Original Function And Transformed Function:Hint: Trace What Happens To The Point (2, 0) In Terms Of Right/left, Up/down, And Rotation.A. − F ( X + 3 ) + 2 -f(x+3)+2 − F ( X + 3 ) + 2 B. F ( X − 3 ) + 2 F(x-3)+2 F ( X − 3 ) + 2 C. − F ( X + 3 ) − 2 -f(x+3)-2 − F ( X + 3 ) − 2 D. F ( X + 3 ) − 2 F(x+3)-2 F ( X + 3 ) − 2

Introduction

In mathematics, functions are used to describe relationships between variables. When a function is transformed, its graph undergoes changes that can be understood by analyzing the effects of the transformation on specific points. In this article, we will explore how the original function and its transformed versions behave in terms of right/left, up/down, and rotation, using the point (2, 0) as a reference.

Understanding Function Transformations

Function transformations involve changing the original function in various ways, such as shifting, scaling, or reflecting. These transformations can be represented algebraically using various notations, including:

• Horizontal shifts: Shifting the graph to the left or right by a certain amount.
• Vertical shifts: Shifting the graph up or down by a certain amount.
• Horizontal stretches or compressions: Stretching or compressing the graph horizontally.
• Vertical stretches or compressions: Stretching or compressing the graph vertically.
• Reflections: Reflecting the graph across the x-axis or y-axis.

Analyzing the Point (2, 0)

To understand the effects of function transformations on the graph, we will analyze how the point (2, 0) behaves under different transformations. This point is chosen because it lies on the x-axis, making it easier to visualize the transformations.

A. $-f(x+3)+2$

When the original function $f(x)$ is transformed into $-f(x+3)+2$, the following changes occur:

• Horizontal shift: The graph is shifted 3 units to the left.
• Vertical reflection: The graph is reflected across the x-axis.
• Vertical shift: The graph is shifted 2 units up.

To analyze the point (2, 0), we substitute $x=2-3=-1$ into the transformed function:

$$-f(-1+3)+2 = -f(2)+2$$

Since the point (2, 0) lies on the original graph, the value of $f(2)$ is 0. Therefore, the transformed function evaluates to:

$$-f(2)+2 = -0+2 = 2$$

This means that the point (2, 0) is transformed into the point (-1, 2).

B. $f(x-3)+2$

When the original function $f(x)$ is transformed into $f(x-3)+2$, the following changes occur:

• Horizontal shift: The graph is shifted 3 units to the right.
• Vertical shift: The graph is shifted 2 units up.

To analyze the point (2, 0), we substitute $x=2-3=-1$ into the transformed function:

$$f(-1-3)+2 = f(-4)+2$$

Since the point (2, 0) lies on the original graph, the value of $f(2)$ is 0. However, the value of $f(-4)$ is not necessarily 0. Therefore, the transformed function evaluates to:

$$f(-4)+2$$

This means that the point (2, 0) is transformed into the point (-4, $f(-4)+2$).

C. $-f(x+3)-2$

When the original function $f(x)$ is transformed into $-f(x+3)-2$, the following changes occur:

• Horizontal shift: The graph is shifted 3 units to the left.
• Vertical reflection: The graph is reflected across the x-axis.
• Vertical shift: The graph is shifted 2 units down.

To analyze the point (2, 0), we substitute $x=2-3=-1$ into the transformed function:

$$-f(-1+3)-2 = -f(2)-2$$

Since the point (2, 0) lies on the original graph, the value of $f(2)$ is 0. Therefore, the transformed function evaluates to:

$$-f(2)-2 = -0-2 = -2$$

This means that the point (2, 0) is transformed into the point (-1, -2).

D. $f(x+3)-2$

When the original function $f(x)$ is transformed into $f(x+3)-2$, the following changes occur:

• Horizontal shift: The graph is shifted 3 units to the left.
• Vertical shift: The graph is shifted 2 units down.

To analyze the point (2, 0), we substitute $x=2-3=-1$ into the transformed function:

$$f(-1+3)-2 = f(2)-2$$

Since the point (2, 0) lies on the original graph, the value of $f(2)$ is 0. Therefore, the transformed function evaluates to:

$$f(2)-2 = 0-2 = -2$$

This means that the point (2, 0) is transformed into the point (-1, -2).

Conclusion

In this article, we analyzed how the original function and its transformed versions behave in terms of right/left, up/down, and rotation, using the point (2, 0) as a reference. We saw that different transformations result in different changes to the graph, including horizontal and vertical shifts, reflections, and vertical stretches or compressions. By understanding these transformations, we can better visualize and analyze the behavior of functions and their graphs.

References

• [1] "Function Transformations" by Khan Academy
• [2] "Graphing Functions" by Math Open Reference
• [3] "Algebraic Notations" by Wolfram MathWorld
  Transforming Functions: A Q&A Guide =====================================

Introduction

In our previous article, we explored how the original function and its transformed versions behave in terms of right/left, up/down, and rotation, using the point (2, 0) as a reference. In this article, we will answer some frequently asked questions about function transformations to help you better understand this concept.

Q&A

Q: What is a function transformation?

A: A function transformation is a change made to the original function, resulting in a new function with a different graph.

Q: What are the different types of function transformations?

A: There are several types of function transformations, including:

• Horizontal shifts: Shifting the graph to the left or right by a certain amount.
• Vertical shifts: Shifting the graph up or down by a certain amount.
• Horizontal stretches or compressions: Stretching or compressing the graph horizontally.
• Vertical stretches or compressions: Stretching or compressing the graph vertically.
• Reflections: Reflecting the graph across the x-axis or y-axis.

Q: How do I determine the type of function transformation?

A: To determine the type of function transformation, look for the following:

• Horizontal shifts: The function is in the form $f(x-h)$, where $h$ is the horizontal shift.
• Vertical shifts: The function is in the form $f(x)+k$ or $f(x)-k$, where $k$ is the vertical shift.
• Horizontal stretches or compressions: The function is in the form $f(ax)$ or $f(\frac{x}{a})$, where $a$ is the horizontal stretch or compression.
• Vertical stretches or compressions: The function is in the form $af(x)$ or $\frac{f(x)}{a}$, where $a$ is the vertical stretch or compression.
• Reflections: The function is in the form $-f(x)$ or $f(-x)$.

Q: How do I apply function transformations to a graph?

A: To apply function transformations to a graph, follow these steps:

1. Identify the type of transformation: Determine the type of transformation by looking at the function notation.
2. Apply the transformation: Apply the transformation to the graph by shifting, stretching, compressing, or reflecting the graph accordingly.
3. Check the result: Check the result to ensure that the transformation was applied correctly.

Q: What are some common function transformations?

A: Some common function transformations include:

• Horizontal shift: $f(x-3)$
• Vertical shift: $f(x)+2$
• Horizontal stretch: $f(2x)$
• Vertical compression: $\frac{f(x)}{2}$
• Reflection: $-f(x)$

Q: How do I graph a function with multiple transformations?

A: To graph a function with multiple transformations, follow these steps:

1. Apply each transformation: Apply each transformation to the graph in the correct order.
2. Check the result: Check the result to ensure that the transformations were applied correctly.
3. Graph the final result: Graph the final result to visualize the transformed function.

Conclusion

In this article, we answered some frequently asked questions about function transformations to help you better understand this concept. By understanding function transformations, you can better visualize and analyze the behavior of functions and their graphs.

References

• [1] "Function Transformations" by Khan Academy
• [2] "Graphing Functions" by Math Open Reference
• [3] "Algebraic Notations" by Wolfram MathWorld