Find The Graph Of The Following Exponential Function:$ Y = -3 \cdot \left( \frac{1}{2} \right)^x $

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Introduction

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Exponential functions are a fundamental concept in mathematics, and graphing them is an essential skill for any math enthusiast. In this article, we will explore the graph of the exponential function $y = -3 \cdot \left( \frac{1}{2} \right)^x$. We will delve into the properties of exponential functions, how to identify key features of the graph, and provide a step-by-step guide on how to graph this specific function.

Understanding Exponential Functions

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Exponential functions have the general form $y = a \cdot b^x$, where $a$ is the coefficient and $b$ is the base. The base $b$ can be any positive real number, and the coefficient $a$ can be any real number. The graph of an exponential function is characterized by its rapid growth or decay, depending on the value of $b$.

Properties of Exponential Functions

• Domain: The domain of an exponential function is all real numbers, $(-\infty, \infty)$.
• Range: The range of an exponential function depends on the value of $a$. If $a > 0$, the range is $(0, \infty)$. If $a < 0$, the range is $(-\infty, 0)$.
• Asymptotes: Exponential functions have no vertical asymptotes, but they may have horizontal asymptotes.
• Intercepts: Exponential functions have a single $y$-intercept, which is the point where $x = 0$.

Graphing the Exponential Function $y = -3 \cdot \left( \frac{1}{2} \right)^x$

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To graph the exponential function $y = -3 \cdot \left( \frac{1}{2} \right)^x$, we need to identify its key features.

Identifying Key Features

• Domain: The domain of this function is all real numbers, $(-\infty, \infty)$.
• Range: Since $a = -3 < 0$, the range is $(-\infty, 0)$.
• Asymptotes: This function has no vertical asymptotes, but it has a horizontal asymptote at $y = 0$.
• Intercepts: The $y$-intercept is the point where $x = 0$. To find the $y$-intercept, we substitute $x = 0$ into the function: $y = -3 \cdot \left( \frac{1}{2} \right)^0 = -3$.

Graphing the Function

To graph the function, we can use the following steps:

1. Plot the $y$-intercept: The $y$-intercept is the point $(0, -3)$.
2. Plot a few points: We can choose a few values of $x$ and calculate the corresponding values of $y$ to plot additional points on the graph.
3. Draw the graph: Using the points we have plotted, we can draw the graph of the function.

Step-by-Step Guide to Graphing the Function

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Here is a step-by-step guide to graphing the function $y = -3 \cdot \left( \frac{1}{2} \right)^x$:

Step 1: Plot the $y$-intercept

The $y$-intercept is the point where $x = 0$. To find the $y$-intercept, we substitute $x = 0$ into the function: $y = -3 \cdot \left( \frac{1}{2} \right)^0 = -3$. The $y$-intercept is the point $(0, -3)$.

Step 2: Plot a few points

We can choose a few values of $x$ and calculate the corresponding values of $y$ to plot additional points on the graph. Let's choose $x = -2, -1, 1,$ and $2$.

• For $x = -2$, we have $y = -3 \cdot \left( \frac{1}{2} \right)^{-2} = -3 \cdot 4 = -12$.
• For $x = -1$, we have $y = -3 \cdot \left( \frac{1}{2} \right)^{-1} = -3 \cdot 2 = -6$.
• For $x = 1$, we have $y = -3 \cdot \left( \frac{1}{2} \right)^1 = -3 \cdot \frac{1}{2} = -\frac{3}{2}$.
• For $x = 2$, we have $y = -3 \cdot \left( \frac{1}{2} \right)^2 = -3 \cdot \frac{1}{4} = -\frac{3}{4}$.

Step 3: Draw the graph

Using the points we have plotted, we can draw the graph of the function. The graph will be a decreasing curve that approaches the horizontal asymptote $y = 0$ as $x$ increases.

Conclusion

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Graphing exponential functions is an essential skill for any math enthusiast. In this article, we explored the graph of the exponential function $y = -3 \cdot \left( \frac{1}{2} \right)^x$. We identified its key features, including its domain, range, asymptotes, and intercepts. We also provided a step-by-step guide on how to graph the function. By following these steps, you can graph any exponential function and gain a deeper understanding of its properties.

Frequently Asked Questions

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Q: What is the domain of the exponential function $y = -3 \cdot \left( \frac{1}{2} \right)^x$?

A: The domain of this function is all real numbers, $(-\infty, \infty)$.

Q: What is the range of the exponential function $y = -3 \cdot \left( \frac{1}{2} \right)^x$?

