Select The Correct Answer.Function \[$ H \$\] Is A Transformation Of The Parent Cubic Function \[$ F \$\].$\[ H(x) = 2 \sqrt[3]{x} + 1 \\]Which Phrase Describes A Transformation Of The Graph Of Function \[$ F \$\] To

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Understanding Function Transformations: A Closer Look at the Graph of h(x) = 2 ∛x + 1

In mathematics, function transformations are a crucial concept in understanding how a parent function can be modified to create a new function. These transformations can be vertical, horizontal, or a combination of both, and they play a vital role in graphing and analyzing functions. In this article, we will delve into the world of function transformations and explore how the graph of a parent cubic function can be transformed to create a new function.

What is a Function Transformation?

A function transformation is a process of modifying a parent function to create a new function. This can be achieved by applying various transformations such as vertical shifts, horizontal shifts, reflections, and stretches. The resulting function will have a different graph than the parent function, and understanding these transformations is essential in graphing and analyzing functions.

The Parent Cubic Function

The parent cubic function is a fundamental function in mathematics, and it is defined as f(x) = x^3. This function has a characteristic "S" shape, and it is used as a building block to create other functions. In this article, we will be working with a transformation of the parent cubic function, specifically the function h(x) = 2 ∛x + 1.

The Transformation Function h(x) = 2 ∛x + 1

The function h(x) = 2 ∛x + 1 is a transformation of the parent cubic function f(x) = x^3. To understand this transformation, let's break it down into its components. The function h(x) has two main components: the cube root of x (∛x) and the constant 2. The cube root of x is a radical function that takes the cube root of x, while the constant 2 is a vertical stretch factor.

Vertical Stretch

The constant 2 in the function h(x) = 2 ∛x + 1 is a vertical stretch factor. This means that the graph of the function h(x) will be stretched vertically by a factor of 2 compared to the graph of the parent cubic function f(x) = x^3. This stretch will result in a taller graph, with the same width as the parent function.

Horizontal Compression

The cube root of x (∛x) in the function h(x) = 2 ∛x + 1 is a horizontal compression factor. This means that the graph of the function h(x) will be compressed horizontally by a factor of 3 compared to the graph of the parent cubic function f(x) = x^3. This compression will result in a narrower graph, with the same height as the parent function.

Reflection

The constant 1 in the function h(x) = 2 ∛x + 1 is a reflection factor. This means that the graph of the function h(x) will be reflected across the x-axis compared to the graph of the parent cubic function f(x) = x^3. This reflection will result in a graph that is a mirror image of the parent function.

Which Phrase Describes a Transformation of the Graph of Function f(x) = x^3 to h(x) = 2 ∛x + 1?

Based on our analysis of the function h(x) = 2 ∛x + 1, we can conclude that the phrase that describes a transformation of the graph of function f(x) = x^3 to h(x) = 2 ∛x + 1 is:

  • The graph of h(x) = 2 ∛x + 1 is a vertical stretch and horizontal compression of the graph of f(x) = x^3.

This phrase accurately describes the transformations that have been applied to the parent cubic function f(x) = x^3 to create the function h(x) = 2 ∛x + 1.

In our previous article, we explored the concept of function transformations and analyzed the function h(x) = 2 ∛x + 1, a transformation of the parent cubic function f(x) = x^3. In this article, we will continue to delve into the world of function transformations and provide a Q&A guide to help you better understand this concept.

Q: What is a function transformation?

A: A function transformation is a process of modifying a parent function to create a new function. This can be achieved by applying various transformations such as vertical shifts, horizontal shifts, reflections, and stretches.

Q: What are the different types of function transformations?

A: There are several types of function transformations, including:

  • Vertical shifts: These transformations involve moving the graph of a function up or down by a certain distance.
  • Horizontal shifts: These transformations involve moving the graph of a function left or right by a certain distance.
  • Reflections: These transformations involve reflecting the graph of a function across the x-axis or y-axis.
  • Stretches: These transformations involve stretching the graph of a function vertically or horizontally.

Q: How do I determine the type of function transformation?

A: To determine the type of function transformation, you need to analyze the function and identify the transformations that have been applied. This can be done by looking at the function equation and identifying the transformations that have been applied.

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

A: A vertical stretch involves stretching the graph of a function vertically by a certain factor, while a horizontal compression involves compressing the graph of a function horizontally by a certain factor.

Q: How do I apply a vertical stretch to a function?

A: To apply a vertical stretch to a function, you need to multiply the function by a certain factor. For example, if you want to apply a vertical stretch of 2 to the function f(x) = x^2, you would multiply the function by 2 to get the new function f(x) = 2x^2.

Q: How do I apply a horizontal compression to a function?

A: To apply a horizontal compression to a function, you need to replace x with x/a, where a is the compression factor. For example, if you want to apply a horizontal compression of 2 to the function f(x) = x^2, you would replace x with x/2 to get the new function f(x) = (x/2)^2.

Q: What is the difference between a reflection across the x-axis and a reflection across the y-axis?

A: A reflection across the x-axis involves reflecting the graph of a function across the x-axis, while a reflection across the y-axis involves reflecting the graph of a function across the y-axis.

Q: How do I apply a reflection across the x-axis to a function?

A: To apply a reflection across the x-axis to a function, you need to multiply the function by -1. For example, if you want to apply a reflection across the x-axis to the function f(x) = x^2, you would multiply the function by -1 to get the new function f(x) = -x^2.

Q: How do I apply a reflection across the y-axis to a function?

A: To apply a reflection across the y-axis to a function, you need to replace x with -x. For example, if you want to apply a reflection across the y-axis to the function f(x) = x^2, you would replace x with -x to get the new function f(x) = (-x)^2.

In conclusion, function transformations are a crucial concept in mathematics, and understanding how a parent function can be modified to create a new function is essential in graphing and analyzing functions. By analyzing the function h(x) = 2 ∛x + 1, we can conclude that it involves a vertical stretch, horizontal compression, and reflection. We hope that this Q&A guide has helped you better understand the concept of function transformations and how to apply them to different functions.