Which Transformation Is Applied If A Quadratic Function Y = X 2 Y = X^2 Y = X 2 Is Changed To Y = 2 X 2 Y = 2x^2 Y = 2 X 2 ?A. Vertical Stretch B. Horizontal Compression C. Horizontal Stretch D. Vertical Shift

Introduction

In mathematics, transformations of functions refer to the process of changing the position, size, or shape of a function's graph. These transformations are essential in understanding various mathematical concepts, including algebra, geometry, and calculus. In this article, we will focus on the transformation applied when a quadratic function $y = x^2$ is changed to $y = 2x^2$.

What is a Quadratic Function?

A quadratic function is a polynomial function of degree two, which means the highest power of the variable is two. The general form of a quadratic function is $y = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants. The graph of a quadratic function is a parabola, which is a U-shaped curve.

The Original Function: $y = x^2$

The original function $y = x^2$ is a quadratic function with a leading coefficient of 1. This means that the parabola opens upwards, and its vertex is at the origin (0, 0). The graph of this function is a simple parabola that passes through the points (0, 0), (1, 1), and (-1, 1).

The Transformed Function: $y = 2x^2$

When the original function $y = x^2$ is changed to $y = 2x^2$, the leading coefficient is multiplied by 2. This transformation affects the graph of the function in a significant way.

Understanding the Transformation

To understand the transformation applied to the function, let's analyze the effect of multiplying the leading coefficient by 2. When the leading coefficient is multiplied by a positive number, the parabola is stretched vertically. This means that the distance between the vertex and the points on the graph increases.

Vertical Stretch vs. Horizontal Stretch

A vertical stretch is a transformation that changes the size of the graph in the vertical direction. In this case, the graph of $y = 2x^2$ is stretched vertically because the leading coefficient is multiplied by 2. On the other hand, a horizontal stretch would change the size of the graph in the horizontal direction.

Conclusion

In conclusion, when a quadratic function $y = x^2$ is changed to $y = 2x^2$, the transformation applied is a vertical stretch. This is because the leading coefficient is multiplied by 2, which affects the size of the graph in the vertical direction.

Key Takeaways

• A quadratic function is a polynomial function of degree two.
• The graph of a quadratic function is a parabola.
• A vertical stretch is a transformation that changes the size of the graph in the vertical direction.
• When the leading coefficient of a quadratic function is multiplied by a positive number, the parabola is stretched vertically.

Frequently Asked Questions

Q: What is the effect of multiplying the leading coefficient by 2 on the graph of a quadratic function?

A: Multiplying the leading coefficient by 2 stretches the graph of the quadratic function vertically.

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

A: A vertical stretch changes the size of the graph in the vertical direction, while a horizontal stretch changes the size of the graph in the horizontal direction.

Q: How does the transformation of a quadratic function affect its vertex?

A: The transformation of a quadratic function does not affect its vertex. The vertex remains at the same point on the graph.

Q: Can a quadratic function be stretched horizontally?

A: No, a quadratic function cannot be stretched horizontally. However, it can be stretched vertically or compressed horizontally.

References

• [1] Algebra, 2nd Edition, Michael Artin, Prentice Hall, 2010.
• [2] Calculus, 3rd Edition, Michael Spivak, Publish or Perish, 2008.
• [3] Geometry, 2nd Edition, Michael Spivak, Publish or Perish, 2008.

Glossary

• Quadratic function: A polynomial function of degree two.
• Parabola: A U-shaped curve that is the graph of a quadratic function.
• Leading coefficient: The coefficient of the highest power of the variable in a polynomial function.
• Vertical stretch: A transformation that changes the size of the graph in the vertical direction.
• Horizontal stretch: A transformation that changes the size of the graph in the horizontal direction.
  Quadratic Function Transformations: A Q&A Guide =====================================================

Introduction

In our previous article, we discussed the transformation applied when a quadratic function $y = x^2$ is changed to $y = 2x^2$. We learned that the transformation applied is a vertical stretch. In this article, we will continue to explore quadratic function transformations and answer some frequently asked questions.

Q&A Guide

Q: What is the effect of multiplying the leading coefficient by a negative number on the graph of a quadratic function?

A: Multiplying the leading coefficient by a negative number reflects the graph of the quadratic function across the x-axis.

Q: What is the effect of multiplying the leading coefficient by a fraction on the graph of a quadratic function?

A: Multiplying the leading coefficient by a fraction compresses the graph of the quadratic function vertically.

Q: What is the effect of adding a constant to the function on the graph of a quadratic function?

A: Adding a constant to the function shifts the graph of the quadratic function vertically.

