Vertical Stretch Transformation: Identifying The Correct Function

Understanding transformations of functions is crucial in mathematics, especially when dealing with graphical representations. One common type of transformation is a vertical stretch, which alters the function's output values, effectively stretching the graph vertically. In this article, we will delve into the concept of vertical stretches, specifically focusing on identifying the function that represents a vertical stretch of a given function ${ f(x) }$ by a factor of 1.25. We will analyze the provided options and explain why one of them correctly represents the transformation.

Understanding Vertical Stretches

In the realm of function transformations, a vertical stretch is a transformation that affects the y-values of a function. When a function ${ f(x) }$ undergoes a vertical stretch by a factor of ${ k }$, where ${ k > 1 }$, the resulting function is { g(x) = k (x) }. This means that each y-value of the original function is multiplied by ${ k }$, causing the graph to stretch vertically away from the x-axis. Conversely, if ${ 0 < k < 1 }$, the transformation is a vertical compression (or shrink). For instance, if we have a function ${ f(x) }$ and we want to stretch it vertically by a factor of 2, the transformed function would be ${ g(x) = 2f(x) }$.

The key to understanding vertical stretches lies in recognizing that the factor ${ k }$ directly multiplies the function's output. This contrasts with horizontal stretches or compressions, where the factor affects the input ${ x }$ inside the function. For example, consider the exponential function ${ f(x) = 10^x }$. If we want to stretch this function vertically by a factor of 1.25, we need to multiply the entire function by 1.25, resulting in { g(x) = 1.25 (x) = 1.25 imes 10^x }. This simple multiplication is the essence of a vertical stretch.

Vertical stretches play a significant role in various mathematical and real-world applications. In physics, for example, understanding how to stretch or compress waveforms is essential in signal processing. In economics, transformations of supply and demand curves can illustrate market changes. The ability to identify and apply vertical stretches is, therefore, a fundamental skill in mathematical analysis and modeling.

Analyzing the Given Options

To determine which function represents a vertical stretch of ${ f(x) }$ by a factor of 1.25, we need to examine the given options and see which one fits the form { g(x) = k (x) }, where ${ k = 1.25 }$. Let's consider the options provided:

• A. ${ g(x) = 0.8 imes 10^x }$
• B. ${ g(x) = 1.25 imes 10^x }$
• C. ${ g(x) = 10^{1.25x} }$
• D. ${ g(x) = 10^{0.8x} }$

Option A represents a vertical compression by a factor of 0.8, as the function ${ 10^x }$ is multiplied by a value less than 1. This means the graph would be compressed towards the x-axis, not stretched away from it. Therefore, option A is not the correct answer.

Option B, ${ g(x) = 1.25 imes 10^x }$, is the most promising candidate. Here, the original function ${ 10^x }$ is multiplied by 1.25, which fits the definition of a vertical stretch by a factor of 1.25. This means that every y-value of the original function is multiplied by 1.25, effectively stretching the graph vertically. Therefore, option B appears to be the correct answer.

Option C, ${ g(x) = 10^{1.25x} }$, involves multiplying the input ${ x }$ by 1.25 within the exponent. This represents a horizontal compression, not a vertical stretch. Horizontal transformations affect the x-values, not the y-values, so this option is incorrect.

Similarly, option D, ${ g(x) = 10^{0.8x} }$, represents a horizontal stretch by a factor of { rac{1}{0.8} = 1.25 }, since the input ${ x }$ is multiplied by 0.8 within the exponent. Again, this is a horizontal transformation, not a vertical stretch, making this option incorrect.

By carefully analyzing each option, we can see that only option B correctly represents a vertical stretch by a factor of 1.25.

The Correct Function: Option B

After careful analysis, it's clear that the correct function representing a vertical stretch of ${ f(x) = 10^x }$ by a factor of 1.25 is Option B: ${ g(x) = 1.25 imes 10^x }$. This function adheres to the principle that a vertical stretch by a factor of ${ k }$ involves multiplying the original function by ${ k }$. In this case, multiplying ${ 10^x }$ by 1.25 stretches the graph vertically, increasing each y-value by a factor of 1.25.

To further illustrate this, let's consider a few points on the graph of ${ f(x) = 10^x }$ and how they transform under the vertical stretch:

• When ${ x = 0 }$, ${ f(0) = 10^0 = 1 }$. For ${ g(x) }$, ${ g(0) = 1.25 imes 10^0 = 1.25 }$. The y-value is stretched from 1 to 1.25.
• When ${ x = 1 }$, ${ f(1) = 10^1 = 10 }$. For ${ g(x) }$, ${ g(1) = 1.25 imes 10^1 = 12.5 }$. The y-value is stretched from 10 to 12.5.
• When ${ x = -1 }$, ${ f(-1) = 10^{-1} = 0.1 }$. For ${ g(x) }$, ${ g(-1) = 1.25 imes 10^{-1} = 0.125 }$. The y-value is stretched from 0.1 to 0.125.

These examples clearly show how the y-values are scaled by a factor of 1.25, confirming that option B correctly represents the vertical stretch.

Options A, C, and D, on the other hand, do not represent a vertical stretch by a factor of 1.25. Option A represents a vertical compression, while options C and D represent horizontal transformations. Understanding the difference between vertical and horizontal transformations is essential for correctly interpreting function transformations.

Importance of Understanding Function Transformations

Function transformations, including vertical stretches, are fundamental concepts in mathematics with wide-ranging applications. A solid grasp of these concepts is crucial for students and professionals alike. Understanding how transformations affect the graph of a function allows for better visualization and analysis of mathematical models.

In calculus, function transformations are used extensively in curve sketching and optimization problems. They help in understanding the behavior of functions and finding critical points. In physics, transformations are used to model changes in physical systems, such as oscillations and waves. In computer graphics, transformations are used to manipulate objects in 2D and 3D space.

Furthermore, function transformations are essential in data analysis and statistics. They are used to normalize data, remove outliers, and make data more suitable for statistical analysis. For example, logarithmic transformations are often used to handle skewed data, and standardization techniques involve shifting and scaling data using linear transformations.

The ability to identify and apply function transformations is also crucial for solving real-world problems. In engineering, understanding how transformations affect the performance of a system is vital for design and optimization. In economics, transformations are used to model economic trends and make predictions.

By mastering function transformations, individuals can develop a deeper understanding of mathematical concepts and their applications in various fields. This knowledge empowers them to solve complex problems and make informed decisions.

Conclusion

Identifying the correct function that represents a vertical stretch transformation requires a clear understanding of how transformations affect the graph of a function. In the case of a vertical stretch by a factor of 1.25, the function must be multiplied by 1.25. Therefore, Option B, ${ g(x) = 1.25 imes 10^x }$, is the correct answer. This function accurately represents the vertical stretch of the original function ${ f(x) = 10^x }$.

Understanding function transformations is a crucial skill in mathematics and has numerous applications in various fields. By mastering these concepts, students and professionals can enhance their problem-solving abilities and gain a deeper appreciation for the power of mathematics.

For further exploration of function transformations, you might find helpful resources on websites like Khan Academy's Function Transformations.