Determining the Type of Translation

Oct 13, 2025 by windailys.com 22 views

Introduction

In mathematics, graph transformations are essential concepts that help us understand how functions change when subjected to various transformations. One of the most common types of transformations is translation, which involves shifting the graph of a function up, down, left, or right. In this article, we will explore the concept of translation and determine which type of translation transforms the graph of $y = x^2$ to the graph of $y = (x+5)^2$ .

Understanding the Graph of $y = x^2$

The graph of $y = x^2$ is a parabola that opens upwards, with its vertex at the origin (0, 0). This graph is a classic example of a quadratic function, and it is symmetric about the y-axis.

Understanding the Graph of $y = (x+5)^2$

The graph of $y = (x+5)^2$ is also a parabola, but it is shifted 5 units to the left compared to the graph of $y = x^2$ . This means that the vertex of the graph of $y = (x+5)^2$ is located at the point (-5, 0).

Determining the Type of Translation

To determine which type of translation transforms the graph of $y = x^2$ to the graph of $y = (x+5)^2$ , we need to analyze the effect of each type of translation on the graph.

A. Translation 5 Units to the Right

If we translate the graph of $y = x^2$ 5 units to the right, the new equation would be $y = (x-5)^2$ . This means that the vertex of the graph would be shifted 5 units to the right, resulting in a new vertex at the point (5, 0). This is not the desired transformation, as the vertex of the graph of $y = (x+5)^2$ is located at the point (-5, 0).

B. Translation 5 Units to the Left

If we translate the graph of $y = x^2$ 5 units to the left, the new equation would be $y = (x+5)^2$ . This means that the vertex of the graph would be shifted 5 units to the left, resulting in a new vertex at the point (-5, 0). This is the desired transformation, as the vertex of the graph of $y = (x+5)^2$ is located at the point (-5, 0).

C. Translation 5 Units Down

If we translate the graph of $y = x^2$ 5 units down, the new equation would be $y = x^2 - 5$ . This means that the vertex of the graph would be shifted 5 units down, resulting in a new vertex at the point (0, -5). This is not the desired transformation, as the vertex of the graph of $y = (x+5)^2$ is located at the point (-5, 0).

D. Translation 5 Units Up

If we translate the graph of $y = x^2$ 5 units up, the new equation would be $y = x^2 + 5$ . This means that the vertex of the graph would be shifted 5 units up, resulting in a new vertex at the point (0, 5). This is not the desired transformation, as the vertex of the graph of $y = (x+5)^2$ is located at the point (-5, 0).

Conclusion

In conclusion, the correct answer is B. A translation 5 units to the left. This type of translation transforms the graph of $y = x^2$ to the graph of $y = (x+5)^2$ , resulting in a new vertex at the point (-5, 0).

Final Answer

Q&A: Transforming Graphs with Translation

In our previous article, we explored the concept of translation and determined which type of translation transforms the graph of $y = x^2$ to the graph of $y = (x+5)^2$ . In this article, we will answer some frequently asked questions about transforming graphs with translation.

Q: What is translation in mathematics?

A: Translation is a type of transformation that involves shifting the graph of a function up, down, left, or right. This means that the graph is moved to a new position without changing its shape or size.

Q: What are the different types of translations?

A: There are four main types of translations:

Horizontal translation: This involves shifting the graph to the left or right.
Vertical translation: This involves shifting the graph up or down.

Q: How do I determine the type of translation?

A: To determine the type of translation, you need to analyze the effect of each type of translation on the graph. You can do this by looking at the equation of the graph and identifying the changes that occur when you apply the translation.

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

A: A horizontal translation involves shifting the graph to the left or right. This means that the x-coordinates of the graph are changed, while the y-coordinates remain the same.

Q: What is the effect of a vertical translation on a graph?

A: A vertical translation involves shifting the graph up or down. This means that the y-coordinates of the graph are changed, while the x-coordinates remain the same.

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

A: To apply a translation to a graph, you need to change the equation of the graph by adding or subtracting a constant value from the x or y variable.

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

A: A translation involves shifting the graph to a new position, while a reflection involves flipping the graph over a line or axis.

Q: Can I apply multiple translations to a graph?

A: Yes, you can apply multiple translations to a graph. This involves applying one translation and then applying another translation to the resulting graph.

Q: How do I determine the order of multiple translations?

A: To determine the order of multiple translations, you need to analyze the effect of each translation on the graph. You can do this by looking at the equation of the graph and identifying the changes that occur when you apply each translation.

Q: What is the effect of a translation on the vertex of a graph?

A: A translation involves shifting the vertex of the graph to a new position. This means that the x and y coordinates of the vertex are changed.

Q: Can I apply a translation to a graph with a hole or a gap?

A: Yes, you can apply a translation to a graph with a hole or a gap. This involves shifting the graph to a new position, while preserving the hole or gap.

Q: How do I determine the type of translation that transforms a graph?

A: To determine the type of translation that transforms a graph, you need to analyze the effect of each type of translation on the graph. You can do this by looking at the equation of the graph and identifying the changes that occur when you apply the translation.

Conclusion

In conclusion, translation is a fundamental concept in mathematics that involves shifting the graph of a function up, down, left, or right. By understanding the effects of translation, you can apply multiple translations to a graph and determine the type of translation that transforms a graph.

Final Answer

The final answer is B.