Which Sequence Of Transformations Takes The Graph Of Y = K ( X Y=k(x Y = K ( X ] To The Graph Of Y = − 1 2 K ( X + 1 Y=-\frac{1}{2} K(x+1 Y = − 2 1 ​ K ( X + 1 ]?A. Translate 1 To The Left, Scale Vertically By 1 2 \frac{1}{2} 2 1 ​ , Then Reflect Over The X X X -axis. B. Translate Left

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Introduction

In mathematics, transformations of graphs are essential concepts that help us understand how functions change under various operations. When we apply a sequence of transformations to a graph, we can obtain a new graph that represents a different function. In this article, we will explore the sequence of transformations that takes the graph of y=k(x)y=k(x) to the graph of y=−12k(x+1)y=-\frac{1}{2} k(x+1).

Understanding the Original Graph

The original graph is represented by the equation y=k(x)y=k(x). This is a linear function with a slope of kk and a y-intercept of 00. The graph of this function is a straight line that passes through the origin.

Understanding the Target Graph

The target graph is represented by the equation y=−12k(x+1)y=-\frac{1}{2} k(x+1). This is also a linear function, but with a slope of −12k-\frac{1}{2} k and a y-intercept of −12k-\frac{1}{2} k. The graph of this function is a straight line that passes through the point (−1,−12k)(-1, -\frac{1}{2} k).

Step 1: Translation

The first step in transforming the original graph to the target graph is to translate it to the left by 11 unit. This means that we need to replace xx with x+1x+1 in the original equation. The new equation becomes y=k(x+1)y=k(x+1).

# Translation
## Translation to the left by 1 unit

The original equation is: y = k(x)
The translated equation is: y = k(x+1)

Step 2: Scaling

The next step is to scale the translated graph vertically by 12\frac{1}{2}. This means that we need to multiply the entire equation by 12\frac{1}{2}. The new equation becomes y=12k(x+1)y=\frac{1}{2} k(x+1).

# Scaling
## Scaling vertically by 1/2

The translated equation is: y = k(x+1)
The scaled equation is: y = 1/2 k(x+1)

Step 3: Reflection

The final step is to reflect the scaled graph over the xx-axis. This means that we need to multiply the entire equation by −1-1. The new equation becomes y=−12k(x+1)y=-\frac{1}{2} k(x+1).

# Reflection
## Reflection over the x-axis

The scaled equation is: y = 1/2 k(x+1)
The reflected equation is: y = -1/2 k(x+1)

Conclusion

In conclusion, the sequence of transformations that takes the graph of y=k(x)y=k(x) to the graph of y=−12k(x+1)y=-\frac{1}{2} k(x+1) is:

  1. Translate to the left by 11 unit.
  2. Scale vertically by 12\frac{1}{2}.
  3. Reflect over the xx-axis.

This sequence of transformations can be represented by the following equation:

y=−12k(x+1)y=-\frac{1}{2} k(x+1)

Answer

The correct answer is:

A. Translate 11 to the left, scale vertically by 12\frac{1}{2}, then reflect over the xx-axis.

Discussion

The discussion category for this problem is mathematics. The problem requires the application of transformation concepts to obtain a new graph. The solution involves a sequence of three transformations: translation, scaling, and reflection. The correct answer is the sequence of transformations that takes the original graph to the target graph.

Additional Resources

For more information on transformations of graphs, please refer to the following resources:

  • Khan Academy: Graphing Lines and Functions
  • Mathway: Graphing Lines and Functions
  • Wolfram Alpha: Graphing Lines and Functions

References

  • [1] "Graphing Lines and Functions" by Khan Academy
  • [2] "Graphing Lines and Functions" by Mathway
  • [3] "Graphing Lines and Functions" by Wolfram Alpha
    Transformations of Graphs: A Q&A Guide =====================================

Introduction

In our previous article, we explored the sequence of transformations that takes the graph of y=k(x)y=k(x) to the graph of y=−12k(x+1)y=-\frac{1}{2} k(x+1). In this article, we will provide a Q&A guide to help you better understand the concepts of transformations of graphs.

Q: What is a transformation of a graph?

A: A transformation of a graph is a change in the position, size, or orientation of the graph. This can include translations, scaling, reflections, and rotations.

Q: What are the different types of transformations?

A: The different types of transformations are:

  • Translation: A translation is a change in the position of the graph. This can be a horizontal or vertical translation.
  • Scaling: A scaling is a change in the size of the graph. This can be a horizontal or vertical scaling.
  • Reflection: A reflection is a change in the orientation of the graph. This can be a reflection over the x-axis or y-axis.
  • Rotation: A rotation is a change in the orientation of the graph. This can be a rotation around the origin or a rotation around a point.

Q: How do I apply a transformation to a graph?

A: To apply a transformation to a graph, you need to follow these steps:

  1. Identify the type of transformation you want to apply.
  2. Determine the direction and magnitude of the transformation.
  3. Apply the transformation to the graph.

Q: What is the order of operations for transformations?

A: The order of operations for transformations is:

  1. Translation: Apply any translations to the graph.
  2. Scaling: Apply any scaling to the graph.
  3. Reflection: Apply any reflections to the graph.
  4. Rotation: Apply any rotations to the graph.

Q: How do I determine the equation of a transformed graph?

A: To determine the equation of a transformed graph, you need to follow these steps:

  1. Identify the original equation of the graph.
  2. Apply the transformation to the equation.
  3. Simplify the equation to obtain the final equation.

Q: What are some common transformations?

A: Some common transformations include:

  • Horizontal translation: y=f(x−h)y = f(x - h)
  • Vertical translation: y=f(x)+ky = f(x) + k
  • Horizontal scaling: y=f(ax)y = f(ax)
  • Vertical scaling: y=af(x)y = af(x)
  • Reflection over the x-axis: y=−f(x)y = -f(x)
  • Reflection over the y-axis: y=f(−x)y = f(-x)

Q: How do I graph a transformed function?

A: To graph a transformed function, you need to follow these steps:

  1. Graph the original function.
  2. Apply the transformation to the graph.
  3. Label the transformed graph.

Conclusion

In conclusion, transformations of graphs are an essential concept in mathematics. By understanding the different types of transformations and how to apply them, you can graph and analyze functions in a more efficient and effective way. We hope this Q&A guide has helped you better understand the concepts of transformations of graphs.

Additional Resources

For more information on transformations of graphs, please refer to the following resources:

  • Khan Academy: Graphing Lines and Functions
  • Mathway: Graphing Lines and Functions
  • Wolfram Alpha: Graphing Lines and Functions

References

  • [1] "Graphing Lines and Functions" by Khan Academy
  • [2] "Graphing Lines and Functions" by Mathway
  • [3] "Graphing Lines and Functions" by Wolfram Alpha