Consider The Function F ( X ) = 2 X − 6 F(x) = 2x - 6 F ( X ) = 2 X − 6 .Match Each Transformation Of F ( X F(x F ( X ] With Its Description. Not All Options Will Be Used.- Shifts F ( X F(x F ( X ] 4 Units Down ⟶ \longrightarrow ⟶ □ \square □ - Compresses

Introduction

In mathematics, transformations of linear functions are essential concepts that help us understand how functions can be manipulated and represented in different ways. A linear function is a function that can be written in the form $f(x) = mx + b$, where $m$ is the slope and $b$ is the y-intercept. In this article, we will explore the transformations of the linear function $f(x) = 2x - 6$ and match each transformation with its description.

Understanding the Original Function

Before we dive into the transformations, let's understand the original function $f(x) = 2x - 6$. This function has a slope of 2 and a y-intercept of -6. The graph of this function is a straight line that passes through the point (0, -6) and has a slope of 2.

Transformations of the Function

Shifts

A shift is a transformation that moves the graph of a function up or down. To shift a function $f(x)$ units down, we subtract $x$ from the function. In this case, we want to shift the function $f(x) = 2x - 6$ 4 units down.

Shifts $f(x)$ 4 units down

To shift the function $f(x) = 2x - 6$ 4 units down, we subtract 4 from the function:

$f(x) = 2x - 6 - 4$ $f(x) = 2x - 10$

This is the correct transformation for shifting the function 4 units down.

Compressions

A compression is a transformation that makes the graph of a function narrower. To compress a function horizontally by a factor of $a$, we multiply the input $x$ by $a$. In this case, we want to compress the function $f(x) = 2x - 6$ horizontally by a factor of 3.

Compresses $f(x)$ horizontally by a factor of 3

To compress the function $f(x) = 2x - 6$ horizontally by a factor of 3, we multiply the input $x$ by 3:

$f(x) = 2(3x) - 6$ $f(x) = 6x - 6$

This is the correct transformation for compressing the function horizontally by a factor of 3.

Stretches

A stretch is a transformation that makes the graph of a function wider. To stretch a function vertically by a factor of $a$, we multiply the output $f(x)$ by $a$. In this case, we want to stretch the function $f(x) = 2x - 6$ vertically by a factor of 2.

Stretches $f(x)$ vertically by a factor of 2

To stretch the function $f(x) = 2x - 6$ vertically by a factor of 2, we multiply the output $f(x)$ by 2:

$f(x) = 2(2x - 6)$ $f(x) = 4x - 12$

This is the correct transformation for stretching the function vertically by a factor of 2.

Reflections

A reflection is a transformation that flips the graph of a function over a line. To reflect a function over the x-axis, we multiply the output $f(x)$ by -1. In this case, we want to reflect the function $f(x) = 2x - 6$ over the x-axis.

Reflects $f(x)$ over the x-axis

To reflect the function $f(x) = 2x - 6$ over the x-axis, we multiply the output $f(x)$ by -1:

$f(x) = -1(2x - 6)$ $f(x) = -2x + 6$

This is the correct transformation for reflecting the function over the x-axis.

Conclusion

In this article, we explored the transformations of the linear function $f(x) = 2x - 6$. We matched each transformation with its description and provided examples of how to perform each transformation. Understanding these transformations is essential for working with linear functions and is a fundamental concept in mathematics.

Key Takeaways

• Shifts: To shift a function $f(x)$ units down, we subtract $x$ from the function.
• Compressions: To compress a function horizontally by a factor of $a$, we multiply the input $x$ by $a$.
• Stretches: To stretch a function vertically by a factor of $a$, we multiply the output $f(x)$ by $a$.
• Reflections: To reflect a function over the x-axis, we multiply the output $f(x)$ by -1.

Practice Problems

1. Shift the function $f(x) = 3x + 2$ 5 units down.
2. Compress the function $f(x) = 2x - 4$ horizontally by a factor of 2.
3. Stretch the function $f(x) = x - 3$ vertically by a factor of 3.
4. Reflect the function $f(x) = x + 1$ over the x-axis.

