When exploring transformations of functions, the concept of a vertical stretch is fundamental. In this article, we will delve deep into the vertical stretching of the absolute value parent function, f(x) = |x|, and thoroughly investigate how multiplying the parent function by a constant affects its graph and equation. We'll specifically focus on understanding what happens when we stretch the absolute value function vertically by a factor of 3, and how to derive the equation of the new function. This exploration will not only help clarify the mechanics behind vertical stretches but also provide a solid foundation for handling more complex function transformations.
The Parent Function: f(x) = |x|
The absolute value parent function, f(x) = |x|, serves as the cornerstone for understanding absolute value functions. Its graph is characterized by a distinctive V-shape, with the vertex located at the origin (0,0). The graph extends symmetrically into the first and second quadrants, as the absolute value of any real number is always non-negative. The function's simplicity belies its importance, as it forms the basis for numerous transformations, including shifts, reflections, and stretches. Understanding the parent function is crucial because all transformations are performed relative to this foundational graph. The equation f(x) = |x| implies that for any input x, the output is the absolute value of x. This means that positive values of x remain unchanged, while negative values are converted to their positive counterparts. The resulting V-shape is a direct consequence of this behavior. Each point on the graph is equidistant from the y-axis, creating a mirror-like symmetry. This symmetry is a key characteristic of the absolute value function and plays a significant role in how transformations affect the graph. Recognizing the basic shape and behavior of f(x) = |x| is the first step in mastering transformations of absolute value functions. This foundational knowledge allows us to predict how the graph will change under various operations, such as vertical and horizontal stretches, shifts, and reflections. By starting with the parent function, we can build a strong understanding of the more complex transformed functions.
Vertical Stretch: A Detailed Explanation
In the realm of function transformations, a vertical stretch plays a crucial role in altering the appearance of a graph. Vertical stretching involves multiplying the function's output (y-value) by a constant factor, effectively elongating the graph along the y-axis. This transformation changes the vertical scale of the graph, making it appear taller or narrower depending on the stretch factor. A vertical stretch by a factor greater than 1 will stretch the graph away from the x-axis, while a stretch by a factor between 0 and 1 will compress the graph towards the x-axis. Understanding how vertical stretches work is essential for analyzing and manipulating functions in various contexts. When you apply a vertical stretch to a function, you are essentially multiplying each y-value of the original function by the stretch factor. This means that if a point (x, y) lies on the original graph, the corresponding point on the stretched graph will be (x, ky), where k is the stretch factor. For example, if k = 2, each y-value is doubled, making the graph twice as tall. If k = 0.5, each y-value is halved, compressing the graph vertically. The vertex of the graph, which is a critical point for many functions, is also affected by vertical stretches. For the absolute value function, the vertex at (0,0) remains unchanged when stretched vertically because multiplying 0 by any factor still results in 0. However, other points on the graph will move farther from or closer to the x-axis, depending on the stretch factor. The visual effect of a vertical stretch is a change in the steepness of the graph. Stretching by a factor greater than 1 makes the graph steeper, while compressing makes it less steep. This change in steepness can significantly alter the function's behavior and characteristics. For instance, a stretched absolute value function will have a narrower V-shape, while a compressed one will have a wider V-shape. — Hormone Imbalance Question - Your Comprehensive Guide To Understanding And Treatment
Stretching f(x) = |x| by a Factor of 3
Consider the absolute value parent function, f(x) = |x|. When we apply a vertical stretch by a factor of 3, we are essentially multiplying the entire function by 3. Mathematically, this transformation can be represented as g(x) = 3|x|. This means that for every x-value, the corresponding y-value of the transformed function g(x) is three times the y-value of the original function f(x). The vertical stretch significantly alters the graph of the absolute value function. To illustrate this, let's consider a few points on the graph of f(x) = |x|. The point (1,1) on f(x) becomes (1,3) on g(x). Similarly, the point (-2,2) on f(x) becomes (-2,6) on g(x). The vertex of the absolute value function, which is at (0,0), remains unchanged by the vertical stretch because 3 * 0 = 0. However, all other points on the graph move farther away from the x-axis. The result is a graph that appears to be stretched vertically, making the V-shape narrower and steeper compared to the original function. The equation g(x) = 3|x| encapsulates this transformation. It clearly shows that the output of the function is three times the absolute value of the input. This vertical stretch changes the slope of the lines forming the V-shape. The original function has slopes of 1 and -1, while the stretched function has slopes of 3 and -3. This change in slope is a direct consequence of the vertical stretch and highlights the impact of multiplying the function by a constant factor. Visualizing this transformation can be aided by plotting both f(x) and g(x) on the same coordinate plane. The difference in steepness and the vertical displacement of points become immediately apparent. This graphical representation provides a clear understanding of how the vertical stretch affects the shape and position of the absolute value function.
Analyzing the Given Options
To accurately identify the equation representing the vertical stretch of f(x) = |x| by a factor of 3, it is crucial to analyze each option provided and understand how different transformations affect the absolute value function. Let's break down the options and determine which one correctly represents the given transformation. — *And Just Like That...* Finale Breakdown
Option A: g(x) = |x - 3|
This option represents a horizontal shift rather than a vertical stretch. The subtraction of 3 inside the absolute value function shifts the graph 3 units to the right. This is because the transformation affects the x-values directly, causing a horizontal translation. A horizontal shift does not change the shape or steepness of the V-shaped graph; it merely repositions it on the coordinate plane. Therefore, g(x) = |x - 3| does not represent a vertical stretch.
Option B: g(x) = (1/3)|x|
This equation represents a vertical compression by a factor of 1/3. Multiplying the absolute value function by a fraction between 0 and 1 compresses the graph towards the x-axis. In this case, each y-value of the original function is multiplied by 1/3, making the graph one-third as tall. This transformation results in a wider and less steep V-shape compared to the parent function. Therefore, g(x) = (1/3)|x| is not the correct answer for a vertical stretch by a factor of 3. — Vancouver Houses For Sale: Find Your Dream Home
Option C: g(x) = |x + 3|
Similar to option A, this equation represents a horizontal shift. The addition of 3 inside the absolute value function shifts the graph 3 units to the left. This transformation, like the subtraction in option A, affects the x-values and results in a horizontal translation of the graph. The shape and steepness of the V-shaped graph remain unchanged. Thus, g(x) = |x + 3| does not represent a vertical stretch.
Correct Answer: g(x) = 3|x|
By multiplying the absolute value function by 3, we achieve the desired vertical stretch. This transformation scales the y-values by a factor of 3, making the graph three times as tall. The V-shape becomes narrower and steeper, accurately representing a vertical stretch by a factor of 3. This transformation directly corresponds to the mathematical definition of a vertical stretch, where each point (x, y) on the original graph is transformed to (x, 3y) on the new graph.
Conclusion
In conclusion, understanding vertical stretches is crucial when working with function transformations, particularly within absolute value functions. Stretching the absolute value parent function, f(x) = |x|, by a factor of 3 results in the equation g(x) = 3|x|. This transformation effectively multiplies the y-values of the original function by 3, creating a narrower and steeper V-shape. By analyzing various options, we can clearly differentiate between vertical stretches, horizontal shifts, and vertical compressions, solidifying our grasp on function transformations. This comprehensive understanding allows us to accurately predict and manipulate graphs, enhancing our mathematical problem-solving skills and overall comprehension of function behavior.
