How Is The Graph Of The Parent Function Of Y = X 3 Y=\sqrt[3]{x} Y = 3 X Transformed To Produce The Graph Y = 1 2 X 3 Y=\sqrt[3]{\frac{1}{2} X} Y = 3 2 1 X ?A. It Is Horizontally Stretched By A Factor Of 1 2 \frac{1}{2} 2 1 .B. It Is Vertically Stretched By A Factor Of

Mar 10, 2025 by ADMIN 278 views

**Understanding Transformations of Parent Functions in Mathematics**

Introduction

In mathematics, parent functions are the basic functions from which other functions can be derived through various transformations. Understanding these transformations is crucial in graphing and analyzing functions. In this article, we will explore how the graph of the parent function $y=\sqrt[3]{x}$ is transformed to produce the graph $y=\sqrt[3]{\frac{1}{2} x}$ .

What are Parent Functions?

Parent functions are the simplest forms of functions that can be transformed to produce other functions. They are the foundation of function families and are used to derive other functions through various transformations. In the case of the cube root function, the parent function is $y=\sqrt[3]{x}$ .

Transformations of Parent Functions

Transformations of parent functions involve changing the position, size, or orientation of the graph of the parent function. There are four types of transformations: horizontal shifts, vertical shifts, horizontal stretches, and vertical stretches.

Horizontal Shifts: A horizontal shift occurs when the graph of the parent function is moved to the left or right. This is achieved by adding or subtracting a constant value from the input variable.
Vertical Shifts: A vertical shift occurs when the graph of the parent function is moved up or down. This is achieved by adding or subtracting a constant value from the output variable.
Horizontal Stretches: A horizontal stretch occurs when the graph of the parent function is stretched horizontally. This is achieved by multiplying the input variable by a constant value greater than 1.
Vertical Stretches: A vertical stretch occurs when the graph of the parent function is stretched vertically. This is achieved by multiplying the output variable by a constant value greater than 1.

Transforming the Graph of $y=\sqrt[3]{x}$

To transform the graph of $y=\sqrt[3]{x}$ to produce the graph $y=\sqrt[3]{\frac{1}{2} x}$ , we need to apply a horizontal stretch by a factor of $\frac{1}{2}$ .

Why is a Horizontal Stretch Applied?

A horizontal stretch is applied because the input variable $x$ is multiplied by $\frac{1}{2}$ . This means that the graph of the parent function is stretched horizontally by a factor of $\frac{1}{2}$ .

How to Apply a Horizontal Stretch

To apply a horizontal stretch, we need to multiply the input variable $x$ by a constant value greater than 1. In this case, we multiply $x$ by $\frac{1}{2}$ .

The Final Graph

The final graph of $y=\sqrt[3]{\frac{1}{2} x}$ is a horizontal stretch of the graph of $y=\sqrt[3]{x}$ by a factor of $\frac{1}{2}$ .

Conclusion

In conclusion, the graph of the parent function $y=\sqrt[3]{x}$ is transformed to produce the graph $y=\sqrt[3]{\frac{1}{2} x}$ by applying a horizontal stretch by a factor of $\frac{1}{2}$ . This transformation involves multiplying the input variable $x$ by $\frac{1}{2}$ , resulting in a horizontal stretch of the graph of the parent function.

References

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

Discussion

What are some other transformations that can be applied to the graph of $y=\sqrt[3]{x}$ ? How do these transformations affect the graph of the parent function?

Answer

Some other transformations that can be applied to the graph of $y=\sqrt[3]{x}$ include:

Vertical Shifts: Adding or subtracting a constant value from the output variable.
Horizontal Shifts: Adding or subtracting a constant value from the input variable.
Vertical Stretches: Multiplying the output variable by a constant value greater than 1.
Horizontal Compressions: Multiplying the input variable by a constant value less than 1.

These transformations can be applied to the graph of $y=\sqrt[3]{x}$ to produce different graphs, each with its own unique characteristics.

Final Thoughts

Q: What is the difference between a horizontal stretch and a horizontal compression?

A: A horizontal stretch occurs when the graph of the parent function is stretched horizontally, resulting in a wider graph. A horizontal compression occurs when the graph of the parent function is compressed horizontally, resulting in a narrower graph.

Q: How do you apply a horizontal stretch to the graph of $y=\sqrt[3]{x}$ ?

A: To apply a horizontal stretch, you need to multiply the input variable $x$ by a constant value greater than 1. For example, to apply a horizontal stretch by a factor of $\frac{1}{2}$ , you would multiply $x$ by $\frac{1}{2}$ , resulting in the graph $y=\sqrt[3]{\frac{1}{2} x}$ .

Q: What is the effect of a vertical stretch on the graph of $y=\sqrt[3]{x}$ ?

A: A vertical stretch occurs when the graph of the parent function is stretched vertically, resulting in a taller graph. To apply a vertical stretch, you need to multiply the output variable by a constant value greater than 1. For example, to apply a vertical stretch by a factor of 2, you would multiply $y$ by 2, resulting in the graph $y=2\sqrt[3]{x}$ .

Q: How do you apply a horizontal shift to the graph of $y=\sqrt[3]{x}$ ?

A: To apply a horizontal shift, you need to add or subtract a constant value from the input variable $x$ . For example, to apply a horizontal shift of 2 units to the right, you would add 2 to $x$ , resulting in the graph $y=\sqrt[3]{x-2}$ .

Q: What is the effect of a vertical shift on the graph of $y=\sqrt[3]{x}$ ?

A: A vertical shift occurs when the graph of the parent function is shifted up or down. To apply a vertical shift, you need to add or subtract a constant value from the output variable $y$ . For example, to apply a vertical shift of 2 units up, you would add 2 to $y$ , resulting in the graph $y=\sqrt[3]{x}+2$ .

Q: Can you give an example of a function that is a combination of multiple transformations?

A: Yes, here is an example of a function that is a combination of multiple transformations:

$y=2\sqrt[3]{\frac{1}{2} (x-2)}+1$

This function is a combination of a horizontal shift of 2 units to the right, a horizontal stretch by a factor of $\frac{1}{2}$ , a vertical stretch by a factor of 2, and a vertical shift of 1 unit up.

Q: How do you determine the order of transformations when applying multiple transformations to a graph?

A: When applying multiple transformations to a graph, it is essential to determine the order of transformations. The general rule is to apply transformations in the following order:

Horizontal shifts
Horizontal stretches and compressions
Vertical shifts
Vertical stretches and compressions

By following this order, you can ensure that the transformations are applied correctly and that the resulting graph is accurate.

Q: What are some common mistakes to avoid when applying transformations to a graph?

A: Some common mistakes to avoid when applying transformations to a graph include:

Applying transformations in the wrong order
Failing to account for the effects of multiple transformations
Not considering the impact of transformations on the graph's asymptotes and intercepts
Not checking the graph for accuracy and completeness

By avoiding these common mistakes, you can ensure that your transformations are accurate and effective.