Understanding the Transformation of f Represented by g

Oct 13, 2025 by windailys.com 54 views

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Introduction

In mathematics, transformations are a crucial concept in understanding how functions behave under different operations. Given a function f and another function g, we are often asked to describe the transformation of f represented by g. This involves understanding how g affects the graph of f. In this article, we will delve into the concept of describing the transformation of f represented by g and write a rule for g.

Understanding the Transformation of f Represented by g

The transformation of f represented by g can be understood by analyzing the effect of g on the graph of f. This involves identifying the type of transformation that g represents, such as a shift, reflection, dilation, or rotation. To describe the transformation of f represented by g, we need to identify the following:

Horizontal shift: If g represents a horizontal shift, we need to determine the direction and magnitude of the shift.
Vertical shift: If g represents a vertical shift, we need to determine the direction and magnitude of the shift.
Reflection: If g represents a reflection, we need to determine the axis of reflection.
Dilation: If g represents a dilation, we need to determine the center of dilation and the scale factor.
Rotation: If g represents a rotation, we need to determine the center of rotation and the angle of rotation.

Writing a Rule for g

Once we have identified the type of transformation that g represents, we can write a rule for g. This involves expressing g as a function of f, taking into account the type of transformation that g represents. For example, if g represents a horizontal shift of 3 units to the right, we can write a rule for g as:

g(x) = f(x - 3)

Similarly, if g represents a vertical shift of 2 units up, we can write a rule for g as:

g(x) = f(x) + 2

If g represents a reflection across the x-axis, we can write a rule for g as:

g(x) = -f(x)

If g represents a dilation with a scale factor of 2, we can write a rule for g as:

g(x) = 2f(x)

If g represents a rotation of 90 degrees counterclockwise, we can write a rule for g as:

g(x) = f(-x)

Example 1: Horizontal Shift

Suppose we have a function f(x) = x^2 and another function g(x) = (x - 3)^2. To describe the transformation of f represented by g, we need to identify the type of transformation that g represents. In this case, g represents a horizontal shift of 3 units to the right.

To write a rule for g, we can express g as a function of f, taking into account the type of transformation that g represents. In this case, we can write a rule for g as:

g(x) = f(x - 3)

Example 2: Vertical Shift

Suppose we have a function f(x) = x^2 and another function g(x) = x^2 + 2. To describe the transformation of f represented by g, we need to identify the type of transformation that g represents. In this case, g represents a vertical shift of 2 units up.

To write a rule for g, we can express g as a function of f, taking into account the type of transformation that g represents. In this case, we can write a rule for g as:

g(x) = f(x) + 2

Example 3: Reflection

Suppose we have a function f(x) = x^2 and another function g(x) = -(x^2). To describe the transformation of f represented by g, we need to identify the type of transformation that g represents. In this case, g represents a reflection across the x-axis.

To write a rule for g, we can express g as a function of f, taking into account the type of transformation that g represents. In this case, we can write a rule for g as:

g(x) = -f(x)

Example 4: Dilation

Suppose we have a function f(x) = x^2 and another function g(x) = 2(x^2). To describe the transformation of f represented by g, we need to identify the type of transformation that g represents. In this case, g represents a dilation with a scale factor of 2.

To write a rule for g, we can express g as a function of f, taking into account the type of transformation that g represents. In this case, we can write a rule for g as:

g(x) = 2f(x)

Example 5: Rotation

Suppose we have a function f(x) = x^2 and another function g(x) = (x^2)'. To describe the transformation of f represented by g, we need to identify the type of transformation that g represents. In this case, g represents a rotation of 90 degrees counterclockwise.

To write a rule for g, we can express g as a function of f, taking into account the type of transformation that g represents. In this case, we can write a rule for g as:

g(x) = f(-x)

Conclusion

In conclusion, describing the transformation of f represented by g involves understanding how g affects the graph of f. This involves identifying the type of transformation that g represents, such as a shift, reflection, dilation, or rotation. Once we have identified the type of transformation that g represents, we can write a rule for g by expressing g as a function of f, taking into account the type of transformation that g represents.

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Frequently Asked Questions

Q1: What is the transformation of f represented by g?

A1: The transformation of f represented by g is the effect of g on the graph of f. This involves identifying the type of transformation that g represents, such as a shift, reflection, dilation, or rotation.

Q2: How do I identify the type of transformation that g represents?

A2: To identify the type of transformation that g represents, you need to analyze the effect of g on the graph of f. This involves looking for changes in the position, size, or orientation of the graph.

Q3: What are the different types of transformations that g can represent?

A3: The different types of transformations that g can represent are:

Horizontal shift: A shift to the left or right
Vertical shift: A shift up or down
Reflection: A flip across an axis
Dilation: A change in size
Rotation: A turn around a point

Q4: How do I write a rule for g?

A4: To write a rule for g, you need to express g as a function of f, taking into account the type of transformation that g represents. For example, if g represents a horizontal shift of 3 units to the right, you can write a rule for g as:

g(x) = f(x - 3)

Q5: What is the difference between a horizontal shift and a vertical shift?

A5: A horizontal shift involves moving the graph of f to the left or right, while a vertical shift involves moving the graph of f up or down.

Q6: How do I determine the magnitude of a shift?

A6: To determine the magnitude of a shift, you need to look at the difference between the x-coordinates of the original and transformed graphs.

Q7: What is the difference between a reflection and a rotation?

A7: A reflection involves flipping the graph of f across an axis, while a rotation involves turning the graph of f around a point.

Q8: How do I determine the angle of rotation?

A8: To determine the angle of rotation, you need to look at the amount of rotation that the graph of f has undergone.

Q9: What is the difference between a dilation and a rotation?

A9: A dilation involves changing the size of the graph of f, while a rotation involves turning the graph of f around a point.

Q10: How do I determine the scale factor of a dilation?

A10: To determine the scale factor of a dilation, you need to look at the ratio of the original and transformed graphs.

Example Questions

Q1: Suppose we have a function f(x) = x^2 and another function g(x) = (x - 3)^2. What type of transformation does g represent?

A1: g represents a horizontal shift of 3 units to the right.

Q2: Suppose we have a function f(x) = x^2 and another function g(x) = x^2 + 2. What type of transformation does g represent?

A2: g represents a vertical shift of 2 units up.

Q3: Suppose we have a function f(x) = x^2 and another function g(x) = -(x^2). What type of transformation does g represent?

A3: g represents a reflection across the x-axis.

Q4: Suppose we have a function f(x) = x^2 and another function g(x) = 2(x^2). What type of transformation does g represent?

A4: g represents a dilation with a scale factor of 2.

Q5: Suppose we have a function f(x) = x^2 and another function g(x) = (x^2)'. What type of transformation does g represent?

A5: g represents a rotation of 90 degrees counterclockwise.

Conclusion

In conclusion, describing the transformation of f represented by g involves understanding how g affects the graph of f. This involves identifying the type of transformation that g represents, such as a shift, reflection, dilation, or rotation. By analyzing the effect of g on the graph of f, you can determine the type of transformation that g represents and write a rule for g.