The Function { K $}$ Is Constructed By Applying Three Transformations To The Graph Of { H $}$ In This Order: A Horizontal Translation By -4 Units, A Vertical Dilation By A Factor Of 2, And A Vertical Translation By 3 Units. Which

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Introduction

In mathematics, functions are used to describe the relationship between variables and their behavior under different transformations. The function { k $}$ is constructed by applying three transformations to the graph of { h $}$, which includes a horizontal translation, a vertical dilation, and a vertical translation. In this article, we will explore the impact of these transformations on the graph of { h $}$ and understand how they affect the function { k $}$.

Horizontal Translation

A horizontal translation is a transformation that shifts the graph of a function to the left or right by a certain number of units. In the case of the function { k $}$, the graph of { h $}$ is translated horizontally by -4 units. This means that for every point (x, y) on the graph of { h $}$, the corresponding point on the graph of { k $}$ is (x + 4, y).

Vertical Dilation

A vertical dilation is a transformation that stretches or compresses the graph of a function vertically by a certain factor. In the case of the function { k $}$, the graph of { h $}$ is dilated vertically by a factor of 2. This means that for every point (x, y) on the graph of { h $}$, the corresponding point on the graph of { k $}$ is (x, 2y).

Vertical Translation

A vertical translation is a transformation that shifts the graph of a function up or down by a certain number of units. In the case of the function { k $}$, the graph of { h $}$ is translated vertically by 3 units. This means that for every point (x, y) on the graph of { h $}$, the corresponding point on the graph of { k $}$ is (x, y + 3).

Combining the Transformations

When we combine the three transformations, we get the following result:

  • The graph of { h $}$ is translated horizontally by -4 units.
  • The resulting graph is then dilated vertically by a factor of 2.
  • The resulting graph is then translated vertically by 3 units.

The Final Result

The final result of the three transformations is a new function { k $}$ that is constructed by applying the transformations to the graph of { h $}$. The graph of { k $}$ is a vertical dilation of the graph of { h $}$ by a factor of 2, followed by a vertical translation by 3 units, and finally a horizontal translation by -4 units.

Example

Let's consider an example to illustrate the impact of the transformations on the graph of { h $}$. Suppose we have a function { h $}$ defined by the equation y = x^2. We can apply the three transformations to the graph of { h $}$ to get the graph of { k $}$.

  • First, we translate the graph of { h $}$ horizontally by -4 units. This gives us the equation y = (x + 4)^2.
  • Next, we dilate the resulting graph vertically by a factor of 2. This gives us the equation y = 2(x + 4)^2.
  • Finally, we translate the resulting graph vertically by 3 units. This gives us the equation y = 2(x + 4)^2 + 3.

Conclusion

In conclusion, the function { k $}$ is constructed by applying three transformations to the graph of { h $}$. The transformations include a horizontal translation by -4 units, a vertical dilation by a factor of 2, and a vertical translation by 3 units. By understanding the impact of these transformations on the graph of { h $}$, we can construct the graph of { k $}$ and analyze its behavior.

Understanding the Graph of { k $}$

The graph of { k $}$ is a vertical dilation of the graph of { h $}$ by a factor of 2, followed by a vertical translation by 3 units, and finally a horizontal translation by -4 units. This means that the graph of { k $}$ is a stretched and shifted version of the graph of { h $}$.

Key Takeaways

  • The function { k $}$ is constructed by applying three transformations to the graph of { h $}$.
  • The transformations include a horizontal translation by -4 units, a vertical dilation by a factor of 2, and a vertical translation by 3 units.
  • The graph of { k $}$ is a vertical dilation of the graph of { h $}$ by a factor of 2, followed by a vertical translation by 3 units, and finally a horizontal translation by -4 units.

Real-World Applications

The function { k $}$ has several real-world applications, including:

  • Physics: The function { k $}$ can be used to model the motion of objects under the influence of gravity.
  • Engineering: The function { k $}$ can be used to design and optimize systems, such as bridges and buildings.
  • Computer Science: The function { k $}$ can be used to develop algorithms and models for data analysis and machine learning.

Conclusion

Frequently Asked Questions

Q: What is the function { k $}$ and how is it constructed? A: The function { k $}$ is constructed by applying three transformations to the graph of { h $}$. The transformations include a horizontal translation by -4 units, a vertical dilation by a factor of 2, and a vertical translation by 3 units.

Q: What is the impact of the horizontal translation on the graph of { h $}$? A: The horizontal translation shifts the graph of { h $}$ to the left by 4 units. This means that for every point (x, y) on the graph of { h $}$, the corresponding point on the graph of { k $}$ is (x + 4, y).

Q: What is the impact of the vertical dilation on the graph of { h $}$? A: The vertical dilation stretches the graph of { h $}$ vertically by a factor of 2. This means that for every point (x, y) on the graph of { h $}$, the corresponding point on the graph of { k $}$ is (x, 2y).

Q: What is the impact of the vertical translation on the graph of { h $}$? A: The vertical translation shifts the graph of { h $}$ up by 3 units. This means that for every point (x, y) on the graph of { h $}$, the corresponding point on the graph of { k $}$ is (x, y + 3).

Q: How do the transformations affect the graph of { h $}$? A: The transformations affect the graph of { h $}$ by stretching and shifting it. The horizontal translation shifts the graph to the left, the vertical dilation stretches the graph vertically, and the vertical translation shifts the graph up.

Q: What is the final result of the three transformations? A: The final result of the three transformations is a new function { k $}$ that is constructed by applying the transformations to the graph of { h $}$. The graph of { k $}$ is a vertical dilation of the graph of { h $}$ by a factor of 2, followed by a vertical translation by 3 units, and finally a horizontal translation by -4 units.

Q: What are some real-world applications of the function { k $}$? A: The function { k $}$ has several real-world applications, including physics, engineering, and computer science. It can be used to model the motion of objects under the influence of gravity, design and optimize systems, and develop algorithms and models for data analysis and machine learning.

Q: How can I visualize the graph of { k $}$? A: You can visualize the graph of { k $}$ by using a graphing calculator or a computer program to plot the function. You can also use a graphing app or a spreadsheet to create a table of values and plot the function.

Q: What are some common mistakes to avoid when working with the function { k $}$? A: Some common mistakes to avoid when working with the function { k $}$ include:

  • Not applying the transformations in the correct order.
  • Not using the correct values for the transformations.
  • Not checking the units and dimensions of the function.
  • Not using the correct notation and terminology.

Q: How can I learn more about the function { k $}$? A: You can learn more about the function { k $}$ by:

  • Reading books and articles on the subject.
  • Taking online courses or tutorials.
  • Practicing problems and exercises.
  • Joining online communities and forums.
  • Seeking help from a teacher or tutor.