How Is The Graph Of $y=(x-1)^2-3$ Transformed To Produce The Graph Of $y=\frac{1}{2}(x+4)^2$?A. The Graph Is Translated Left 5 Units, Compressed Vertically By A Factor Of $\frac{1}{2}$, And Translated Up 3 Units.B. The Graph

Introduction

Graph transformations are a crucial concept in mathematics, particularly in algebra and geometry. They involve changing the position, size, or orientation of a graph to create a new graph. In this article, we will explore how the graph of $y=(x-1)^2-3$ is transformed to produce the graph of $y=\frac{1}{2}(x+4)^2$. We will break down the transformation process into manageable steps and analyze the effects of each transformation on the graph.

Step 1: Horizontal Translation

The first step in transforming the graph of $y=(x-1)^2-3$ to $y=\frac{1}{2}(x+4)^2$ is to perform a horizontal translation. To do this, we need to compare the two equations and identify the differences in their x-coordinates.

The equation $y=(x-1)^2-3$ has an x-coordinate of $x-1$, while the equation $y=\frac{1}{2}(x+4)^2$ has an x-coordinate of $x+4$. This means that the graph of $y=\frac{1}{2}(x+4)^2$ is shifted to the left by 5 units compared to the graph of $y=(x-1)^2-3$.

Step 2: Vertical Compression

The next step is to analyze the coefficients of the squared terms in the two equations. In the equation $y=(x-1)^2-3$, the coefficient of the squared term is 1, while in the equation $y=\frac{1}{2}(x+4)^2$, the coefficient is $\frac{1}{2}$.

When the coefficient of the squared term is less than 1, the graph is compressed vertically. In this case, the graph of $y=\frac{1}{2}(x+4)^2$ is compressed vertically by a factor of $\frac{1}{2}$ compared to the graph of $y=(x-1)^2-3$.

Step 3: Vertical Translation

The final step is to analyze the constant terms in the two equations. In the equation $y=(x-1)^2-3$, the constant term is -3, while in the equation $y=\frac{1}{2}(x+4)^2$, the constant term is 0.

When the constant term is positive, the graph is shifted up. In this case, the graph of $y=\frac{1}{2}(x+4)^2$ is shifted up by 3 units compared to the graph of $y=(x-1)^2-3$.

Conclusion

In conclusion, the graph of $y=(x-1)^2-3$ is transformed to produce the graph of $y=\frac{1}{2}(x+4)^2$ by performing a horizontal translation to the left by 5 units, a vertical compression by a factor of $\frac{1}{2}$, and a vertical translation up by 3 units.

Step 1: Horizontal Translation

The first step in transforming the graph of $y=(x-1)^2-3$ to $y=\frac{1}{2}(x+4)^2$ is to perform a horizontal translation. To do this, we need to compare the two equations and identify the differences in their x-coordinates.

The equation $y=(x-1)^2-3$ has an x-coordinate of $x-1$, while the equation $y=\frac{1}{2}(x+4)^2$ has an x-coordinate of $x+4$. This means that the graph of $y=\frac{1}{2}(x+4)^2$ is shifted to the left by 5 units compared to the graph of $y=(x-1)^2-3$.

Step 2: Vertical Compression

The next step is to analyze the coefficients of the squared terms in the two equations. In the equation $y=(x-1)^2-3$, the coefficient of the squared term is 1, while in the equation $y=\frac{1}{2}(x+4)^2$, the coefficient is $\frac{1}{2}$.

When the coefficient of the squared term is less than 1, the graph is compressed vertically. In this case, the graph of $y=\frac{1}{2}(x+4)^2$ is compressed vertically by a factor of $\frac{1}{2}$ compared to the graph of $y=(x-1)^2-3$.

Step 3: Vertical Translation

The final step is to analyze the constant terms in the two equations. In the equation $y=(x-1)^2-3$, the constant term is -3, while in the equation $y=\frac{1}{2}(x+4)^2$, the constant term is 0.

