Describe The Translation Of The Following Function. If The Function Did Not Shift Vertically Or Horizontally In A Particular Direction, Enter 0. Y = X 2 − 7 Y = X^2 - 7 Y = X 2 − 7 Vertical Shift (UP) = _____ UnitsVertical Shift (DOWN) = _____ UnitsHorizontal Shift

Introduction

In mathematics, particularly in algebra and geometry, the translation of a function refers to the process of shifting the graph of the function in a particular direction. This can be done vertically or horizontally, and it is an essential concept in understanding the behavior of functions. In this article, we will discuss the translation of a quadratic function, specifically the function $y = x^2 - 7$. We will explore how to determine the vertical and horizontal shifts of this function and provide examples to illustrate the concept.

Vertical Shift

A vertical shift of a function occurs when the graph of the function is moved up or down. This type of shift is denoted by the symbol $y = f(x) + c$, where $c$ is the amount of shift. If the graph is shifted up, the value of $c$ is positive, and if it is shifted down, the value of $c$ is negative.

Vertical Shift (UP)

To determine the vertical shift (up) of the function $y = x^2 - 7$, we need to identify the value of $c$ that represents the amount of shift. In this case, the function is already in the form $y = f(x) + c$, where $c = -7$. Therefore, the vertical shift (up) is equal to the absolute value of $c$, which is $|-7| = 7$ units.

Vertical Shift (DOWN)

To determine the vertical shift (down) of the function $y = x^2 - 7$, we need to identify the value of $c$ that represents the amount of shift. In this case, the function is already in the form $y = f(x) + c$, where $c = -7$. Therefore, the vertical shift (down) is equal to the absolute value of $c$, which is $|-7| = 7$ units.

Horizontal Shift

A horizontal shift of a function occurs when the graph of the function is moved left or right. This type of shift is denoted by the symbol $y = f(x - h)$, where $h$ is the amount of shift. If the graph is shifted to the right, the value of $h$ is positive, and if it is shifted to the left, the value of $h$ is negative.

Horizontal Shift

To determine the horizontal shift of the function $y = x^2 - 7$, we need to identify the value of $h$ that represents the amount of shift. In this case, the function is already in the form $y = f(x)$, which means that there is no horizontal shift. Therefore, the horizontal shift is equal to 0 units.

Conclusion

In conclusion, the translation of a quadratic function refers to the process of shifting the graph of the function in a particular direction. We have discussed the vertical and horizontal shifts of the function $y = x^2 - 7$ and provided examples to illustrate the concept. The vertical shift (up) is equal to 7 units, the vertical shift (down) is equal to 7 units, and the horizontal shift is equal to 0 units.

Examples

Here are some examples to illustrate the concept of translation:

• Example 1: The function $y = x^2 + 3$ has a vertical shift (up) of 3 units and a horizontal shift of 0 units.
• Example 2: The function $y = x^2 - 2$ has a vertical shift (down) of 2 units and a horizontal shift of 0 units.
• Example 3: The function $y = (x - 2)^2 + 1$ has a horizontal shift of 2 units and a vertical shift (up) of 1 unit.

Practice Problems

Here are some practice problems to help you understand the concept of translation:

• Problem 1: Determine the vertical and horizontal shifts of the function $y = x^2 + 5$.
• Problem 2: Determine the vertical and horizontal shifts of the function $y = x^2 - 4$.
• Problem 3: Determine the vertical and horizontal shifts of the function $y = (x + 3)^2 - 2$.

Answer Key

Here is the answer key for the practice problems:

• Problem 1: The vertical shift (up) is equal to 5 units, and the horizontal shift is equal to 0 units.
• Problem 2: The vertical shift (down) is equal to 4 units, and the horizontal shift is equal to 0 units.
• Problem 3: The horizontal shift is equal to 3 units, and the vertical shift (down) is equal to 2 units.

Final Thoughts

Introduction

In our previous article, we discussed the translation of quadratic functions, including the vertical and horizontal shifts of the function $y = x^2 - 7$. In this article, we will provide a Q&A section to help you better understand the concept of translation and how to apply it to different functions.

Q1: What is the vertical shift (up) of the function $y = x^2 + 2$?

A1: The vertical shift (up) of the function $y = x^2 + 2$ is equal to 2 units.

Q2: What is the horizontal shift of the function $y = (x - 3)^2 - 1$?

A2: The horizontal shift of the function $y = (x - 3)^2 - 1$ is equal to 3 units.

Q3: What is the vertical shift (down) of the function $y = x^2 - 4$?

A3: The vertical shift (down) of the function $y = x^2 - 4$ is equal to 4 units.

Q4: How do you determine the vertical shift (up) of a function?

A4: To determine the vertical shift (up) of a function, you need to identify the value of $c$ that represents the amount of shift. If the function is in the form $y = f(x) + c$, then the vertical shift (up) is equal to the absolute value of $c$.

Q5: How do you determine the horizontal shift of a function?

A5: To determine the horizontal shift of a function, you need to identify the value of $h$ that represents the amount of shift. If the function is in the form $y = f(x - h)$, then the horizontal shift is equal to the absolute value of $h$.

Q6: What is the difference between a vertical shift and a horizontal shift?

A6: A vertical shift occurs when the graph of a function is moved up or down, while a horizontal shift occurs when the graph of a function is moved left or right.

Q7: Can a function have both a vertical and horizontal shift?

A7: Yes, a function can have both a vertical and horizontal shift. For example, the function $y = (x - 2)^2 + 3$ has a horizontal shift of 2 units and a vertical shift (up) of 3 units.

Q8: How do you write a function with a vertical shift (up) of 5 units?

A8: To write a function with a vertical shift (up) of 5 units, you need to add 5 to the original function. For example, if the original function is $y = x^2$, then the function with a vertical shift (up) of 5 units is $y = x^2 + 5$.

Q9: How do you write a function with a horizontal shift of 3 units?

A9: To write a function with a horizontal shift of 3 units, you need to subtract 3 from the input variable $x$. For example, if the original function is $y = x^2$, then the function with a horizontal shift of 3 units is $y = (x - 3)^2$.

Q10: What is the importance of understanding the translation of quadratic functions?

A10: Understanding the translation of quadratic functions is important because it helps you to analyze and graph functions, which is a crucial skill in mathematics and science. It also helps you to identify the key features of a function, such as its vertex and axis of symmetry.

Conclusion

In conclusion, the translation of quadratic functions is an essential concept in mathematics and science. We hope that this Q&A article has provided you with a better understanding of the concept of translation and how to apply it to different functions. If you have any further questions or need additional help, please don't hesitate to ask.