Q: What is the effect of subtracting a constant from the function on the graph of a quadratic function?

A: Subtracting a constant from the function shifts the graph of the quadratic function vertically.

Q: Can a quadratic function be transformed in both the x and y directions?

A: Yes, a quadratic function can be transformed in both the x and y directions. For example, the function $y = 2(x - 1)^2 + 3$ is a quadratic function that has been transformed in both the x and y directions.

Q: How do you determine the type of transformation applied to a quadratic function?

A: To determine the type of transformation applied to a quadratic function, you need to analyze the changes made to the function. If the leading coefficient is multiplied by a positive number, the graph is stretched vertically. If the leading coefficient is multiplied by a negative number, the graph is reflected across the x-axis. If the function is shifted vertically, the graph is shifted up or down.

Q: Can a quadratic function be transformed in a way that changes its shape?

A: Yes, a quadratic function can be transformed in a way that changes its shape. For example, the function $y = -x^2$ is a quadratic function that has been transformed in a way that changes its shape.

Q: How do you graph a quadratic function that has been transformed?

A: To graph a quadratic function that has been transformed, you need to apply the transformations to the original graph. For example, if the function $y = x^2$ is transformed into $y = 2(x - 1)^2 + 3$, you need to apply the transformations of stretching, reflecting, and shifting to the original graph.

Common Transformations of Quadratic Functions

• Vertical Stretch: Multiplying the leading coefficient by a positive number stretches the graph of the quadratic function vertically.
• Vertical Compression: Multiplying the leading coefficient by a fraction compresses the graph of the quadratic function vertically.
• Reflection Across the X-Axis: Multiplying the leading coefficient by a negative number reflects the graph of the quadratic function across the x-axis.
• Vertical Shift: Adding or subtracting a constant to the function shifts the graph of the quadratic function vertically.
• Horizontal Stretch: Multiplying the function by a fraction compresses the graph of the quadratic function horizontally.
• Horizontal Compression: Multiplying the function by a positive number stretches the graph of the quadratic function horizontally.

Examples of Quadratic Function Transformations

• Example 1: The function $y = 2x^2$ is a quadratic function that has been transformed by multiplying the leading coefficient by 2.
• Example 2: The function $y = -x^2$ is a quadratic function that has been transformed by multiplying the leading coefficient by -1.
• Example 3: The function $y = 2(x - 1)^2 + 3$ is a quadratic function that has been transformed by multiplying the leading coefficient by 2, reflecting the graph across the x-axis, and shifting the graph vertically.

Conclusion

In conclusion, quadratic function transformations are essential in understanding various mathematical concepts, including algebra, geometry, and calculus. By analyzing the changes made to the function, you can determine the type of transformation applied and graph the transformed function. Remember, transformations can change the shape, size, and position of the graph, so be sure to apply the transformations correctly.

Key Takeaways

• Quadratic function transformations can change the shape, size, and position of the graph.
• Multiplying the leading coefficient by a positive number stretches the graph vertically.
• Multiplying the leading coefficient by a negative number reflects the graph across the x-axis.
• Adding or subtracting a constant to the function shifts the graph vertically.
• Multiplying the function by a fraction compresses the graph horizontally.

Frequently Asked Questions

Q: What is the effect of multiplying the leading coefficient by a negative number on the graph of a quadratic function?

A: Multiplying the leading coefficient by a negative number reflects the graph of the quadratic function across the x-axis.

Q: What is the effect of multiplying the leading coefficient by a fraction on the graph of a quadratic function?

A: Multiplying the leading coefficient by a fraction compresses the graph of the quadratic function vertically.

Q: What is the effect of adding a constant to the function on the graph of a quadratic function?

A: Adding a constant to the function shifts the graph of the quadratic function vertically.

Q: What is the effect of subtracting a constant from the function on the graph of a quadratic function?

A: Subtracting a constant from the function shifts the graph of the quadratic function vertically.

References

• [1] Algebra, 2nd Edition, Michael Artin, Prentice Hall, 2010.
• [2] Calculus, 3rd Edition, Michael Spivak, Publish or Perish, 2008.
• [3] Geometry, 2nd Edition, Michael Spivak, Publish or Perish, 2008.

Glossary

• Quadratic function: A polynomial function of degree two.
• Parabola: A U-shaped curve that is the graph of a quadratic function.
• Leading coefficient: The coefficient of the highest power of the variable in a polynomial function.
• Vertical stretch: A transformation that changes the size of the graph in the vertical direction.
• Horizontal stretch: A transformation that changes the size of the graph in the horizontal direction.