Function Transformation: Understanding F(x) = √(5x)

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How has the function been transformed?

Let's break down how the function f(x) = √(5x) has been transformed from its parent function, which is f(x) = √x. Understanding these transformations is crucial in mathematics for visualizing and manipulating functions effectively.

Understanding the Original Function

Before we dive into the transformed function, let's quickly recap the parent function, f(x) = √x. This is the basic square root function, and its graph starts at the origin (0, 0) and increases gradually as x increases. The key here is to understand its fundamental shape and behavior so we can compare it with the transformed function.

When we talk about transformations, we're generally looking at how the graph of the function is altered. This can involve shifts, stretches, compressions, and reflections. Each of these transformations has a specific impact on the function's equation and, consequently, on its graph. For instance, adding a constant to the function shifts it vertically, while multiplying x by a constant affects it horizontally. Recognizing these changes is fundamental to understanding function transformations.

Moreover, knowing the parent function's characteristics, such as its domain and range, is also crucial. For f(x) = √x, the domain is x ≥ 0 (since we can't take the square root of a negative number) and the range is y ≥ 0 (since the square root of a non-negative number is always non-negative). Keeping these details in mind helps in predicting how the transformations will affect these properties. So, with the parent function firmly in mind, let's move on to analyzing the transformed function.

Analyzing the Transformed Function: f(x) = √(5x)

Now, let's focus on the transformed function: f(x) = √(5x). The critical change here is the 5 inside the square root, multiplying x. This indicates a horizontal transformation. Remember, changes inside the function (affecting x) deal with horizontal alterations, while changes outside the function (affecting the entire f(x)) deal with vertical alterations.

So, what does multiplying x by 5 do? It affects the x-values directly. To understand this, consider what x-value in the transformed function gives the same result as a particular x-value in the original function. For example, to get √5 as the output in the transformed function, we need x to be 1 (√(51) = √5*). However, to get the same output in the original function, we need x to be 5 (√5). This means that the transformed function reaches the same y-value at an x-value that is 1/5 of the x-value in the original function. This is horizontal compression.

To further clarify, think of it this way: the function is "squeezed" horizontally towards the y-axis. The factor by which it's compressed is determined by the constant multiplying x. In this case, since x is multiplied by 5, the compression factor is 1/5. Therefore, the graph of f(x) = √(5x) is the graph of f(x) = √x compressed horizontally by a factor of 1/5. It's a subtle but crucial distinction. Many people mix up horizontal and vertical transformations, so always pay close attention to where the constant is in relation to x and the entire function.

Identifying the Correct Transformation

Based on our analysis, the correct transformation is a horizontal compression by a factor of 1/5. Let's look at why the other options are incorrect:

  • A. Vertical compression by a factor of 1/5: This would involve multiplying the entire function by 1/5, resulting in (1/5)√x, not √(5x).
  • B. Vertical stretch by a factor of 5: This would involve multiplying the entire function by 5, resulting in 5√x, not √(5x).
  • D. Horizontal stretch by a factor of 5: This would involve replacing x with (1/5)x, resulting in √(5(1/5)x) = √x*, which is the opposite of what we have.

Therefore, option C. Horizontal compression by a factor of 1/5 is the correct answer. Understanding the subtle differences between horizontal and vertical transformations is essential for accurately identifying how a function has been altered.

Deep Dive into Horizontal Compression

Let's explore horizontal compression a bit more deeply. Horizontal compression occurs when the x-values of a function are "squeezed" towards the y-axis. Mathematically, this is represented by replacing x with kx, where k > 1. In our case, k = 5, so we have f(x) = √(5x). The effect is that the function completes its "cycle" or reaches certain y-values more quickly than the original function.

To visualize this, consider a few points on the original function f(x) = √x. For example, at x = 1, f(x) = 1; at x = 4, f(x) = 2; and at x = 9, f(x) = 3. Now, let's look at the transformed function f(x) = √(5x). To get f(x) = 1, we need 5x = 1, so x = 1/5. To get f(x) = 2, we need 5x = 4, so x = 4/5. And to get f(x) = 3, we need 5x = 9, so x = 9/5. Notice that all the x-values are 1/5 of the original x-values, demonstrating the horizontal compression by a factor of 1/5.

Moreover, it's helpful to think about how this affects the domain of the function. The original function f(x) = √x has a domain of x ≥ 0. The transformed function f(x) = √(5x) also has a domain of x ≥ 0, because 5x must be non-negative, which means x must be non-negative. However, the rate at which the function increases is significantly altered due to the compression.

Common Mistakes and How to Avoid Them

One of the most common mistakes when dealing with function transformations is confusing horizontal and vertical transformations. Remember, anything that happens inside the function (affecting x) is a horizontal transformation, and it often behaves counterintuitively. For example, multiplying x by a constant greater than 1 results in a compression, not a stretch. Conversely, multiplying x by a constant between 0 and 1 results in a stretch.

Another common mistake is misinterpreting the factor of compression or stretch. Always consider how the x-values are changing relative to the original function. If the new x-values are smaller, it's a compression; if they're larger, it's a stretch.

To avoid these mistakes, it's helpful to graph both the original and transformed functions. Visualizing the transformations can make it much easier to understand what's happening. You can also use specific points to compare the x and y-values of the two functions.

Practical Applications of Function Transformations

Function transformations aren't just abstract mathematical concepts; they have numerous practical applications in various fields. For example, in computer graphics, transformations are used to manipulate images and objects, such as scaling, rotating, and translating them. Understanding these transformations is essential for creating realistic and visually appealing graphics.

In physics, transformations are used to model various phenomena, such as wave propagation and oscillations. By understanding how functions are transformed, physicists can better analyze and predict the behavior of these systems. For instance, understanding how the amplitude and frequency of a wave change can be crucial in designing effective communication systems.

Moreover, in economics, transformations are used to model economic trends and patterns. For example, transformations can be used to adjust data for inflation or seasonal variations, allowing economists to make more accurate predictions about future economic conditions. The ability to manipulate and understand functions is a valuable tool for anyone working with data and models.

Conclusion

In summary, the function f(x) = √(5x) is a horizontal compression of the parent function f(x) = √x by a factor of 1/5. Understanding the principles of function transformations is crucial for success in mathematics and its applications. By remembering the rules for horizontal and vertical transformations and visualizing the effects on the graph, you can confidently analyze and manipulate functions in various contexts. So keep practicing, and you'll become a pro at spotting those transformations in no time!