Select All The Correct Answers.Consider Functions F F F And G G G : F ( X ) = − 2 Cos ⁡ ( X − 1 ) + 3 F(x) = -2 \cos (x-1) + 3 F ( X ) = − 2 Cos ( X − 1 ) + 3 G ( X ) = 2 Cos ⁡ ( X + 3 ) − 3 G(x) = 2 \cos (x+3) - 3 G ( X ) = 2 Cos ( X + 3 ) − 3 Which Statements Describe Transformations Of The Graph Of F F F Resulting In The Graph Of

Introduction

In mathematics, transformations of graphs are essential concepts that help us understand how functions can be manipulated to create new graphs. In this article, we will explore the transformations of the graph of function $f$ to obtain the graph of function $g$. We will analyze the given functions $f$ and $g$ and identify the correct statements that describe the transformations of the graph of $f$ resulting in the graph of $g$.

Given Functions

The given functions are:

$f(x) = -2 \cos (x-1) + 3$

$g(x) = 2 \cos (x+3) - 3$

Understanding the Graphs

To understand the transformations of the graph of $f$ to obtain the graph of $g$, we need to analyze the given functions. The function $f(x)$ is a cosine function with a period of $2\pi$, amplitude of $2$, phase shift of $1$, and vertical shift of $3$. The function $g(x)$ is also a cosine function with a period of $2\pi$, amplitude of $2$, phase shift of $-3$, and vertical shift of $-3$.

Transformations of Graphs

The graph of $g(x)$ can be obtained by applying the following transformations to the graph of $f(x)$:

• Horizontal Shift: The graph of $g(x)$ is obtained by shifting the graph of $f(x)$ to the left by $4$ units.
• Vertical Shift: The graph of $g(x)$ is obtained by shifting the graph of $f(x)$ down by $6$ units.
• Amplitude: The graph of $g(x)$ has an amplitude of $2$, which is the same as the amplitude of the graph of $f(x)$.
• Phase Shift: The graph of $g(x)$ has a phase shift of $-3$, which is different from the phase shift of the graph of $f(x)$.

Correct Statements

The following statements describe the transformations of the graph of $f$ resulting in the graph of $g$:

• Statement 1: The graph of $g(x)$ is obtained by shifting the graph of $f(x)$ to the left by $4$ units.
• Statement 2: The graph of $g(x)$ is obtained by shifting the graph of $f(x)$ down by $6$ units.
• Statement 3: The graph of $g(x)$ has an amplitude of $2$, which is the same as the amplitude of the graph of $f(x)$.
• Statement 4: The graph of $g(x)$ has a phase shift of $-3$, which is different from the phase shift of the graph of $f(x)$.

Conclusion

In conclusion, the graph of $g(x)$ can be obtained by applying the following transformations to the graph of $f(x)$: horizontal shift to the left by $4$ units, vertical shift down by $6$ units, amplitude of $2$, and phase shift of $-3$. The correct statements that describe the transformations of the graph of $f$ resulting in the graph of $g$ are:

• The graph of $g(x)$ is obtained by shifting the graph of $f(x)$ to the left by $4$ units.
• The graph of $g(x)$ is obtained by shifting the graph of $f(x)$ down by $6$ units.
• The graph of $g(x)$ has an amplitude of $2$, which is the same as the amplitude of the graph of $f(x)$.
• The graph of $g(x)$ has a phase shift of $-3$, which is different from the phase shift of the graph of $f(x)$.

References

• [1] "Graphing Functions" by Math Open Reference
• [2] "Transformations of Graphs" by Khan Academy

Final Answer

The final answer is:

• The graph of $g(x)$ is obtained by shifting the graph of $f(x)$ to the left by $4$ units.
• The graph of $g(x)$ is obtained by shifting the graph of $f(x)$ down by $6$ units.
• The graph of $g(x)$ has an amplitude of $2$, which is the same as the amplitude of the graph of $f(x)$.
• The graph of $g(x)$ has a phase shift of $-3$, which is different from the phase shift of the graph of $f(x)$.
  Q&A: Transformations of Graphs =====================================

Introduction

In our previous article, we discussed the transformations of the graph of function $f$ to obtain the graph of function $g$. We analyzed the given functions and identified the correct statements that describe the transformations of the graph of $f$ resulting in the graph of $g$. In this article, we will provide a Q&A section to help you better understand the concepts of transformations of graphs.

