The Graph Of $f(x) = \cos(x$\] Is Transformed To A New Function, $g(x$\], By Reflecting It Over The $x$-axis, Stretching It Horizontally By A Factor Of 3, And Shifting It 4 Units Down. What Is The Equation Of The New Function

In mathematics, transformations of functions are essential concepts that help us understand how functions can be manipulated to create new functions. In this article, we will explore the transformation of the function $f(x) = \cos(x)$ to a new function $g(x)$ by reflecting it over the $x$-axis, stretching it horizontally by a factor of 3, and shifting it 4 units down.

Reflection Over the $x$-Axis

When a function is reflected over the $x$-axis, its graph is flipped upside down. This means that the $y$-coordinates of the points on the graph are negated. In other words, if the original function is $f(x)$, the reflected function is $-f(x)$.

Stretching Horizontally by a Factor of 3

When a function is stretched horizontally by a factor of 3, its graph is compressed vertically by a factor of 3. This means that the $x$-coordinates of the points on the graph are multiplied by 3, while the $y$-coordinates remain the same. In other words, if the original function is $f(x)$, the stretched function is $f(3x)$.

Shifting 4 Units Down

When a function is shifted 4 units down, its graph is moved 4 units down the $y$-axis. This means that the $y$-coordinates of the points on the graph are decreased by 4. In other words, if the original function is $f(x)$, the shifted function is $f(x) - 4$.

Combining the Transformations

Now that we have discussed the individual transformations, let's combine them to find the equation of the new function $g(x)$. We will start with the original function $f(x) = \cos(x)$ and apply the transformations one by one.

1. Reflection Over the $x$-Axis: The reflected function is $-f(x) = -\cos(x)$.
2. Stretching Horizontally by a Factor of 3: The stretched function is $f(3x) = \cos(3x)$.
3. Shifting 4 Units Down: The shifted function is $f(3x) - 4 = \cos(3x) - 4$.

Therefore, the equation of the new function $g(x)$ is:

g(x) = \cos(3x) - 4

Graph of the Transformed Function

To visualize the graph of the transformed function, we can use a graphing calculator or software. The graph of $g(x) = \cos(3x) - 4$ is shown below:

[Insert graph of g(x) = cos(3x) - 4]

Conclusion

In this article, we have discussed the transformation of the function $f(x) = \cos(x)$ to a new function $g(x)$ by reflecting it over the $x$-axis, stretching it horizontally by a factor of 3, and shifting it 4 units down. We have combined the individual transformations to find the equation of the new function $g(x)$, which is $g(x) = \cos(3x) - 4$. The graph of the transformed function is also shown.

References

• [1] "Functions" by Khan Academy
• [2] "Graphing Functions" by Mathway
• [3] "Transformations of Functions" by Purplemath

Further Reading

• "Functions and Graphs" by James Stewart
• "Calculus: Early Transcendentals" by James Stewart
• "Mathematics for the Nonmathematician" by Morris Kline
  Q&A: Transformations of Functions =====================================

In our previous article, we discussed the transformation of the function $f(x) = \cos(x)$ to a new function $g(x)$ by reflecting it over the $x$-axis, stretching it horizontally by a factor of 3, and shifting it 4 units down. In this article, we will answer some frequently asked questions about transformations of functions.

Q: What is the difference between a reflection and a translation?

A: A reflection is a transformation that flips a function over a line, while a translation is a transformation that moves a function up or down. For example, if we reflect the function $f(x) = \cos(x)$ over the $x$-axis, we get the function $-f(x) = -\cos(x)$. On the other hand, if we translate the function $f(x) = \cos(x)$ 4 units down, we get the function $f(x) - 4 = \cos(x) - 4$.

Q: How do I determine the equation of a transformed function?

A: To determine the equation of a transformed function, you need to apply the individual transformations one by one. For example, if we want to reflect the function $f(x) = \cos(x)$ over the $x$-axis, stretch it horizontally by a factor of 3, and shift it 4 units down, we would first reflect the function to get $-f(x) = -\cos(x)$, then stretch it to get $f(3x) = \cos(3x)$, and finally shift it to get $f(3x) - 4 = \cos(3x) - 4$.

Q: What is the effect of stretching a function horizontally by a factor of 3?

A: When a function is stretched horizontally by a factor of 3, its graph is compressed vertically by a factor of 3. This means that the $x$-coordinates of the points on the graph are multiplied by 3, while the $y$-coordinates remain the same. For example, if we stretch the function $f(x) = \cos(x)$ horizontally by a factor of 3, we get the function $f(3x) = \cos(3x)$.

Q: How do I graph a transformed function?

A: To graph a transformed function, you can use a graphing calculator or software. You can also use the individual transformations to graph the function. For example, if we want to graph the function $g(x) = \cos(3x) - 4$, we can first graph the function $f(x) = \cos(x)$, then stretch it horizontally by a factor of 3 to get $f(3x) = \cos(3x)$, and finally shift it 4 units down to get $g(x) = \cos(3x) - 4$.

Q: What are some common transformations of functions?

A: Some common transformations of functions include:

• Reflection over the $x$-axis: $-f(x)$
• Reflection over the $y$-axis: $f(-x)$
• Stretching horizontally by a factor of $a$: $f(ax)$
• Stretching vertically by a factor of $a$: $af(x)$
• Shifting $c$ units up: $f(x) + c$
• Shifting $c$ units down: $f(x) - c$

Q: How do I determine the equation of a function that has been transformed in multiple ways?

A: To determine the equation of a function that has been transformed in multiple ways, you need to apply the individual transformations one by one. For example, if we want to reflect the function $f(x) = \cos(x)$ over the $x$-axis, stretch it horizontally by a factor of 3, and shift it 4 units down, we would first reflect the function to get $-f(x) = -\cos(x)$, then stretch it to get $f(3x) = \cos(3x)$, and finally shift it to get $f(3x) - 4 = \cos(3x) - 4$.

Conclusion

In this article, we have answered some frequently asked questions about transformations of functions. We have discussed the individual transformations, including reflection, stretching, and shifting, and have provided examples of how to apply these transformations to determine the equation of a transformed function. We have also discussed how to graph a transformed function and have listed some common transformations of functions.