Select The Correct Answer From Each Drop-down Menu.The Parent Cosine Function Is Transformed To Create Function $d$.$d(x) = \cos(2x - 1) + 5$To Create Function $d$, The Graph Of The Parent Cosine Function Undergoes These

The parent cosine function is a fundamental function in mathematics, and it can be transformed in various ways to create new functions. In this article, we will discuss how the parent cosine function is transformed to create function $d$, which is given by $d(x) = \cos(2x - 1) + 5$.

Understanding the Parent Cosine Function

The parent cosine function is defined as $f(x) = \cos x$. This function has a period of $2\pi$, which means that it repeats itself every $2\pi$ units. The graph of the parent cosine function is a smooth, continuous curve that oscillates between $-1$ and $1$.

Transforming the Parent Cosine Function to Create Function $d$

To create function $d$, the graph of the parent cosine function undergoes several transformations. The first transformation is a horizontal stretch by a factor of $\frac{1}{2}$. This means that the graph of the parent cosine function is stretched horizontally by a factor of $\frac{1}{2}$, which results in a new function $g(x) = \cos(2x)$.

Horizontal Stretch

The horizontal stretch is a type of transformation that changes the width of the graph. In this case, the horizontal stretch by a factor of $\frac{1}{2}$ results in a new function $g(x) = \cos(2x)$.

Effect of Horizontal Stretch on the Graph

The horizontal stretch has a significant effect on the graph of the parent cosine function. The graph is stretched horizontally, which means that the period of the function is doubled. The new period of the function is $2\pi$, which is twice the original period.

Vertical Shift

The next transformation is a vertical shift by $5$ units. This means that the graph of the function $g(x) = \cos(2x)$ is shifted vertically by $5$ units, resulting in a new function $h(x) = \cos(2x) + 5$.

Effect of Vertical Shift on the Graph

The vertical shift has a significant effect on the graph of the function. The graph is shifted vertically by $5$ units, which means that the amplitude of the function is increased by $5$ units. The new amplitude of the function is $6$, which is the sum of the original amplitude and the vertical shift.

Final Transformation

The final transformation is a horizontal shift by $1$ unit to the right. This means that the graph of the function $h(x) = \cos(2x) + 5$ is shifted horizontally by $1$ unit to the right, resulting in the final function $d(x) = \cos(2x - 1) + 5$.

Effect of Horizontal Shift on the Graph

The horizontal shift has a significant effect on the graph of the function. The graph is shifted horizontally by $1$ unit to the right, which means that the phase of the function is changed. The new phase of the function is $1$ unit to the right, which is the sum of the original phase and the horizontal shift.

Conclusion

In conclusion, the parent cosine function is transformed to create function $d$ by undergoing several transformations. The transformations include a horizontal stretch by a factor of $\frac{1}{2}$, a vertical shift by $5$ units, and a horizontal shift by $1$ unit to the right. The final function $d(x) = \cos(2x - 1) + 5$ is a transformed version of the parent cosine function, and it has a period of $2\pi$, an amplitude of $6$, and a phase of $1$ unit to the right.

Key Takeaways

• The parent cosine function is a fundamental function in mathematics.
• The parent cosine function can be transformed in various ways to create new functions.
• The transformations include horizontal and vertical shifts, as well as horizontal stretches.
• The final function $d(x) = \cos(2x - 1) + 5$ is a transformed version of the parent cosine function.

Practice Problems

1. What is the period of the function $d(x) = \cos(2x - 1) + 5$?
2. What is the amplitude of the function $d(x) = \cos(2x - 1) + 5$?
3. What is the phase of the function $d(x) = \cos(2x - 1) + 5$?

Answer Key

1. The period of the function $d(x) = \cos(2x - 1) + 5$ is $2\pi$.
2. The amplitude of the function $d(x) = \cos(2x - 1) + 5$ is $6$.
3. The phase of the function $d(x) = \cos(2x - 1) + 5$ is $1$ unit to the right.
   Q&A: Transforming the Parent Cosine Function =====================================================

In our previous article, we discussed how the parent cosine function is transformed to create function $d$, which is given by $d(x) = \cos(2x - 1) + 5$. In this article, we will answer some frequently asked questions about transforming the parent cosine function.

