Understanding Vertical Stretching

Oct 13, 2025 by leolavanderia.com.br 66 views

Introduction

In mathematics, quadratic functions are a fundamental concept in algebra and geometry. These functions are represented by the equation f(x) = 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. In this article, we will explore the concept of vertical stretching in quadratic functions and how it affects the graph of the function.

Understanding Vertical Stretching

Vertical stretching is a transformation that changes the scale of a function. When a function is vertically stretched, its graph is stretched vertically, meaning that the y-coordinates of the points on the graph are multiplied by a factor. This transformation affects the shape and position of the graph.

Shawn's Perspective

Shawn thinks that the function f(x) = x^2 was vertically stretched by a factor of 9. This means that Shawn believes that the graph of the function f(x) = x^2 has been stretched vertically by a factor of 9. To understand this, let's analyze the given points on the graph: (1, 9), (0, 0), and (-1, 9). If the function f(x) = x^2 was vertically stretched by a factor of 9, then the y-coordinates of these points would be multiplied by 9.

Brielle's Perspective

Brielle thinks that the function f(x) = x^2 is the same as the function f(x) = 9x^2. This means that Brielle believes that the graph of the function f(x) = x^2 is the same as the graph of the function f(x) = 9x^2. To understand this, let's analyze the given points on the graph: (1, 9), (0, 0), and (-1, 9). If the function f(x) = x^2 is the same as the function f(x) = 9x^2, then the y-coordinates of these points would be multiplied by 9.

Analyzing the Points

Let's analyze the given points on the graph: (1, 9), (0, 0), and (-1, 9). If the function f(x) = x^2 was vertically stretched by a factor of 9, then the y-coordinates of these points would be multiplied by 9. This means that the points on the graph would be (1, 81), (0, 0), and (-1, 81).

However, if the function f(x) = x^2 is the same as the function f(x) = 9x^2, then the y-coordinates of these points would be multiplied by 9. This means that the points on the graph would be (1, 9), (0, 0), and (-1, 9).

Conclusion

In conclusion, the graph of a quadratic function contains the points (1, 9), (0, 0), and (-1, 9). Shawn thinks that the function f(x) = x^2 was vertically stretched by a factor of 9, while Brielle thinks that the function f(x) = x^2 is the same as the function f(x) = 9x^2. By analyzing the given points on the graph, we can see that Brielle's perspective is correct. The function f(x) = x^2 is the same as the function f(x) = 9x^2, and the graph of the function f(x) = x^2 is the same as the graph of the function f(x) = 9x^2.

Understanding the Concept of Vertical Stretching

Examples of Vertical Stretching

Here are some examples of vertical stretching:

Example 1: The function f(x) = x^2 is vertically stretched by a factor of 2. The graph of the function f(x) = x^2 is stretched vertically by a factor of 2, resulting in the graph of the function f(x) = 2x^2.
Example 2: The function f(x) = x^2 is vertically stretched by a factor of 3. The graph of the function f(x) = x^2 is stretched vertically by a factor of 3, resulting in the graph of the function f(x) = 3x^2.
Example 3: The function f(x) = x^2 is vertically stretched by a factor of 4. The graph of the function f(x) = x^2 is stretched vertically by a factor of 4, resulting in the graph of the function f(x) = 4x^2.

Conclusion

In conclusion, vertical stretching is a transformation that changes the scale of a function. When a function is vertically stretched, its graph is stretched vertically, meaning that the y-coordinates of the points on the graph are multiplied by a factor. This transformation affects the shape and position of the graph. By understanding the concept of vertical stretching, we can analyze the given points on the graph and determine the correct function.

Real-World Applications of Vertical Stretching

Vertical stretching has many real-world applications in various fields, including:

Physics: Vertical stretching is used to model the motion of objects under the influence of gravity.
Engineering: Vertical stretching is used to design and analyze the structural integrity of buildings and bridges.
Computer Science: Vertical stretching is used to develop algorithms for image processing and computer vision.

Conclusion

Q&A: Understanding Vertical Stretching

Q: What is vertical stretching in mathematics?

A: Vertical stretching is a transformation that changes the scale of a function. When a function is vertically stretched, its graph is stretched vertically, meaning that the y-coordinates of the points on the graph are multiplied by a factor.

Q: How does vertical stretching affect the graph of a function?

A: Vertical stretching affects the shape and position of the graph. When a function is vertically stretched, its graph is stretched vertically, resulting in a change in the y-coordinates of the points on the graph.

Q: What are some examples of vertical stretching?

A: Here are some examples of vertical stretching:

Example 1: The function f(x) = x^2 is vertically stretched by a factor of 2. The graph of the function f(x) = x^2 is stretched vertically by a factor of 2, resulting in the graph of the function f(x) = 2x^2.
Example 2: The function f(x) = x^2 is vertically stretched by a factor of 3. The graph of the function f(x) = x^2 is stretched vertically by a factor of 3, resulting in the graph of the function f(x) = 3x^2.
Example 3: The function f(x) = x^2 is vertically stretched by a factor of 4. The graph of the function f(x) = x^2 is stretched vertically by a factor of 4, resulting in the graph of the function f(x) = 4x^2.

Q: How can I determine if a function has been vertically stretched?

A: To determine if a function has been vertically stretched, you can analyze the given points on the graph. If the y-coordinates of the points on the graph are multiplied by a factor, then the function has been vertically stretched.

Q: What are some real-world applications of vertical stretching?

A: Vertical stretching has many real-world applications in various fields, including:

Physics: Vertical stretching is used to model the motion of objects under the influence of gravity.
Engineering: Vertical stretching is used to design and analyze the structural integrity of buildings and bridges.
Computer Science: Vertical stretching is used to develop algorithms for image processing and computer vision.

Q: How can I use vertical stretching in my daily life?

A: Vertical stretching can be used in many ways in your daily life, such as:

Modeling real-world phenomena: Vertical stretching can be used to model real-world phenomena, such as the motion of objects under the influence of gravity.
Designing and analyzing structures: Vertical stretching can be used to design and analyze the structural integrity of buildings and bridges.
Developing algorithms: Vertical stretching can be used to develop algorithms for image processing and computer vision.

Conclusion

In conclusion, vertical stretching is a fundamental concept in mathematics that has many real-world applications. By understanding the concept of vertical stretching, you can analyze the given points on the graph and determine the correct function.