Q1: What is a transformation of a graph?

A transformation of a graph is a change in the position, size, or shape of the graph. It can be a horizontal shift, vertical shift, reflection, dilation, or rotation of the graph.

Q2: What are the different types of transformations?

There are several types of transformations, including:

• Horizontal Shift: A horizontal shift is a change in the position of the graph along the x-axis.
• Vertical Shift: A vertical shift is a change in the position of the graph along the y-axis.
• Reflection: A reflection is a change in the orientation of the graph.
• Dilation: A dilation is a change in the size of the graph.
• Rotation: A rotation is a change in the orientation of the graph.

Q3: How do I determine the type of transformation?

To determine the type of transformation, you need to analyze the function and identify the changes in the position, size, or shape of the graph. You can use the following steps:

1. Identify the function and its components (e.g., amplitude, period, phase shift).
2. Compare the function with the original graph to identify the changes.
3. Determine the type of transformation based on the changes.

Q4: What is the difference between a horizontal shift and a vertical shift?

A horizontal shift is a change in the position of the graph along the x-axis, while a vertical shift is a change in the position of the graph along the y-axis.

Q5: How do I apply a horizontal shift to a graph?

To apply a horizontal shift to a graph, you need to add or subtract a value from the x-coordinate of the function. For example, if you want to shift the graph of $f(x)$ to the left by $2$ units, you need to replace $x$ with $x+2$.

Q6: How do I apply a vertical shift to a graph?

To apply a vertical shift to a graph, you need to add or subtract a value from the y-coordinate of the function. For example, if you want to shift the graph of $f(x)$ down by $3$ units, you need to replace $f(x)$ with $f(x)-3$.

Q7: What is the amplitude of a graph?

The amplitude of a graph is the maximum distance from the x-axis to the graph. It is a measure of the size of the graph.

Q8: How do I determine the amplitude of a graph?

To determine the amplitude of a graph, you need to analyze the function and identify the maximum distance from the x-axis to the graph. You can use the following steps:

1. Identify the function and its components (e.g., amplitude, period, phase shift).
2. Determine the maximum distance from the x-axis to the graph.
3. The amplitude is the absolute value of the maximum distance.

Q9: What is the period of a graph?

The period of a graph is the distance between two consecutive points on the graph that have the same y-coordinate. It is a measure of the frequency of the graph.

Q10: How do I determine the period of a graph?

To determine the period of a graph, you need to analyze the function and identify the distance between two consecutive points on the graph that have the same y-coordinate. You can use the following steps:

1. Identify the function and its components (e.g., amplitude, period, phase shift).
2. Determine the distance between two consecutive points on the graph that have the same y-coordinate.
3. The period is the absolute value of the distance.

Conclusion

In conclusion, transformations of graphs are essential concepts in mathematics that help us understand how functions can be manipulated to create new graphs. By analyzing the function and identifying the changes in the position, size, or shape of the graph, we can determine the type of transformation and apply it to the graph. We hope that this Q&A article has helped you better understand the concepts of transformations of graphs.

References

• [1] "Graphing Functions" by Math Open Reference
• [2] "Transformations of Graphs" by Khan Academy

Final Answer

The final answer is:

• A transformation of a graph is a change in the position, size, or shape of the graph.
• There are several types of transformations, including horizontal shift, vertical shift, reflection, dilation, and rotation.
• To determine the type of transformation, you need to analyze the function and identify the changes in the position, size, or shape of the graph.
• A horizontal shift is a change in the position of the graph along the x-axis, while a vertical shift is a change in the position of the graph along the y-axis.
• To apply a horizontal shift to a graph, you need to add or subtract a value from the x-coordinate of the function.
• To apply a vertical shift to a graph, you need to add or subtract a value from the y-coordinate of the function.
• The amplitude of a graph is the maximum distance from the x-axis to the graph.
• The period of a graph is the distance between two consecutive points on the graph that have the same y-coordinate.