Unit 6 Review5. Create A Graph To Represent $A = 240 \cdot \left(\frac{1}{3}\right)^t$ When $t$ Is $0, 1, 2, 3,$ And $4$. Think Carefully About How You Choose The Scale For The Axes. If You Get Stuck, Consider

Introduction

In this review, we will focus on understanding exponential functions and how to represent them graphically. We will use the given function $A = 240 \cdot \left(\frac{1}{3}\right)^t$ and explore its behavior when $t$ takes on the values $0, 1, 2, 3,$ and $4$. We will also discuss the importance of choosing an appropriate scale for the axes when creating a graph.

Understanding Exponential Functions

Exponential functions are a type of mathematical function that describes a relationship between two variables, typically denoted as $x$ and $y$. The general form of an exponential function is $y = a \cdot b^x$, where $a$ is the initial value, $b$ is the base, and $x$ is the exponent. In our given function, $A = 240 \cdot \left(\frac{1}{3}\right)^t$, we have $a = 240$, $b = \frac{1}{3}$, and $x = t$.

Graphing Exponential Functions

To create a graph of the given function, we need to calculate the values of $A$ for each value of $t$. We will use the following values of $t$: $0, 1, 2, 3,$ and $4$.

Choosing the Scale for the Axes

When creating a graph, it is essential to choose an appropriate scale for the axes. In this case, we need to consider the range of values for both $t$ and $A$. For $t$, we have a range of $0$ to $4$, which is a relatively small range. For $A$, we have a range of $2.97$ to $240$, which is a relatively large range.

To choose an appropriate scale for the axes, we need to consider the following factors:

• The range of values for both $t$ and $A$
• The units of measurement for both $t$ and $A$
• The type of graph we want to create (e.g., linear, logarithmic, etc.)

Creating the Graph

Based on the values we calculated earlier, we can create a graph of the given function. We will use a logarithmic scale for the $y$-axis to better represent the exponential behavior of the function.

**Graph of A = 240 \* (1/3)^t**
================================

| t | A |
| --- | --- |
| 0 | 240 |
| 1 | 80 |
| 2 | 26.67 |
| 3 | 8.89 |
| 4 | 2.97 |

**Logarithmic Scale for the y-axis**
-----------------------------------

| y | log(y) |
| --- | --- |
| 2.97 | -0.53 |
| 8.89 | -0.05 |
| 26.67 | 1.43 |
| 80 | 1.90 |
| 240 | 2.38 |

**Interpreting the Graph**
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The graph of the given function shows an exponential decay as $t$ increases. The function starts at $240$ when $t = 0$ and decreases rapidly as $t$ increases. The logarithmic scale for the $y$-axis helps to better represent the exponential behavior of the function.

**Conclusion**
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In this review, we have explored the concept of exponential functions and how to represent them graphically. We have used the given function $A = 240 \cdot \left(\frac{1}{3}\right)^t$ and calculated its values for $t = 0, 1, 2, 3,$ and $4$. We have also discussed the importance of choosing an appropriate scale for the axes when creating a graph. By using a logarithmic scale for the $y$-axis, we can better represent the exponential behavior of the function.

**Key Takeaways**
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* Exponential functions describe a relationship between two variables, typically denoted as $x$ and $y$.
* The general form of an exponential function is $y = a \cdot b^x$, where $a$ is the initial value, $b$ is the base, and $x$ is the exponent.
* To create a graph of an exponential function, we need to calculate the values of the function for a range of values of the exponent.
* Choosing an appropriate scale for the axes is essential when creating a graph.
* A logarithmic scale for the $y$-axis can help to better represent the exponential behavior of a function.<br/>
**Unit 6 Review: Understanding Exponential Functions and Graphs - Q&A**
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**Q: What is an exponential function?**
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A: An exponential function is a type of mathematical function that describes a relationship between two variables, typically denoted as $x$ and $y$. The general form of an exponential function is $y = a \cdot b^x$, where $a$ is the initial value, $b$ is the base, and $x$ is the exponent.

**Q: What is the difference between an exponential function and a linear function?**
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A: An exponential function grows or decays at a rate that is proportional to its current value, whereas a linear function grows or decays at a constant rate. For example, the function $y = 2^x$ is an exponential function, while the function $y = 2x$ is a linear function.

**Q: How do I choose the scale for the axes when creating a graph?**
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A: When creating a graph, it is essential to choose an appropriate scale for the axes. Consider the following factors:

* The range of values for both $x$ and $y$
* The units of measurement for both $x$ and $y$
* The type of graph you want to create (e.g., linear, logarithmic, etc.)

**Q: What is a logarithmic scale, and how is it used in graphing?**
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A: A logarithmic scale is a scale that uses logarithmic units instead of linear units. It is used to represent exponential functions in a more intuitive way. When using a logarithmic scale, the $y$-axis is divided into equal intervals, with each interval representing a power of the base.

**Q: How do I create a graph of an exponential function?**
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A: To create a graph of an exponential function, follow these steps:

1. Calculate the values of the function for a range of values of the exponent.
2. Choose an appropriate scale for the axes.
3. Plot the points on the graph.
4. Use a logarithmic scale for the $y$-axis if the function is exponential.

**Q: What is the significance of the base in an exponential function?**
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A: The base of an exponential function determines the rate at which the function grows or decays. A base greater than 1 will result in a function that grows exponentially, while a base less than 1 will result in a function that decays exponentially.

**Q: Can I use a logarithmic scale for the $x$-axis as well?**
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A: Yes, you can use a logarithmic scale for the $x$-axis as well. However, this is less common and is typically used when the function is periodic or has a specific symmetry.

**Q: How do I determine the type of graph to use for an exponential function?**
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A: The type of graph to use for an exponential function depends on the specific function and the range of values for the exponent. A linear scale is typically used for small ranges of values, while a logarithmic scale is used for larger ranges.

**Q: Can I use a graphing calculator to create a graph of an exponential function?**
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A: Yes, you can use a graphing calculator to create a graph of an exponential function. Graphing calculators can automatically choose the scale for the axes and plot the points on the graph.

**Q: What are some common applications of exponential functions in real-life situations?**
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A: Exponential functions have many applications in real-life situations, including:

* Population growth and decay
* Financial calculations (e.g., compound interest)
* Chemical reactions
* Electrical circuits
* Medical modeling (e.g., disease spread)

**Conclusion**
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In this Q&A article, we have explored the concept of exponential functions and graphing. We have discussed the importance of choosing an appropriate scale for the axes, the use of logarithmic scales, and the significance of the base in an exponential function. We have also provided examples of common applications of exponential functions in real-life situations.