The Effect of Horizontal Scaling on the Graph

Oct 13, 2025 by windailys.com 92 views

Introduction

In mathematics, graph transformation is a crucial concept that helps us understand how functions behave under different transformations. In this article, we will explore the graph transformation of $g(x) = f(4x)$ , where $f(x) = x^2$ . We will delve into the concept of horizontal scaling and its effects on the graph of a function.

What is Horizontal Scaling?

Horizontal scaling is a type of transformation that affects the graph of a function by stretching or compressing it horizontally. When a function $f(x)$ is scaled horizontally by a factor of $a$ , the new function becomes $f(ax)$ . In this case, we are scaling the function $f(x) = x^2$ horizontally by a factor of $4$ , resulting in the function $g(x) = f(4x)$ .

The Effect of Horizontal Scaling on the Graph

When we scale a function horizontally, the graph of the function is stretched or compressed in the horizontal direction. In the case of $g(x) = f(4x)$ , the graph of $f(x) = x^2$ is compressed horizontally by a factor of $4$ . This means that the graph of $g(x)$ will be narrower than the graph of $f(x)$ .

Visualizing the Graph of $g(x) = f(4x)$

To visualize the graph of $g(x) = f(4x)$ , we can start by plotting the graph of $f(x) = x^2$ . The graph of $f(x)$ is a parabola that opens upwards, with its vertex at the origin $(0, 0)$ . When we scale the graph of $f(x)$ horizontally by a factor of $4$ , the graph of $g(x)$ will be compressed in the horizontal direction.

Key Features of the Graph of $g(x) = f(4x)$

The graph of $g(x) = f(4x)$ has several key features that are worth noting:

Vertex: The vertex of the graph of $g(x)$ is at the point $(0, 0)$ , just like the graph of $f(x)$ .
Axis of Symmetry: The axis of symmetry of the graph of $g(x)$ is the y-axis, just like the graph of $f(x)$ .
Intercepts: The x-intercepts of the graph of $g(x)$ are at the points $(0, 0)$ and $(\frac{1}{4}, 0)$ .
End Behavior: The end behavior of the graph of $g(x)$ is the same as the graph of $f(x)$ , with the graph approaching the x-axis as x approaches positive or negative infinity.

Conclusion

In conclusion, the graph of $g(x) = f(4x)$ is a compressed version of the graph of $f(x) = x^2$ . The graph of $g(x)$ has a narrower shape than the graph of $f(x)$ , with the same vertex, axis of symmetry, intercepts, and end behavior. Understanding the concept of horizontal scaling and its effects on the graph of a function is crucial in mathematics, and this article has provided a comprehensive overview of the graph transformation of $g(x) = f(4x)$ .

Example Problems

Problem 1

Suppose $f(x) = x^2$ . What is the graph of $g(x) = f(2x)$ ?

Solution

The graph of $g(x) = f(2x)$ is a compressed version of the graph of $f(x) = x^2$ . The graph of $g(x)$ has a narrower shape than the graph of $f(x)$ , with the same vertex, axis of symmetry, intercepts, and end behavior.

Problem 2

Suppose $f(x) = x^2$ . What is the graph of $g(x) = f(\frac{x}{2})$ ?

Solution

The graph of $g(x) = f(\frac{x}{2})$ is a stretched version of the graph of $f(x) = x^2$ . The graph of $g(x)$ has a wider shape than the graph of $f(x)$ , with the same vertex, axis of symmetry, intercepts, and end behavior.

Practice Problems

Problem 1

Suppose $f(x) = x^2$ . What is the graph of $g(x) = f(3x)$ ?

Problem 2

Suppose $f(x) = x^2$ . What is the graph of $g(x) = f(\frac{x}{3})$ ?

Problem 3

Suppose $f(x) = x^2$ . What is the graph of $g(x) = f(5x)$ ?

Problem 4

Suppose $f(x) = x^2$ . What is the graph of $g(x) = f(\frac{x}{5})$ ?

Answer Key

Problem 1

The graph of $g(x) = f(3x)$ is a compressed version of the graph of $f(x) = x^2$ . The graph of $g(x)$ has a narrower shape than the graph of $f(x)$ , with the same vertex, axis of symmetry, intercepts, and end behavior.

Problem 2

The graph of $g(x) = f(\frac{x}{3})$ is a stretched version of the graph of $f(x) = x^2$ . The graph of $g(x)$ has a wider shape than the graph of $f(x)$ , with the same vertex, axis of symmetry, intercepts, and end behavior.