Answer Key

1. $f(x) = 3x + 2 - 5$ $f(x) = 3x - 3$
2. $f(x) = 2(2x) - 4$ $f(x) = 4x - 4$
3. $f(x) = 3(x - 3)$ $f(x) = 3x - 9$
4. $f(x) = -1(x + 1)$ $f(x) = -x - 1$
   Transformations of Linear Functions: Q&A =============================================

Introduction

In our previous article, we explored the transformations of the linear function $f(x) = 2x - 6$. We matched each transformation with its description and provided examples of how to perform each transformation. In this article, we will answer some frequently asked questions about transformations of linear functions.

Q&A

Q: What is the difference between a shift and a compression?

A: A shift is a transformation that moves the graph of a function up or down, while a compression is a transformation that makes the graph of a function narrower.

Q: How do I shift a function $f(x)$ units down?

A: To shift a function $f(x)$ units down, we subtract $x$ from the function. For example, to shift the function $f(x) = 2x - 6$ 4 units down, we subtract 4 from the function:

$f(x) = 2x - 6 - 4$ $f(x) = 2x - 10$

Q: How do I compress a function horizontally by a factor of $a$?

A: To compress a function horizontally by a factor of $a$, we multiply the input $x$ by $a$. For example, to compress the function $f(x) = 2x - 6$ horizontally by a factor of 3, we multiply the input $x$ by 3:

$f(x) = 2(3x) - 6$ $f(x) = 6x - 6$

Q: How do I stretch a function vertically by a factor of $a$?

A: To stretch a function vertically by a factor of $a$, we multiply the output $f(x)$ by $a$. For example, to stretch the function $f(x) = 2x - 6$ vertically by a factor of 2, we multiply the output $f(x)$ by 2:

$f(x) = 2(2x - 6)$ $f(x) = 4x - 12$

Q: How do I reflect a function over the x-axis?

A: To reflect a function over the x-axis, we multiply the output $f(x)$ by -1. For example, to reflect the function $f(x) = 2x - 6$ over the x-axis, we multiply the output $f(x)$ by -1:

$f(x) = -1(2x - 6)$ $f(x) = -2x + 6$

Q: What are some common transformations of linear functions?

A: Some common transformations of linear functions include:

• Shifts: moving the graph of a function up or down
• Compressions: making the graph of a function narrower
• Stretches: making the graph of a function wider
• Reflections: flipping the graph of a function over a line

Q: How do I determine the type of transformation that has been applied to a function?

A: To determine the type of transformation that has been applied to a function, we need to look at the equation of the function and identify the changes that have been made. For example, if a function has been shifted 4 units down, we can see that the constant term has been decreased by 4.

Q: Can I apply multiple transformations to a function?

A: Yes, we can apply multiple transformations to a function. For example, we can shift a function 4 units down and then compress it horizontally by a factor of 3.

Conclusion

In this article, we answered some frequently asked questions about transformations of linear functions. We covered topics such as shifts, compressions, stretches, and reflections, and provided examples of how to perform each transformation. Understanding these transformations is essential for working with linear functions and is a fundamental concept in mathematics.

Key Takeaways

• Shifts: moving the graph of a function up or down
• Compressions: making the graph of a function narrower
• Stretches: making the graph of a function wider
• Reflections: flipping the graph of a function over a line
• Multiple transformations: applying multiple transformations to a function

Practice Problems

1. Shift the function $f(x) = 3x + 2$ 5 units down.
2. Compress the function $f(x) = 2x - 4$ horizontally by a factor of 2.
3. Stretch the function $f(x) = x - 3$ vertically by a factor of 3.
4. Reflect the function $f(x) = x + 1$ over the x-axis.
5. Apply multiple transformations to the function $f(x) = 2x - 6$.

Answer Key

1. $f(x) = 3x + 2 - 5$ $f(x) = 3x - 3$
2. $f(x) = 2(2x) - 4$ $f(x) = 4x - 4$
3. $f(x) = 3(x - 3)$ $f(x) = 3x - 9$
4. $f(x) = -1(x + 1)$ $f(x) = -x - 1$
5. $f(x) = 2(3x - 6) - 4$ $f(x) = 6x - 12 - 4$ $f(x) = 6x - 16$