Describe The Transformation(s) Made From F ( X ) = X 3 + X 2 + 1 F(x)=x^3+x^2+1 F ( X ) = X 3 + X 2 + 1 To G ( X ) = 3 ⋅ F ( X ) + 2 G(x)=3 \cdot F(x)+2 G ( X ) = 3 ⋅ F ( X ) + 2 .A. Translation Left 2 B. Translation Right 2 C. Vertical Stretch By A Factor Of 3 D. Translation Up 2 E. Vertical Shrink By A Factor Of 3 F.

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Introduction

In mathematics, functions are used to describe the relationship between variables. When we transform a function, we are essentially changing its shape or position. In this article, we will explore the transformation made from the function f(x)=x3+x2+1f(x)=x^3+x^2+1 to g(x)=3⋅f(x)+2g(x)=3 \cdot f(x)+2. We will analyze the changes made to the original function and determine the type of transformation that occurred.

Understanding the Original Function

The original function is given by f(x)=x3+x2+1f(x)=x^3+x^2+1. This is a cubic function, which means it has a cubic term (x3x^3) as its highest degree term. The function has a positive leading coefficient, which means it opens upwards.

Understanding the Transformed Function

The transformed function is given by g(x)=3⋅f(x)+2g(x)=3 \cdot f(x)+2. To understand the transformation, we need to break it down into two parts: the vertical stretch and the vertical translation.

Vertical Stretch

The vertical stretch is represented by the factor of 3 in front of the function f(x)f(x). This means that the function f(x)f(x) is stretched vertically by a factor of 3. In other words, the function f(x)f(x) is multiplied by 3, resulting in a new function that is 3 times the height of the original function.

Vertical Translation

The vertical translation is represented by the addition of 2 to the function f(x)f(x). This means that the function f(x)f(x) is shifted upwards by 2 units. In other words, the function f(x)f(x) is translated vertically by 2 units.

Determining the Type of Transformation

Now that we have analyzed the changes made to the original function, we can determine the type of transformation that occurred. Based on our analysis, we can conclude that the transformation made from f(x)=x3+x2+1f(x)=x^3+x^2+1 to g(x)=3⋅f(x)+2g(x)=3 \cdot f(x)+2 is a vertical stretch by a factor of 3 followed by a vertical translation up 2.

Conclusion

In conclusion, the transformation made from f(x)=x3+x2+1f(x)=x^3+x^2+1 to g(x)=3⋅f(x)+2g(x)=3 \cdot f(x)+2 is a vertical stretch by a factor of 3 followed by a vertical translation up 2. This type of transformation is commonly used in mathematics to describe the relationship between variables.

Key Takeaways

  • The transformation made from f(x)=x3+x2+1f(x)=x^3+x^2+1 to g(x)=3⋅f(x)+2g(x)=3 \cdot f(x)+2 is a vertical stretch by a factor of 3 followed by a vertical translation up 2.
  • The vertical stretch is represented by the factor of 3 in front of the function f(x)f(x).
  • The vertical translation is represented by the addition of 2 to the function f(x)f(x).
  • The type of transformation that occurred is a vertical stretch by a factor of 3 followed by a vertical translation up 2.

Frequently Asked Questions

Q: What is the type of transformation made from f(x)=x3+x2+1f(x)=x^3+x^2+1 to g(x)=3⋅f(x)+2g(x)=3 \cdot f(x)+2?

A: The transformation made from f(x)=x3+x2+1f(x)=x^3+x^2+1 to g(x)=3⋅f(x)+2g(x)=3 \cdot f(x)+2 is a vertical stretch by a factor of 3 followed by a vertical translation up 2.

Q: What is the vertical stretch represented by in the transformed function?

A: The vertical stretch is represented by the factor of 3 in front of the function f(x)f(x).

Q: What is the vertical translation represented by in the transformed function?

A: The vertical translation is represented by the addition of 2 to the function f(x)f(x).

Q: What is the type of transformation that occurred?

A: The type of transformation that occurred is a vertical stretch by a factor of 3 followed by a vertical translation up 2.