A: Since $a = -3 < 0$, the range is $(-\infty, 0)$.

Q: What is the $y$-intercept of the exponential function $y = -3 \cdot \left( \frac{1}{2} \right)^x$?

A: The $y$-intercept is the point $(0, -3)$.

Q: How do I graph the exponential function $y = -3 \cdot \left( \frac{1}{2} \right)^x$?

A: To graph the function, you can follow the steps outlined in this article: plot the $y$-intercept, plot a few points, and draw the graph using the points you have plotted.

References

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• [1] "Exponential Functions." Math Open Reference, mathopenref.com/expfunc.html.
• [2] "Graphing Exponential Functions." Math Is Fun, mathisfun.com/algebra/exponential-functions-graph.html.
  

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Introduction

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Graphing exponential functions is an essential skill for any math enthusiast. In our previous article, we explored the graph of the exponential function $y = -3 \cdot \left( \frac{1}{2} \right)^x$. In this article, we will provide a Q&A guide to help you better understand the graphing of exponential functions.

Q&A Guide

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Q: What is the domain of the exponential function $y = a \cdot b^x$?

A: The domain of the exponential function $y = a \cdot b^x$ is all real numbers, $(-\infty, \infty)$.

Q: What is the range of the exponential function $y = a \cdot b^x$?

A: The range of the exponential function $y = a \cdot b^x$ depends on the value of $a$. If $a > 0$, the range is $(0, \infty)$. If $a < 0$, the range is $(-\infty, 0)$.

Q: What is the $y$-intercept of the exponential function $y = a \cdot b^x$?

A: The $y$-intercept of the exponential function $y = a \cdot b^x$ is the point $(0, a)$.

Q: How do I graph the exponential function $y = a \cdot b^x$?

A: To graph the exponential function $y = a \cdot b^x$, you can follow these steps:

1. Plot the $y$-intercept: The $y$-intercept is the point $(0, a)$.
2. Plot a few points: Choose a few values of $x$ and calculate the corresponding values of $y$ to plot additional points on the graph.
3. Draw the graph: Using the points you have plotted, draw the graph of the function.

Q: What is the difference between a linear and exponential function?

A: A linear function has a constant rate of change, while an exponential function has a constant ratio of change. This means that a linear function will always increase or decrease at a constant rate, while an exponential function will increase or decrease at a rate that is proportional to the current value.

Q: Can an exponential function have a horizontal asymptote?

A: Yes, an exponential function can have a horizontal asymptote. If the base $b$ is between 0 and 1, the function will approach the horizontal asymptote $y = 0$ as $x$ increases.

Q: Can an exponential function have a vertical asymptote?

A: No, an exponential function cannot have a vertical asymptote. However, it can have a hole in the graph if the base $b$ is a rational number and the exponent $x$ is an integer.

Q: How do I determine the type of exponential function (growth or decay)?

A: To determine the type of exponential function, you can look at the coefficient $a$. If $a > 0$, the function is a growth function. If $a < 0$, the function is a decay function.

Conclusion

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Graphing exponential functions is an essential skill for any math enthusiast. In this article, we provided a Q&A guide to help you better understand the graphing of exponential functions. We covered topics such as the domain and range of exponential functions, the $y$-intercept, and how to graph exponential functions. We also discussed the differences between linear and exponential functions, and how to determine the type of exponential function (growth or decay).

Frequently Asked Questions

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Q: What is the domain of the exponential function $y = a \cdot b^x$?

A: The domain of the exponential function $y = a \cdot b^x$ is all real numbers, $(-\infty, \infty)$.

Q: What is the range of the exponential function $y = a \cdot b^x$?

A: The range of the exponential function $y = a \cdot b^x$ depends on the value of $a$. If $a > 0$, the range is $(0, \infty)$. If $a < 0$, the range is $(-\infty, 0)$.

Q: What is the $y$-intercept of the exponential function $y = a \cdot b^x$?

A: The $y$-intercept of the exponential function $y = a \cdot b^x$ is the point $(0, a)$.

Q: How do I graph the exponential function $y = a \cdot b^x$?

A: To graph the exponential function $y = a \cdot b^x$, you can follow the steps outlined in this article: plot the $y$-intercept, plot a few points, and draw the graph using the points you have plotted.

References

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• [1] "Exponential Functions." Math Open Reference, mathopenref.com/expfunc.html.
• [2] "Graphing Exponential Functions." Math Is Fun, mathisfun.com/algebra/exponential-functions-graph.html.