Problem 3

The graph of $g(x) = f(5x)$ is a compressed version of the graph of $f(x) = x^2$ . The graph of $g(x)$ has a narrower shape than the graph of $f(x)$ , with the same vertex, axis of symmetry, intercepts, and end behavior.

Problem 4

$The graph of $g(x) = f(\frac{x}{5})$ is a stretched version of the graph of $f(x) = x^2$. The graph of $g(x)$ has a wider shape than the graph of $f(x)$, with the same vertex, axis of symmetry, intercepts, and end behavior.<br/> **Q&A: Understanding the Graph Transformation of $g(x) = f(4x)$**$

Q: What is the graph of $g(x) = f(4x)$ ?

A: The graph of $g(x) = f(4x)$ is a compressed version of the graph of $f(x) = x^2$ . The graph of $g(x)$ has a narrower shape than the graph of $f(x)$ , with the same vertex, axis of symmetry, intercepts, and end behavior.

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

A: Horizontal scaling is a type of transformation that affects the graph of a function by stretching or compressing it horizontally. When a function $f(x)$ is scaled horizontally by a factor of $a$ , the new function becomes $f(ax)$ . In the case of $g(x) = f(4x)$ , the graph of $f(x) = x^2$ is compressed horizontally by a factor of $4$ .

Q: What are the key features of the graph of $g(x) = f(4x)$ ?

A: The graph of $g(x) = f(4x)$ has several key features that are worth noting:

Vertex: The vertex of the graph of $g(x)$ is at the point $(0, 0)$ , just like the graph of $f(x)$ .
Axis of Symmetry: The axis of symmetry of the graph of $g(x)$ is the y-axis, just like the graph of $f(x)$ .
Intercepts: The x-intercepts of the graph of $g(x)$ are at the points $(0, 0)$ and $(\frac{1}{4}, 0)$ .
End Behavior: The end behavior of the graph of $g(x)$ is the same as the graph of $f(x)$ , with the graph approaching the x-axis as x approaches positive or negative infinity.

Q: How does the graph of $g(x) = f(4x)$ compare to the graph of $f(x) = x^2$ ?

Q: What is the relationship between the graph of $g(x) = f(4x)$ and the graph of $f(x) = x^2$ ?

A: The graph of $g(x) = f(4x)$ is a transformation of the graph of $f(x) = x^2$ . The graph of $g(x)$ is obtained by compressing the graph of $f(x)$ horizontally by a factor of $4$ .

Q: How can I visualize the graph of $g(x) = f(4x)$ ?

A: To visualize the graph of $g(x) = f(4x)$ , you can start by plotting the graph of $f(x) = x^2$ . The graph of $f(x)$ is a parabola that opens upwards, with its vertex at the origin $(0, 0)$ . When you scale the graph of $f(x)$ horizontally by a factor of $4$ , the graph of $g(x)$ will be compressed in the horizontal direction.

Q: What are some common mistakes to avoid when working with graph transformations?

A: Some common mistakes to avoid when working with graph transformations include:

Not considering the axis of symmetry: When scaling a function horizontally, it's essential to consider the axis of symmetry of the original function.
Not accounting for the vertex: The vertex of the original function is crucial when scaling a function horizontally.
Not checking the intercepts: The x-intercepts of the original function are also important when scaling a function horizontally.

Q: How can I practice graph transformations?

A: You can practice graph transformations by working through example problems and exercises. Some tips for practicing graph transformations include:

Start with simple transformations: Begin with simple transformations, such as scaling a function horizontally by a factor of $2$ or $3$ .
Use graph paper: Graph paper can be helpful when visualizing the graph of a function and its transformations.
Check your work: Always check your work to ensure that you have correctly applied the transformation.

Q: What are some real-world applications of graph transformations?

A: Graph transformations have numerous real-world applications, including:

Physics: Graph transformations are used to model the motion of objects in physics.
Engineering: Graph transformations are used to design and optimize systems in engineering.
Computer Science: Graph transformations are used in computer science to model and analyze complex systems.

Conclusion

In conclusion, graph transformations are a fundamental concept in mathematics that have numerous real-world applications. By understanding how to apply graph transformations, you can model and analyze complex systems in various fields. Remember to practice graph transformations regularly to build your skills and confidence.