References

Glossary

  • Vertical Stretch: A transformation that changes the height of a function.
  • Vertical Translation: A transformation that changes the position of a function.
  • Cubic Function: A function with a cubic term as its highest degree term.
  • Leading Coefficient: The coefficient of the highest degree term in a function.
    Q&A: Transformations of Functions =====================================

Introduction

In our previous article, we explored the transformation made from the function f(x)=x3+x2+1f(x)=x^3+x^2+1 to g(x)=3⋅f(x)+2g(x)=3 \cdot f(x)+2. We analyzed the changes made to the original function and determined the type of transformation that occurred. In this article, we will answer some frequently asked questions about transformations of functions.

Q&A

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

A: A vertical stretch is a transformation that increases the height of a function, while a vertical shrink is a transformation that decreases the height of a function.

Q: How do you determine the type of transformation that occurred?

A: To determine the type of transformation that occurred, you need to analyze the changes made to the original function. Look for any changes in the height, width, or position of the function.

Q: What is the effect of a vertical translation on a function?

A: A vertical translation changes the position of a function. It shifts the function up or down by a certain number of units.

Q: What is the effect of a horizontal translation on a function?

A: A horizontal translation changes the position of a function. It shifts the function left or right by a certain number of units.

Q: Can a function undergo multiple transformations?

A: Yes, a function can undergo multiple transformations. Each transformation can change the function in a different way, resulting in a new function.

Q: How do you represent a vertical stretch in a function?

A: A vertical stretch is represented by a factor in front of the function. For example, if a function is stretched vertically by a factor of 3, it would be represented as 3⋅f(x)3 \cdot f(x).

Q: How do you represent a vertical shrink in a function?

A: A vertical shrink is represented by a factor in front of the function, but the factor is less than 1. For example, if a function is shrunk vertically by a factor of 1/3, it would be represented as (1/3)⋅f(x)(1/3) \cdot f(x).

Q: What is the effect of a reflection on a function?

A: A reflection changes the direction of a function. It can be a reflection across the x-axis or the y-axis.

Q: Can a function undergo a reflection and a translation at the same time?

A: Yes, a function can undergo a reflection and a translation at the same time. Each transformation can change the function in a different way, resulting in a new function.

Q: How do you represent a reflection across the x-axis in a function?

A: A reflection across the x-axis is represented by multiplying the function by -1. For example, if a function is reflected across the x-axis, it would be represented as −f(x)-f(x).

Q: How do you represent a reflection across the y-axis in a function?

A: A reflection across the y-axis is represented by replacing x with -x in the function. For example, if a function is reflected across the y-axis, it would be represented as f(−x)f(-x).

Conclusion

In conclusion, transformations of functions are an important concept in mathematics. Understanding how to determine the type of transformation that occurred and how to represent different transformations in a function is crucial for solving problems and analyzing functions.

Key Takeaways

  • A vertical stretch increases the height of a function.
  • A vertical shrink decreases the height of a function.
  • A vertical translation changes the position of a function.
  • A horizontal translation changes the position of a function.
  • A reflection changes the direction of a function.
  • A function can undergo multiple transformations.

Frequently Asked Questions

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

A: A vertical stretch is a transformation that increases the height of a function, while a vertical shrink is a transformation that decreases the height of a function.

Q: How do you determine the type of transformation that occurred?

A: To determine the type of transformation that occurred, you need to analyze the changes made to the original function. Look for any changes in the height, width, or position of the function.

Q: What is the effect of a vertical translation on a function?

A: A vertical translation changes the position of a function. It shifts the function up or down by a certain number of units.

Q: What is the effect of a horizontal translation on a function?

A: A horizontal translation changes the position of a function. It shifts the function left or right by a certain number of units.

Q: Can a function undergo multiple transformations?

A: Yes, a function can undergo multiple transformations. Each transformation can change the function in a different way, resulting in a new function.

References

Glossary

  • Vertical Stretch: A transformation that increases the height of a function.
  • Vertical Shrink: A transformation that decreases the height of a function.
  • Vertical Translation: A transformation that changes the position of a function.
  • Horizontal Translation: A transformation that changes the position of a function.
  • Reflection: A transformation that changes the direction of a function.