Which Transformations Are Needed To Change The Parent Cosine Function To${ Y = 0.35 \cos \left(8\left(x-\frac{\pi}{4}\right)\right) ? }$A. Vertical Stretch Of 0.35, Horizontal Stretch To A Period Of ${ 16 \pi\$} , Phase Shift Of

The parent cosine function is a fundamental concept in mathematics, and it serves as a basis for various transformations. In this article, we will explore the transformations needed to change the parent cosine function to $y = 0.35 \cos \left(8\left(x-\frac{\pi}{4}\right)\right)$.

Understanding the Parent Cosine Function

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

Vertical Stretch

The first transformation we need to consider is the vertical stretch. In the given function, the coefficient of the cosine term is $0.35$. This means that the graph of the function will be vertically stretched by a factor of $0.35$ compared to the parent cosine function.

To understand the effect of the vertical stretch, let's consider the amplitude of the function. The amplitude of the parent cosine function is $1$, which means that the graph oscillates between $-1$ and $1$. However, in the given function, the amplitude is reduced to $0.35$, which means that the graph will oscillate between $-0.35$ and $0.35$.

Horizontal Stretch

The next transformation we need to consider is the horizontal stretch. In the given function, the coefficient of the $x$ term is $8$. This means that the graph of the function will be horizontally stretched by a factor of $8$ compared to the parent cosine function.

To understand the effect of the horizontal stretch, let's consider the period of the function. The period of the parent cosine function is $2\pi$, which means that the graph repeats itself every $2\pi$ units. However, in the given function, the period is increased to $\frac{2\pi}{8} = \frac{\pi}{4}$, which means that the graph will repeat itself every $\frac{\pi}{4}$ units.

Phase Shift

The final transformation we need to consider is the phase shift. In the given function, the term $\frac{\pi}{4}$ is subtracted from the $x$ term. This means that the graph of the function will be shifted to the right by $\frac{\pi}{4}$ units compared to the parent cosine function.

To understand the effect of the phase shift, let's consider the graph of the function. The graph of the function will be shifted to the right by $\frac{\pi}{4}$ units, which means that the graph will start at $x = \frac{\pi}{4}$ instead of $x = 0$.

Conclusion

In conclusion, the transformations needed to change the parent cosine function to $y = 0.35 \cos \left(8\left(x-\frac{\pi}{4}\right)\right)$ are:

• A vertical stretch of $0.35$
• A horizontal stretch to a period of $16\pi$
• A phase shift of $\frac{\pi}{4}$ units to the right

These transformations will result in a graph that is vertically stretched, horizontally stretched, and phase-shifted compared to the parent cosine function.

Graph of the Function

The graph of the function $y = 0.35 \cos \left(8\left(x-\frac{\pi}{4}\right)\right)$ is shown below:

[Insert graph here]

Key Takeaways

• The parent cosine function is a fundamental concept in mathematics.
• The transformations needed to change the parent cosine function to $y = 0.35 \cos \left(8\left(x-\frac{\pi}{4}\right)\right)$ are a vertical stretch of $0.35$, a horizontal stretch to a period of $16\pi$, and a phase shift of $\frac{\pi}{4}$ units to the right.
• The graph of the function will be vertically stretched, horizontally stretched, and phase-shifted compared to the parent cosine function.

Frequently Asked Questions

Q: What is the parent cosine function? A: The parent cosine function is given by $y = \cos x$.

Q: What is the effect of the vertical stretch on the graph of the function? A: The vertical stretch reduces the amplitude of the graph from $1$ to $0.35$.

Q: What is the effect of the horizontal stretch on the graph of the function? A: The horizontal stretch increases the period of the graph from $2\pi$ to $16\pi$.

Q: What is the effect of the phase shift on the graph of the function? A: The phase shift shifts the graph to the right by $\frac{\pi}{4}$ units.

References

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Further Reading

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Conclusion

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In our previous article, we explored the transformations needed to change the parent cosine function to $y = 0.35 \cos \left(8\left(x-\frac{\pi}{4}\right)\right)$. 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 given by $y = \cos x$. This function has a period of $2\pi$, which means it repeats itself every $2\pi$ units. The graph of the parent cosine function is a smooth, continuous curve that oscillates between $-1$ and $1$.

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

A: The vertical stretch reduces the amplitude of the graph from $1$ to $0.35$. This means that the graph will oscillate between $-0.35$ and $0.35$ instead of $-1$ and $1$.

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

A: The horizontal stretch increases the period of the graph from $2\pi$ to $16\pi$. This means that the graph will repeat itself every $16\pi$ units instead of every $2\pi$ units.

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

A: The phase shift shifts the graph to the right by $\frac{\pi}{4}$ units. This means that the graph will start at $x = \frac{\pi}{4}$ instead of $x = 0$.

Q: How do I determine the period of the graph after a horizontal stretch?

A: To determine the period of the graph after a horizontal stretch, you need to divide the original period by the coefficient of the $x$ term. In this case, the original period is $2\pi$, and the coefficient of the $x$ term is $8$. Therefore, the new period is $\frac{2\pi}{8} = \frac{\pi}{4}$.

Q: How do I determine the amplitude of the graph after a vertical stretch?

A: To determine the amplitude of the graph after a vertical stretch, you need to multiply the original amplitude by the coefficient of the cosine term. In this case, the original amplitude is $1$, and the coefficient of the cosine term is $0.35$. Therefore, the new amplitude is $0.35$.

Q: Can I apply multiple transformations to the parent cosine function?

A: Yes, you can apply multiple transformations to the parent cosine function. For example, you can apply a vertical stretch, a horizontal stretch, and a phase shift to the parent cosine function. The order in which you apply the transformations does not matter, but the final result will be the same.

Q: How do I graph the function after applying multiple transformations?

A: To graph the function after applying multiple transformations, you need to apply each transformation in the correct order. For example, if you apply a vertical stretch, a horizontal stretch, and a phase shift to the parent cosine function, you need to apply the vertical stretch first, followed by the horizontal stretch, and finally the phase shift.

Q: Can I use the parent cosine function as a building block for other functions?

A: Yes, you can use the parent cosine function as a building block for other functions. By applying various transformations to the parent cosine function, you can create a wide range of functions, including sine, cosine, and tangent functions.

Q: What are some common applications of the parent cosine function?

A: The parent cosine function has many common applications in mathematics, physics, and engineering. Some examples include:

• Modeling periodic phenomena, such as sound waves and light waves
• Describing the motion of objects, such as pendulums and springs
• Analyzing the behavior of electrical circuits and electronic systems
• Solving problems in optics and photonics

Conclusion

In conclusion, the parent cosine function is a fundamental concept in mathematics, and it serves as a building block for many other functions. By applying various transformations to the parent cosine function, you can create a wide range of functions, including sine, cosine, and tangent functions. The parent cosine function has many common applications in mathematics, physics, and engineering, and it is an essential tool for solving problems in these fields.