Select The Correct Answer.Consider The Function $f(x)=10^x$ And Function $g(x)=f(x+4$\].How Will The Graph Of Function $g$ Differ From The Graph Of Function $f$?A. The Graph Of Function $g$ Is The Graph Of

When dealing with functions, understanding how they behave and how their graphs are affected by transformations is crucial. In this article, we will explore how the graph of a function changes when it undergoes a horizontal shift. We will consider the function $f(x)=10^x$ and its transformation $g(x)=f(x+4)$ to understand how the graph of $g$ differs from the graph of $f$.

What is a Horizontal Shift?

A horizontal shift is a transformation that moves the graph of a function to the left or right. This type of shift is achieved by changing the input of the function, which in turn affects the output. In the case of the function $f(x)=10^x$, a horizontal shift to the left by 4 units would result in the function $g(x)=f(x+4)$.

Understanding the Function $f(x)=10^x$

The function $f(x)=10^x$ is an exponential function that represents a curve that increases rapidly as $x$ increases. This function has a base of 10 and an exponent of $x$. The graph of this function is a continuous, smooth curve that passes through the point (0, 1).

Understanding the Function $g(x)=f(x+4)$

The function $g(x)=f(x+4)$ is a transformation of the function $f(x)=10^x$. This transformation shifts the graph of $f(x)$ to the left by 4 units. In other words, the input of the function $f(x)$ is increased by 4, resulting in a new function $g(x)$.

How Does the Graph of $g$ Differ from the Graph of $f$?

The graph of $g(x)=f(x+4)$ differs from the graph of $f(x)=10^x$ in that it is shifted to the left by 4 units. This means that for every point $(x, y)$ on the graph of $f(x)$, there is a corresponding point $(x-4, y)$ on the graph of $g(x)$. In other words, the graph of $g(x)$ is the same as the graph of $f(x)$, but shifted to the left by 4 units.

Visualizing the Shift

To visualize the shift, let's consider a few points on the graph of $f(x)=10^x$. For example, the point (1, 10) is on the graph of $f(x)$. This means that the point (1-4, 10) = (-3, 10) is on the graph of $g(x)=f(x+4)$. Similarly, the point (2, 100) is on the graph of $f(x)$, which means that the point (2-4, 100) = (-2, 100) is on the graph of $g(x)$.

Conclusion

In conclusion, the graph of the function $g(x)=f(x+4)$ differs from the graph of the function $f(x)=10^x$ in that it is shifted to the left by 4 units. This transformation changes the input of the function, resulting in a new function $g(x)$ that is identical to $f(x)$ but shifted to the left by 4 units.

Key Takeaways

• A horizontal shift is a transformation that moves the graph of a function to the left or right.
• The function $g(x)=f(x+4)$ is a transformation of the function $f(x)=10^x$ that shifts the graph of $f(x)$ to the left by 4 units.
• The graph of $g(x)$ is the same as the graph of $f(x)$, but shifted to the left by 4 units.

Frequently Asked Questions

Q: What is a horizontal shift?

A: A horizontal shift is a transformation that moves the graph of a function to the left or right.

Q: How does the graph of $g(x)=f(x+4)$ differ from the graph of $f(x)=10^x$?

A: The graph of $g(x)$ is the same as the graph of $f(x)$, but shifted to the left by 4 units.

Q: What is the effect of a horizontal shift on the graph of a function?

A: A horizontal shift changes the input of the function, resulting in a new function that is identical to the original function but shifted to the left or right.

Q: How can I visualize the shift in the graph of a function?

A: You can visualize the shift by considering a few points on the graph of the original function and finding the corresponding points on the graph of the transformed function.

References

• [1] "Functions and Graphs" by Michael Corral. Available at https://math.libretexts.org/Bookshelves/Calculus/Book%3A_Calculus_(Corral)/02%3A_Functions/2.1%3A_Functions_and_Graphs
• [2] "Graphing Functions" by Paul's Online Math Notes. Available at https://tutorial.math.lamar.edu/Classes/CalcI/GraphingFunctions.aspx

Additional Resources

• Khan Academy: Functions and Graphs
• MIT OpenCourseWare: Calculus
• Wolfram Alpha: Graphing Functions
  Q&A: Understanding the Shift in Graphs of Functions =====================================================

In our previous article, we explored how the graph of a function changes when it undergoes a horizontal shift. We considered the function $f(x)=10^x$ and its transformation $g(x)=f(x+4)$ to understand how the graph of $g$ differs from the graph of $f$. In this article, we will answer some frequently asked questions about horizontal shifts and provide additional resources for further learning.

