Graph The Exponential Function $y=0.5(6)^x$. Identify The $y$-intercept. A. 1 B. 0 C. -4 D. 0.5

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Introduction

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 exponential function $y=0.5(6)^x$ and identify its $y$-intercept. We will also provide a step-by-step guide on how to graph this function and analyze its properties.

What is an Exponential Function?

An exponential function is a mathematical function of the form $y=a^x$, where $a$ is a positive constant and $x$ is the variable. The graph of an exponential function is a curve that increases or decreases rapidly as $x$ increases or decreases. The graph of an exponential function can be either increasing or decreasing, depending on the value of $a$.

Graphing the Exponential Function

To graph the exponential function $y=0.5(6)^x$, we need to follow these steps:

  1. Identify the base and the coefficient: The base of the exponential function is $6$, and the coefficient is $0.5$.
  2. Determine the direction of the graph: Since the coefficient is positive, the graph of the exponential function will be increasing.
  3. Find the $y$-intercept: The $y$-intercept is the point where the graph intersects the $y$-axis. To find the $y$-intercept, we need to substitute $x=0$ into the equation.
  4. Plot the graph: Once we have the $y$-intercept, we can plot the graph by using a table of values or by using a graphing calculator.

Finding the $y$-Intercept

To find the $y$-intercept, we need to substitute $x=0$ into the equation:

y=0.5(6)0y=0.5(6)^0

Since any number raised to the power of $0$ is equal to $1$, we have:

y=0.5(1)y=0.5(1)

y=0.5y=0.5

Therefore, the $y$-intercept of the graph of the exponential function $y=0.5(6)^x$ is $0.5$.

Analyzing the Graph

The graph of the exponential function $y=0.5(6)^x$ is an increasing curve that passes through the point $(0, 0.5)$. As $x$ increases, the value of $y$ increases rapidly. The graph has a horizontal asymptote at $y=0$, which means that as $x$ approaches infinity, the value of $y$ approaches $0$.

Conclusion

In conclusion, graphing the exponential function $y=0.5(6)^x$ involves identifying the base and the coefficient, determining the direction of the graph, finding the $y$-intercept, and plotting the graph. The $y$-intercept of the graph is $0.5$, and the graph is an increasing curve that passes through the point $(0, 0.5)$. We hope that this article has provided a comprehensive guide on how to graph the exponential function and analyze its properties.

Frequently Asked Questions

  • What is the base of the exponential function?
    • The base of the exponential function is $6$.
  • What is the coefficient of the exponential function?
    • The coefficient of the exponential function is $0.5$.
  • What is the $y$-intercept of the graph of the exponential function?
    • The $y$-intercept of the graph of the exponential function is $0.5$.

References

  • [1] "Exponential Functions" by Math Open Reference
  • [2] "Graphing Exponential Functions" by Khan Academy

Additional Resources

  • [1] "Exponential Functions" by Wolfram MathWorld
  • [2] "Graphing Exponential Functions" by Purplemath
    Graphing the Exponential Function: A Comprehensive Guide ===========================================================

Q&A: Graphing the Exponential Function

Q: What is the base of the exponential function?

A: The base of the exponential function is 6.

Q: What is the coefficient of the exponential function?

A: The coefficient of the exponential function is 0.5.

Q: What is the y-intercept of the graph of the exponential function?

A: The y-intercept of the graph of the exponential function is 0.5.

Q: How do I determine the direction of the graph of the exponential function?

A: To determine the direction of the graph of the exponential function, you need to look at the coefficient. If the coefficient is positive, the graph will be increasing. If the coefficient is negative, the graph will be decreasing.

Q: How do I find the y-intercept of the graph of the exponential function?

A: To find the y-intercept of the graph of the exponential function, you need to substitute x=0 into the equation.

Q: What is the horizontal asymptote of the graph of the exponential function?

A: The horizontal asymptote of the graph of the exponential function is y=0.

Q: How do I graph the exponential function?

A: To graph the exponential function, you need to follow these steps:

  1. Identify the base and the coefficient.
  2. Determine the direction of the graph.
  3. Find the y-intercept.
  4. Plot the graph.

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

A: An exponential function is a function of the form y=a^x, where a is a positive constant and x is the variable. A linear function is a function of the form y=mx+b, where m is the slope and b is the y-intercept.

Q: Can I use a graphing calculator to graph the exponential function?

A: Yes, you can use a graphing calculator to graph the exponential function.

Q: What are some common mistakes to avoid when graphing the exponential function?

A: Some common mistakes to avoid when graphing the exponential function include:

  • Not identifying the base and the coefficient correctly.
  • Not determining the direction of the graph correctly.
  • Not finding the y-intercept correctly.
  • Not plotting the graph correctly.

Q: How do I check my work when graphing the exponential function?

A: To check your work when graphing the exponential function, you can use the following steps:

  1. Check that you have identified the base and the coefficient correctly.
  2. Check that you have determined the direction of the graph correctly.
  3. Check that you have found the y-intercept correctly.
  4. Check that you have plotted the graph correctly.

Conclusion

In conclusion, graphing the exponential function involves identifying the base and the coefficient, determining the direction of the graph, finding the y-intercept, and plotting the graph. We hope that this article has provided a comprehensive guide on how to graph the exponential function and answer some of the most frequently asked questions.

Frequently Asked Questions

  • What is the base of the exponential function?
    • The base of the exponential function is 6.
  • What is the coefficient of the exponential function?
    • The coefficient of the exponential function is 0.5.
  • What is the y-intercept of the graph of the exponential function?
    • The y-intercept of the graph of the exponential function is 0.5.

References

  • [1] "Exponential Functions" by Math Open Reference
  • [2] "Graphing Exponential Functions" by Khan Academy

Additional Resources

  • [1] "Exponential Functions" by Wolfram MathWorld
  • [2] "Graphing Exponential Functions" by Purplemath