When we talk about stretching a function vertically by a factor of 4, we are referring to a transformation that affects the y-coordinates of the points on the graph. This type of transformation elongates the graph vertically, making it taller or shorter depending on the multiplier.
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Understanding Vertical Stretch
A vertical stretch by a factor of 4 means that every y-coordinate of the function is multiplied by 4. This results in a vertical elongation of the graph. Visually, the graph becomes steeper and stretches away from the horizontal axis. The shape of the original function remains the same, but it gets distorted vertically.
Vertical stretching is a common transformation in mathematics and is often used to adjust the scale of a graph or to emphasize certain features of a function.
Effects of Vertical Stretch By A Factor Of 4
When a function is vertically stretched by a factor of 4, several key effects can be observed:
- Increased steepness: The slope of the function becomes four times steeper, leading to a faster rate of change.
- Change in amplitude: For trigonometric functions, the amplitude is quadrupled, resulting in a larger swing between the maximum and minimum values.
- Heightened peaks and valleys: The local extrema of the function are accentuated, making the peaks and valleys more pronounced.
- Vertical asymptotes: If the original function had vertical asymptotes, the vertical stretch can alter their positions or create new asymptotes.
Mathematical Representation
The mathematical representation of a vertical stretch by a factor of 4 is given by the function:
f(x) = 4 * g(x)
Where g(x) is the original function. This formula denotes that every y-coordinate of the original function is multiplied by 4 to achieve the vertical stretch.
Examples of Vertical Stretch By A Factor Of 4
Let’s explore a few examples to better understand the concept of vertical stretching by a factor of 4:
1. Linear Function:
Consider the linear function f(x) = 2x + 1. If we apply a vertical stretch by a factor of 4 to this function, the new function becomes:
f(x) = 4(2x + 1) = 8x + 4
The graph of the original function y = 2x + 1 is stretched vertically by a factor of 4, resulting in the new function y = 8x + 4.
2. Quadratic Function:
Let’s take the quadratic function f(x) = x^2. If we apply a vertical stretch by a factor of 4 to this function, the new function becomes:
f(x) = 4(x^2) = 4x^2
The graph of the original quadratic function is vertically stretched by a factor of 4, leading to the new function y = 4x^2.
3. Trigonometric Function:
For a trigonometric function such as f(x) = sin(x), applying a vertical stretch by a factor of 4 results in:
f(x) = 4sin(x)
In this case, the amplitude of the sine function is quadrupled, causing the graph to stretch vertically by a factor of 4.
Applications of Vertical Stretch By A Factor Of 4
Vertical stretching by a factor of 4 has various applications in mathematics, physics, engineering, and other fields. Some common applications include:
- Signal Processing: In signal processing, vertical stretching can amplify or attenuate certain frequencies in a signal.
- Function Analysis: Vertical stretching helps in analyzing the behavior of functions and understanding their transformations.
- Optimization: By adjusting the scale of a graph through vertical stretching, optimization problems can be solved more effectively.
- Visual Representation: Vertical stretch by a factor of 4 can be used to visually highlight specific features of a graph or function.
Conclusion
Vertical stretch by a factor of 4 is a powerful mathematical transformation that elongates the graph of a function vertically. By multiplying every y-coordinate by 4, the original function is stretched and its key properties are altered. Understanding the effects of vertical stretching can provide valuable insights into the behavior of functions and their visual representation.
