The Graph Of The Parent Function $f(x)=x^3$ Is Transformed Such That $g(x)=f\left(-2x^3\right$\]. How Does The Graph Of $g(x$\] Compare To The Graph Of $f(x$\]?A. $g(x$\] Is Stretched Horizontally And Reflected

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Introduction

In mathematics, the transformation of a parent function is a crucial concept in understanding the behavior of various functions. The parent function f(x)=x3f(x)=x^3 is a fundamental cubic function that serves as a basis for understanding the properties of cubic functions. In this article, we will explore how the graph of the parent function f(x)=x3f(x)=x^3 is transformed to obtain the function g(x)=f(−2x3)g(x)=f\left(-2x^3\right).

Understanding the Parent Function f(x)=x3f(x)=x^3

The parent function f(x)=x3f(x)=x^3 is a cubic function that exhibits a characteristic "S" shape. This function has a root at x=0x=0, and its graph passes through the origin. The function is also symmetric about the origin, meaning that for every point (x,y)(x, y) on the graph, the point (−x,−y)(-x, -y) is also on the graph.

The Transformation of the Parent Function

The transformation of the parent function f(x)=x3f(x)=x^3 to obtain the function g(x)=f(−2x3)g(x)=f\left(-2x^3\right) involves two main steps:

  1. Horizontal Stretching: The function g(x)g(x) is obtained by replacing xx with −2x3-2x^3 in the parent function f(x)=x3f(x)=x^3. This replacement causes the graph of g(x)g(x) to be stretched horizontally by a factor of 23=82^3=8.
  2. Reflection: The negative sign in front of 2x32x^3 causes the graph of g(x)g(x) to be reflected about the y-axis.

Comparing the Graphs of f(x)f(x) and g(x)g(x)

The graph of g(x)=f(−2x3)g(x)=f\left(-2x^3\right) is compared to the graph of f(x)=x3f(x)=x^3 in the following ways:

  • Horizontal Stretching: The graph of g(x)g(x) is stretched horizontally by a factor of 88 compared to the graph of f(x)f(x).
  • Reflection: The graph of g(x)g(x) is reflected about the y-axis compared to the graph of f(x)f(x).
  • Amplification: The graph of g(x)g(x) is amplified vertically by a factor of 23=82^3=8 compared to the graph of f(x)f(x).

Conclusion

In conclusion, the graph of the parent function f(x)=x3f(x)=x^3 is transformed to obtain the function g(x)=f(−2x3)g(x)=f\left(-2x^3\right) by applying a horizontal stretching factor of 88 and a reflection about the y-axis. The resulting graph of g(x)g(x) is compared to the graph of f(x)f(x) in terms of horizontal stretching, reflection, and amplification.

Key Takeaways

  • The transformation of the parent function f(x)=x3f(x)=x^3 to obtain the function g(x)=f(−2x3)g(x)=f\left(-2x^3\right) involves horizontal stretching and reflection.
  • The graph of g(x)g(x) is stretched horizontally by a factor of 88 compared to the graph of f(x)f(x).
  • The graph of g(x)g(x) is reflected about the y-axis compared to the graph of f(x)f(x).
  • The graph of g(x)g(x) is amplified vertically by a factor of 23=82^3=8 compared to the graph of f(x)f(x).

Final Thoughts

The transformation of the parent function f(x)=x3f(x)=x^3 to obtain the function g(x)=f(−2x3)g(x)=f\left(-2x^3\right) is a crucial concept in understanding the behavior of various functions. By applying horizontal stretching and reflection, the resulting graph of g(x)g(x) exhibits unique properties compared to the graph of f(x)f(x). This article provides a comprehensive understanding of the transformation of the parent function and its implications on the resulting graph.

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Introduction

In our previous article, we explored how the graph of the parent function f(x)=x3f(x)=x^3 is transformed to obtain the function g(x)=f(−2x3)g(x)=f\left(-2x^3\right). In this article, we will answer some frequently asked questions related to the transformation of the parent function.

Q&A

Q1: What is the effect of the horizontal stretching factor of 8 on the graph of g(x)g(x)?

A1: The horizontal stretching factor of 8 causes the graph of g(x)g(x) to be stretched horizontally by a factor of 8 compared to the graph of f(x)f(x). This means that the graph of g(x)g(x) will be wider than the graph of f(x)f(x).

Q2: How does the reflection about the y-axis affect the graph of g(x)g(x)?

A2: The reflection about the y-axis causes the graph of g(x)g(x) to be reflected about the y-axis compared to the graph of f(x)f(x). This means that the graph of g(x)g(x) will be a mirror image of the graph of f(x)f(x) about the y-axis.

Q3: What is the effect of the amplification factor of 8 on the graph of g(x)g(x)?

A3: The amplification factor of 8 causes the graph of g(x)g(x) to be amplified vertically by a factor of 8 compared to the graph of f(x)f(x). This means that the graph of g(x)g(x) will be taller than the graph of f(x)f(x).

Q4: How does the transformation of the parent function affect the roots of the graph?

A4: The transformation of the parent function causes the roots of the graph to change. The roots of the graph of g(x)g(x) will be different from the roots of the graph of f(x)f(x).

Q5: Can the transformation of the parent function be applied to other functions?

A5: Yes, the transformation of the parent function can be applied to other functions. The transformation involves replacing the variable with a function of the variable and applying the necessary transformations.

Conclusion

In conclusion, the transformation of the parent function f(x)=x3f(x)=x^3 to obtain the function g(x)=f(−2x3)g(x)=f\left(-2x^3\right) involves horizontal stretching, reflection, and amplification. The resulting graph of g(x)g(x) exhibits unique properties compared to the graph of f(x)f(x). This article provides a comprehensive understanding of the transformation of the parent function and its implications on the resulting graph.

Key Takeaways

  • The transformation of the parent function f(x)=x3f(x)=x^3 to obtain the function g(x)=f(−2x3)g(x)=f\left(-2x^3\right) involves horizontal stretching, reflection, and amplification.
  • The graph of g(x)g(x) is stretched horizontally by a factor of 8 compared to the graph of f(x)f(x).
  • The graph of g(x)g(x) is reflected about the y-axis compared to the graph of f(x)f(x).
  • The graph of g(x)g(x) is amplified vertically by a factor of 8 compared to the graph of f(x)f(x).

Final Thoughts

The transformation of the parent function f(x)=x3f(x)=x^3 to obtain the function g(x)=f(−2x3)g(x)=f\left(-2x^3\right) is a crucial concept in understanding the behavior of various functions. By applying horizontal stretching, reflection, and amplification, the resulting graph of g(x)g(x) exhibits unique properties compared to the graph of f(x)f(x). This article provides a comprehensive understanding of the transformation of the parent function and its implications on the resulting graph.

Additional Resources

For further reading on the transformation of the parent function, we recommend the following resources:

  • Math Open Reference: A comprehensive online reference for mathematics.
  • Khan Academy: A free online learning platform for mathematics and other subjects.
  • Wolfram Alpha: A computational knowledge engine for mathematics and other subjects.