When the constant term is positive, the graph is shifted up. In this case, the graph of $y=\frac{1}{2}(x+4)^2$ is shifted up by 3 units compared to the graph of $y=(x-1)^2-3$.

Conclusion

In conclusion, the graph of $y=(x-1)^2-3$ is transformed to produce the graph of $y=\frac{1}{2}(x+4)^2$ by performing a horizontal translation to the left by 5 units, a vertical compression by a factor of $\frac{1}{2}$, and a vertical translation up by 3 units.

Real-World Applications

Graph transformations have numerous real-world applications in fields such as engineering, physics, and computer science. For example, in engineering, graph transformations can be used to model the motion of objects, while in physics, they can be used to describe the behavior of particles in a system.

Tips and Tricks

When working with graph transformations, it's essential to remember the following tips and tricks:

• Always compare the two equations and identify the differences in their x-coordinates, coefficients, and constant terms.
• Use the correct notation and terminology when describing graph transformations.
• Practice, practice, practice! The more you practice graph transformations, the more comfortable you'll become with the process.

Conclusion

Frequently Asked Questions

Q: What is a graph transformation?

A: A graph transformation is a process of changing the position, size, or orientation of a graph to create a new graph. This can involve shifting the graph horizontally or vertically, compressing or expanding it vertically, or reflecting it across a line.

Q: How do I perform a horizontal translation?

A: To perform a horizontal translation, you need to compare the two equations and identify the differences in their x-coordinates. If the x-coordinate in the new equation is greater than the x-coordinate in the original equation, the graph is shifted to the right. If the x-coordinate in the new equation is less than the x-coordinate in the original equation, the graph is shifted to the left.

Q: How do I perform a vertical compression?

A: To perform a vertical compression, you need to compare the coefficients of the squared terms in the two equations. If the coefficient in the new equation is less than 1, the graph is compressed vertically. If the coefficient in the new equation is greater than 1, the graph is expanded vertically.

Q: How do I perform a vertical translation?

A: To perform a vertical translation, you need to compare the constant terms in the two equations. If the constant term in the new equation is greater than the constant term in the original equation, the graph is shifted up. If the constant term in the new equation is less than the constant term in the original equation, the graph is shifted down.

Q: What are some real-world applications of graph transformations?

A: Graph transformations have numerous real-world applications in fields such as engineering, physics, and computer science. For example, in engineering, graph transformations can be used to model the motion of objects, while in physics, they can be used to describe the behavior of particles in a system.

Q: How can I practice graph transformations?

A: You can practice graph transformations by working through examples and exercises in your textbook or online resources. You can also try creating your own graph transformations by modifying the equations of existing graphs.

Q: What are some common mistakes to avoid when performing graph transformations?

A: Some common mistakes to avoid when performing graph transformations include:

• Failing to compare the x-coordinates, coefficients, and constant terms in the two equations.
• Misinterpreting the direction of the translation or compression.
• Failing to account for the effects of multiple transformations on the graph.

Q: How can I use graph transformations to solve real-world problems?

A: Graph transformations can be used to solve real-world problems by modeling the behavior of complex systems. For example, you can use graph transformations to model the motion of a projectile, the behavior of a population of animals, or the flow of a fluid.

Conclusion

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Graph transformations are a fundamental concept in mathematics that have numerous real-world applications. By understanding how to perform horizontal translations, vertical compressions, and vertical translations, you can transform graphs to create new and interesting shapes. Remember to practice regularly and use the tips and tricks outlined in this article to become proficient in graph transformations.

Additional Resources

• Graph Transformations Tutorial
• Graph Transformations Examples
• Graph Transformations Practice Problems

Tips and Tricks

• Always compare the x-coordinates, coefficients, and constant terms in the two equations.
• Use the correct notation and terminology when describing graph transformations.
• Practice, practice, practice! The more you practice graph transformations, the more comfortable you'll become with the process.