Q: What is a horizontal shift?

A: A horizontal shift is a transformation that moves the graph of a function to the left or right. This type of shift is achieved by changing the input of the function, which in turn affects the output.

Q: How does the graph of $g(x)=f(x+4)$ differ from the graph of $f(x)=10^x$?

A: The graph of $g(x)$ is the same as the graph of $f(x)$, but shifted to the left by 4 units. This means that for every point $(x, y)$ on the graph of $f(x)$, there is a corresponding point $(x-4, y)$ on the graph of $g(x)$.

Q: What is the effect of a horizontal shift on the graph of a function?

A: A horizontal shift changes the input of the function, resulting in a new function that is identical to the original function but shifted to the left or right.

Q: How can I visualize the shift in the graph of a function?

A: You can visualize the shift by considering a few points on the graph of the original function and finding the corresponding points on the graph of the transformed function.

Q: What are some common types of horizontal shifts?

A: There are two common types of horizontal shifts:

• Left shift: This type of shift moves the graph of a function to the left by a certain number of units.
• Right shift: This type of shift moves the graph of a function to the right by a certain number of units.

Q: How do I determine the type of horizontal shift?

A: To determine the type of horizontal shift, you need to look at the transformation of the function. If the input of the function is increased by a certain number of units, the graph of the function will be shifted to the left. If the input of the function is decreased by a certain number of units, the graph of the function will be shifted to the right.

Q: Can I apply multiple horizontal shifts to a function?

A: Yes, you can apply multiple horizontal shifts to a function. Each horizontal shift will move the graph of the function to the left or right by a certain number of units.

Q: How do I graph a function with multiple horizontal shifts?

A: To graph a function with multiple horizontal shifts, you need to apply each shift in sequence. For example, if you want to graph the function $f(x+2)+3$, you need to first apply the horizontal shift $f(x+2)$ and then apply the vertical shift $+3$.

Q: What are some real-world applications of horizontal shifts?

A: Horizontal shifts have many real-world applications, including:

• Physics: Horizontal shifts are used to model the motion of objects in physics.
• Engineering: Horizontal shifts are used to design and optimize systems in engineering.
• Computer Science: Horizontal shifts are used in computer graphics and game development.

Conclusion

In conclusion, horizontal shifts are an important concept in mathematics that can be used to model and analyze real-world phenomena. By understanding how to apply horizontal shifts to functions, you can gain a deeper understanding of the behavior of functions and their graphs.

Additional Resources

• Khan Academy: Functions and Graphs
• MIT OpenCourseWare: Calculus
• Wolfram Alpha: Graphing Functions
• "Functions and Graphs" by Michael Corral
• "Graphing Functions" by Paul's Online Math Notes

Frequently Asked Questions

Q: What is a horizontal shift?

A: A horizontal shift is a transformation that moves the graph of a function to the left or right.

Q: How does the graph of $g(x)=f(x+4)$ differ from the graph of $f(x)=10^x$?

A: The graph of $g(x)$ is the same as the graph of $f(x)$, but shifted to the left by 4 units.

Q: What is the effect of a horizontal shift on the graph of a function?

A: A horizontal shift changes the input of the function, resulting in a new function that is identical to the original function but shifted to the left or right.

Q: How can I visualize the shift in the graph of a function?

A: You can visualize the shift by considering a few points on the graph of the original function and finding the corresponding points on the graph of the transformed function.

Q: What are some common types of horizontal shifts?

A: There are two common types of horizontal shifts: left shift and right shift.

Q: How do I determine the type of horizontal shift?

A: To determine the type of horizontal shift, you need to look at the transformation of the function.

Q: Can I apply multiple horizontal shifts to a function?

A: Yes, you can apply multiple horizontal shifts to a function.

Q: How do I graph a function with multiple horizontal shifts?

A: To graph a function with multiple horizontal shifts, you need to apply each shift in sequence.

Q: What are some real-world applications of horizontal shifts?

A: Horizontal shifts have many real-world applications, including physics, engineering, and computer science.