

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