Q: What is the parent cosine function?

A: The parent cosine function is a fundamental function in mathematics, defined as $f(x) = \cos x$. This function has a period of $2\pi$, which means that it repeats itself every $2\pi$ units.

Q: What are the transformations that occur when creating function $d$?

A: The transformations that occur when creating function $d$ include a horizontal stretch by a factor of $\frac{1}{2}$, a vertical shift by $5$ units, and a horizontal shift by $1$ unit to the right.

Q: What is the effect of the horizontal stretch on the graph of the parent cosine function?

A: The horizontal stretch has a significant effect on the graph of the parent cosine function. The graph is stretched horizontally, which means that the period of the function is doubled. The new period of the function is $2\pi$, which is twice the original period.

Q: What is the effect of the vertical shift on the graph of the function?

A: The vertical shift has a significant effect on the graph of the function. The graph is shifted vertically by $5$ units, which means that the amplitude of the function is increased by $5$ units. The new amplitude of the function is $6$, which is the sum of the original amplitude and the vertical shift.

Q: What is the effect of the horizontal shift on the graph of the function?

A: The horizontal shift has a significant effect on the graph of the function. The graph is shifted horizontally by $1$ unit to the right, which means that the phase of the function is changed. The new phase of the function is $1$ unit to the right, which is the sum of the original phase and the horizontal shift.

Q: What is the period of the function $d(x) = \cos(2x - 1) + 5$?

A: The period of the function $d(x) = \cos(2x - 1) + 5$ is $2\pi$.

Q: What is the amplitude of the function $d(x) = \cos(2x - 1) + 5$?

A: The amplitude of the function $d(x) = \cos(2x - 1) + 5$ is $6$.

Q: What is the phase of the function $d(x) = \cos(2x - 1) + 5$?

A: The phase of the function $d(x) = \cos(2x - 1) + 5$ is $1$ unit to the right.

Q: How do I determine the period, amplitude, and phase of a transformed function?

A: To determine the period, amplitude, and phase of a transformed function, you need to analyze the transformations that occurred to create the function. The period is determined by the horizontal stretch, the amplitude is determined by the vertical shift, and the phase is determined by the horizontal shift.

Q: What are some common transformations that occur when creating new functions?

A: Some common transformations that occur when creating new functions include horizontal and vertical shifts, as well as horizontal stretches. These transformations can change the period, amplitude, and phase of the function.

Q: How do I apply transformations to create new functions?

A: To apply transformations to create new functions, you need to analyze the original function and determine the transformations that need to occur to create the new function. You can then apply these transformations to the original function to create the new function.

Conclusion

In conclusion, transforming the parent cosine function is a fundamental concept in mathematics. By understanding the transformations that occur when creating new functions, you can analyze and create new functions with ease. Remember to analyze the period, amplitude, and phase of the function to determine the transformations that occurred to create the function.

Key Takeaways

• The parent cosine function is a fundamental function in mathematics.
• The parent cosine function can be transformed in various ways to create new functions.
• The transformations include horizontal and vertical shifts, as well as horizontal stretches.
• The period, amplitude, and phase of the function are determined by the transformations that occurred to create the function.

Practice Problems

1. What is the period of the function $f(x) = \cos(3x + 2) + 4$?
2. What is the amplitude of the function $f(x) = \cos(3x + 2) + 4$?
3. What is the phase of the function $f(x) = \cos(3x + 2) + 4$?

Answer Key

1. The period of the function $f(x) = \cos(3x + 2) + 4$ is $\frac{2\pi}{3}$.
2. The amplitude of the function $f(x) = \cos(3x + 2) + 4$ is $5$.
3. The phase of the function $f(x) = \cos(3x + 2) + 4$ is $-\frac{2}{3}$ units to the left.