Question 2 (1 Point)Which Of The Following Is The Correct Transformation Of The Graph Y = ( X − 2 ) 2 Y=(x-2)^2 Y = ( X − 2 ) 2 To Get The Graph Of Y = ( X − 2 ) 2 + 5 Y=(x-2)^2+5 Y = ( X − 2 ) 2 + 5 ?A. Translate 5 Units Down B. Translate 5 Units Up C. Translate 2 Units To The Right D. Translate 2

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 focus on understanding the correct transformation of the graph $y=(x-2)^2$ to get the graph of $y=(x-2)^2+5$.

What are Graph Transformations?

Graph transformations are a way of modifying a graph to create a new graph. There are several types of graph transformations, including:

• Translation: This involves moving the graph to a new position without changing its size or orientation.
• Dilation: This involves changing the size of the graph while keeping its position and orientation the same.
• Rotation: This involves rotating the graph around a fixed point.

Understanding the Graph $y=(x-2)^2$

The graph $y=(x-2)^2$ is a parabola with its vertex at the point (2, 0). The equation $(x-2)^2$ represents a quadratic function that is shifted 2 units to the right. The graph of this function is a parabola that opens upwards.

Transforming the Graph $y=(x-2)^2$ to $y=(x-2)^2+5$

To transform the graph $y=(x-2)^2$ to $y=(x-2)^2+5$, we need to understand the effect of adding 5 to the equation. When we add 5 to the equation, we are essentially shifting the graph 5 units up.

Why is this the Correct Transformation?

The correct transformation of the graph $y=(x-2)^2$ to $y=(x-2)^2+5$ is to translate it 5 units up. This is because adding 5 to the equation shifts the graph 5 units up, resulting in a new graph with the same shape and size as the original graph, but with a different position.

Analyzing the Options

Let's analyze the options given in the question:

• A. Translate 5 units down: This is incorrect because translating the graph 5 units down would result in a new graph with the same shape and size as the original graph, but with a different position. However, the new graph would be 5 units below the original graph, not above it.
• B. Translate 5 units up: This is the correct answer because translating the graph 5 units up would result in a new graph with the same shape and size as the original graph, but with a different position. The new graph would be 5 units above the original graph.
• C. Translate 2 units to the right: This is incorrect because translating the graph 2 units to the right would result in a new graph with the same shape and size as the original graph, but with a different position. However, the new graph would be 2 units to the right of the original graph, not above it.
• D. Translate 2 units to the left: This is incorrect because translating the graph 2 units to the left would result in a new graph with the same shape and size as the original graph, but with a different position. However, the new graph would be 2 units to the left of the original graph, not above it.

Conclusion

In conclusion, the correct transformation of the graph $y=(x-2)^2$ to get the graph of $y=(x-2)^2+5$ is to translate it 5 units up. This is because adding 5 to the equation shifts the graph 5 units up, resulting in a new graph with the same shape and size as the original graph, but with a different position.

Key Takeaways

• Graph transformations are a crucial concept in mathematics, particularly in algebra and geometry.
• There are several types of graph transformations, including translation, dilation, and rotation.
• The correct transformation of the graph $y=(x-2)^2$ to get the graph of $y=(x-2)^2+5$ is to translate it 5 units up.
• Adding 5 to the equation shifts the graph 5 units up, resulting in a new graph with the same shape and size as the original graph, but with a different position.
  Graph Transformations Q&A ==========================

Q: What is the difference between a translation and a dilation?

A: A translation involves moving a graph to a new position without changing its size or orientation, while a dilation involves changing the size of a graph while keeping its position and orientation the same.

Q: How do you determine the type of transformation that has occurred?

A: To determine the type of transformation that has occurred, you need to analyze the equation and the resulting graph. If the equation has changed, but the graph has not, then a translation has occurred. If the equation has not changed, but the graph has, then a dilation has occurred.

Q: What is the effect of adding a constant to the equation of a graph?

A: Adding a constant to the equation of a graph shifts the graph up or down, depending on the sign of the constant. If the constant is positive, the graph is shifted up. If the constant is negative, the graph is shifted down.

Q: What is the effect of subtracting a constant from the equation of a graph?

A: Subtracting a constant from the equation of a graph shifts the graph down or up, depending on the sign of the constant. If the constant is positive, the graph is shifted down. If the constant is negative, the graph is shifted up.

Q: How do you determine the direction of a translation?

A: To determine the direction of a translation, you need to analyze the equation and the resulting graph. If the equation has changed, but the graph has not, then a translation has occurred. If the equation has changed, and the graph has moved to the right or left, then the translation is horizontal. If the equation has changed, and the graph has moved up or down, then the translation is vertical.

Q: What is the effect of multiplying or dividing the equation of a graph by a constant?

A: Multiplying or dividing the equation of a graph by a constant changes the size of the graph, while keeping its position and orientation the same. If the constant is greater than 1, the graph is enlarged. If the constant is less than 1, the graph is shrunk.

Q: How do you determine the type of dilation that has occurred?

A: To determine the type of dilation that has occurred, you need to analyze the equation and the resulting graph. If the equation has changed, but the graph has not, then a dilation has occurred. If the equation has changed, and the graph has been enlarged or shrunk, then the dilation is vertical. If the equation has changed, and the graph has been stretched or compressed, then the dilation is horizontal.

Q: What is the effect of rotating a graph around a fixed point?

A: Rotating a graph around a fixed point changes the orientation of the graph, while keeping its position and size the same. The fixed point is called the center of rotation.

Q: How do you determine the angle of rotation?

A: To determine the angle of rotation, you need to analyze the equation and the resulting graph. If the equation has changed, but the graph has not, then a rotation has occurred. If the equation has changed, and the graph has been rotated, then the angle of rotation can be determined by analyzing the equation.

Q: What is the effect of reflecting a graph across a line?

A: Reflecting a graph across a line changes the orientation of the graph, while keeping its position and size the same. The line of reflection is called the axis of reflection.

Q: How do you determine the axis of reflection?

A: To determine the axis of reflection, you need to analyze the equation and the resulting graph. If the equation has changed, but the graph has not, then a reflection has occurred. If the equation has changed, and the graph has been reflected, then the axis of reflection can be determined by analyzing the equation.

Conclusion

In conclusion, graph transformations are a crucial concept in mathematics, particularly in algebra and geometry. Understanding the different types of graph transformations, including translation, dilation, rotation, and reflection, is essential for analyzing and solving problems involving graphs. By analyzing the equation and the resulting graph, you can determine the type of transformation that has occurred and the direction or angle of the transformation.