The Graph Of $y=\sqrt[3]{x}$ Is Reflected Over The $y$-axis And Then Translated Down 2 Units To Form $ F ( X ) F(x) F ( X ) [/tex]. Which Is The Graph Of $f(x)$?

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Introduction

In mathematics, graph transformations are essential concepts that help us understand how functions behave under various operations. One such operation is the reflection of a graph over the y-axis, followed by a translation down 2 units. In this article, we will explore how the graph of $y=\sqrt[3]{x}$ is transformed when reflected over the y-axis and then translated down 2 units to form $f(x)$.

Reflection Over the y-Axis

When a graph is reflected over the y-axis, the x-coordinates of the points on the graph are negated. This means that if a point (x, y) is on the original graph, the corresponding point on the reflected graph is (-x, y). In the case of the graph of $y=\sqrt[3]{x}$, we can reflect it over the y-axis by replacing x with -x.

The Original Graph

The original graph of $y=\sqrt[3]{x}$ is a cubic root function, which can be represented as:

y=x3y = \sqrt[3]{x}

This graph is a curve that passes through the origin (0, 0) and has a positive slope.

The Reflected Graph

When we reflect the original graph over the y-axis, we get a new graph where the x-coordinates are negated. This can be represented as:

y=โˆ’x3y = \sqrt[3]{-x}

The reflected graph is a mirror image of the original graph across the y-axis.

Translation Down 2 Units

After reflecting the graph over the y-axis, we need to translate it down 2 units to form $f(x)$. This means that for every point (x, y) on the reflected graph, the corresponding point on the translated graph is (x, y - 2).

The Translated Graph

The translated graph of $y=\sqrt[3]{-x}$ down 2 units can be represented as:

f(x)=โˆ’x3โˆ’2f(x) = \sqrt[3]{-x} - 2

This graph is a cubic root function that has been reflected over the y-axis and then translated down 2 units.

Graph of $f(x)$

To determine the graph of $f(x)$, we need to visualize the reflected and translated graph of $y=\sqrt[3]{x}$. Since the reflected graph is a mirror image of the original graph across the y-axis, and the translated graph is 2 units below the reflected graph, we can conclude that the graph of $f(x)$ is a cubic root function that has been reflected over the y-axis and then translated down 2 units.

Graphical Representation

The graph of $f(x)$ can be represented as:

f(x) = โˆ›(-x) - 2

This graph is a cubic root function that has been reflected over the y-axis and then translated down 2 units.

Conclusion

In conclusion, the graph of $y=\sqrt[3]{x}$ is reflected over the y-axis and then translated down 2 units to form $f(x)$. The graph of $f(x)$ is a cubic root function that has been reflected over the y-axis and then translated down 2 units. This transformation can be represented as:

f(x)=โˆ’x3โˆ’2f(x) = \sqrt[3]{-x} - 2

The graph of $f(x)$ is a cubic root function that has been transformed through reflection and translation, resulting in a unique graph that is different from the original graph of $y=\sqrt[3]{x}$.

References

Keywords

  • Graph transformations
  • Reflection over the y-axis
  • Translation down 2 units
  • Cubic root function
  • Graph of $f(x)$
  • Reflection and translation of graphs

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Introduction

In our previous article, we explored how the graph of $y=\sqrt[3]{x}$ is transformed when reflected over the y-axis and then translated down 2 units to form $f(x)$. In this article, we will answer some frequently asked questions related to this topic.

Q&A

Q: What is the reflection of the graph of $y=\sqrt[3]{x}$ over the y-axis?

A: The reflection of the graph of $y=\sqrt[3]{x}$ over the y-axis is obtained by replacing x with -x. This can be represented as:

y=โˆ’x3y = \sqrt[3]{-x}

Q: What is the translation of the reflected graph down 2 units?

A: The translation of the reflected graph down 2 units is obtained by subtracting 2 from the y-coordinate of each point on the reflected graph. This can be represented as:

f(x)=โˆ’x3โˆ’2f(x) = \sqrt[3]{-x} - 2

Q: What is the graph of $f(x)$?

A: The graph of $f(x)$ is a cubic root function that has been reflected over the y-axis and then translated down 2 units. It can be represented as:

f(x) = โˆ›(-x) - 2

Q: How do I visualize the graph of $f(x)$?

A: To visualize the graph of $f(x)$, you can start with the original graph of $y=\sqrt[3]{x}$ and reflect it over the y-axis. Then, translate the reflected graph down 2 units to obtain the graph of $f(x)$.

Q: What are some common graph transformations?

A: Some common graph transformations include:

  • Reflection over the x-axis
  • Reflection over the y-axis
  • Translation up or down
  • Translation left or right
  • Rotation

Q: How do I apply graph transformations to a function?

A: To apply graph transformations to a function, you can use the following steps:

  1. Identify the type of transformation you want to apply (e.g. reflection, translation, rotation).
  2. Determine the direction and magnitude of the transformation.
  3. Apply the transformation to the function using the appropriate mathematical operations.

Conclusion

In conclusion, the graph of $y=\sqrt[3]{x}$ is reflected over the y-axis and then translated down 2 units to form $f(x)$. We have answered some frequently asked questions related to this topic, including the reflection and translation of the graph, the graph of $f(x)$, and common graph transformations.

References

Keywords

  • Graph transformations
  • Reflection over the y-axis
  • Translation down 2 units
  • Cubic root function
  • Graph of $f(x)$
  • Reflection and translation of graphs
  • Graph transformations
  • Reflection and translation of functions