Understanding Vertical Stretches

Oct 13, 2025 by magicalceylontours.com 112 views

Introduction

In mathematics, understanding the effects of transformations on functions is crucial for graphing and analyzing functions. One common transformation is the vertical stretch, which is achieved by multiplying the function by a constant factor. In this article, we will analyze the effect of a vertical stretch on the graph of $$f(x) = 2^x$$ when $$f(x)$$ is replaced by $$a f(x)$$ and $$a = \frac{8}{5}$$.

Understanding Vertical Stretches

A vertical stretch is a transformation that stretches the graph of a function vertically by a factor of $$a$$. This means that for every point $$(x, y)$$ on the original graph, the corresponding point on the stretched graph is $$(x, ay)$$. In other words, the vertical distance between the points on the graph is multiplied by $$a$$.

The Transformation $$f(x) = a f(x)$$

When we replace $$f(x)$$ with $$a f(x)$$, we are essentially applying a vertical stretch to the graph of $$f(x)$$. The constant factor $$a$$ determines the amount of stretching. In this case, $$a = \frac{8}{5}$$, which means that the graph of $$f(x)$$ will be stretched vertically by a factor of $$\frac{8}{5}$$.

Analyzing the Effect on the Graph

To analyze the effect of the vertical stretch on the graph of $$f(x) = 2^x$$, let's consider the following:

The original graph of $$f(x) = 2^x$$ is an exponential curve that passes through the point $$(0, 1)$$.
When we apply the vertical stretch $$f(x) = \frac{8}{5} f(x)$$, the graph of $$f(x)$$ will be stretched vertically by a factor of $$\frac{8}{5}$$.
This means that for every point $$(x, y)$$ on the original graph, the corresponding point on the stretched graph is $$(x, \frac{8}{5} y)$$.

Visualizing the Transformation

To visualize the transformation, let's consider the following:

The original graph of $$f(x) = 2^x$$ is a curve that passes through the point $$(0, 1)$$.
When we apply the vertical stretch $$f(x) = \frac{8}{5} f(x)$$, the graph of $$f(x)$$ will be stretched vertically by a factor of $$\frac{8}{5}$$.
This means that the graph of $$f(x)$$ will be stretched upwards, resulting in a new curve that passes through the point $$(0, \frac{8}{5})$$.

Conclusion

In conclusion, when we replace $$f(x)$$ with $$a f(x)$$ and $$a = \frac{8}{5}$$, the graph of $$f(x) = 2^x$$ will be stretched vertically by a factor of $$\frac{8}{5}$$. This means that the graph of $$f(x)$$ will be stretched upwards, resulting in a new curve that passes through the point $$(0, \frac{8}{5})$$.

Which of the Following Statements is True?

Based on our analysis, we can conclude that the following statement is true:

The transformation describes a vertical stretch.

This statement is true because the transformation $$f(x) = a f(x)$$ with $$a = \frac{8}{5}$$ stretches the graph of $$f(x) = 2^x$$ vertically by a factor of $$\frac{8}{5}$$.

References

[1] "Functions and Graphs" by Michael Corral.
Q&A: Analyzing the Effect of Vertical Stretches on the Graph of $$f(x) = 2^x$$ ====================================================================

Introduction

In our previous article, we analyzed the effect of a vertical stretch on the graph of $$f(x) = 2^x$$ when $$f(x)$$ is replaced by $$a f(x)$$ and $$a = \frac{8}{5}$$. In this article, we will answer some frequently asked questions about vertical stretches and their effects on graphs.

Q: What is a vertical stretch?

A: A vertical stretch is a transformation that stretches the graph of a function vertically by a factor of $$a$$. This means that for every point $$(x, y)$$ on the original graph, the corresponding point on the stretched graph is $$(x, ay)$$.

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

A: A vertical stretch affects the graph of a function by stretching it vertically by a factor of $$a$$. This means that the vertical distance between the points on the graph is multiplied by $$a$$.

Q: What is the effect of a vertical stretch on the graph of $$f(x) = 2^x$$ when $$a = \frac{8}{5}$$?

A: When we replace $$f(x)$$ with $$a f(x)$$ and $$a = \frac{8}{5}$$, the graph of $$f(x) = 2^x$$ will be stretched vertically by a factor of $$\frac{8}{5}$$. This means that the graph of $$f(x)$$ will be stretched upwards, resulting in a new curve that passes through the point $$(0, \frac{8}{5})$$.

Q: How can I visualize the effect of a vertical stretch on the graph of a function?

A: You can visualize the effect of a vertical stretch on the graph of a function by considering the following:

The original graph of $$f(x)$$ is a curve that passes through the point $$(0, 1)$$.
When we apply the vertical stretch $$f(x) = a f(x)$$, the graph of $$f(x)$$ will be stretched vertically by a factor of $$a$$.
This means that the graph of $$f(x)$$ will be stretched upwards, resulting in a new curve that passes through the point $$(0, a)$$.

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

A: A vertical stretch and a vertical compression are two different types of transformations that affect the graph of a function. A vertical stretch stretches the graph of a function vertically by a factor of $$a$$, while a vertical compression compresses the graph of a function vertically by a factor of $$\frac{1}{a}$$.

Q: How can I apply a vertical stretch to the graph of a function?

A: You can apply a vertical stretch to the graph of a function by replacing $$f(x)$$ with $$a f(x)$$, where $$a$$ is the factor by which you want to stretch the graph.

Q: What are some common applications of vertical stretches in mathematics?

A: Vertical stretches have many applications in mathematics, including:

Graphing functions
Analyzing the behavior of functions
Solving equations and inequalities
Modeling real-world phenomena

Conclusion

In conclusion, vertical stretches are an important concept in mathematics that can be used to analyze and graph functions. By understanding how vertical stretches affect the graph of a function, you can better visualize and analyze the behavior of functions.