Stretching and Compressing
Function stretching and compressing are types of dilations that change the shape of a function's graph without altering its basic structure. These transformations can be vertical or horizontal, impacting the y-values and x-values of a function, respectively.
Vertical Stretching and Compressing
- Definition: Vertical transformations adjust the y-values of a function. When a function, say f(x), is multiplied by a factor greater than 1, it undergoes a vertical stretch. If multiplied by a factor between 0 and 1, it experiences a vertical compression.
- Interpretation: A vertical stretch pulls the graph away from the x-axis, making it appear taller, while a vertical compression pushes the graph towards the x-axis, making it appear shorter.
- Example: Consider the function f(x) = x2. If we multiply it by 3, the resulting function g(x) = 3x2 represents a vertical stretch of the original parabola. Conversely, h(x) = 0.5x2 represents a vertical compression.
Horizontal Stretching and Compressing
- Definition: Horizontal transformations influence the x-values of a function. When the x in a function f(x) is divided by a factor greater than 1, it undergoes a horizontal compression. If multiplied by a factor between 0 and 1, it experiences a horizontal stretch.
- Interpretation: A horizontal stretch pulls the graph away from the y-axis, while a horizontal compression pushes it closer to the y-axis.Example: For the sine function f(x) = sin(x), the graph of f(0.5x) represents a horizontal stretch, whereas f(2x) represents a horizontal compression.
Real-World Contexts of Transformations
Function transformations aren't just theoretical; they have tangible applications in various real-world scenarios:
Economics
In economics, the behaviour of supply and demand curves can be modelled using function transformations. These curves can shift or stretch based on external factors, such as government policies or global events.
Example: Suppose a government introduces a subsidy for a particular product. This could lead to an increase in its supply, which can be modelled as a vertical stretch of the supply curve. Similarly, a sudden surge in a product's popularity can be represented as a horizontal shift of the demand curve.
Physics
Function transformations are prevalent in physics, especially when representing changes in energy states or wave behaviours.
Example: Consider a pendulum swinging back and forth. The motion of the pendulum can be modelled using a sine or cosine function. If the pendulum is displaced further from its equilibrium position, it will swing with a larger amplitude. This change in amplitude can be represented as a vertical stretch of the sine or cosine function.
Biology
In biology, growth curves, such as those for populations, can be modelled using functions. External factors, like food availability, can lead to shifts or stretches in these curves.
Example: If a bacterial population in a petri dish is given an abundant food source, its growth might accelerate. This rapid growth can be modelled as a vertical stretch of the population growth curve.
Example Questions
1. A city's population, in thousands, over a decade can be modelled by the function P(t) = 4t3. If due to a surge in job opportunities, the population is expected to triple and the growth starts 3 years earlier than predicted, how would the new population function appear?
Answer: The tripling of the population can be modelled by multiplying the function by 3, leading to a vertical stretch. The earlier growth of 3 years can be represented by a horizontal translation to the left by 3 units. Thus, the new population function would be P(t) = 12(t + 3)3.
2. The height of water waves in a pond can be modelled using a sine function. If a stone is thrown into the pond, causing larger ripples but with the same frequency, how would this affect the function?
Answer: The larger ripples caused by the stone would increase the amplitude of the waves. This can be represented as a vertical stretch in the sine function. If the original function was h(t) = sin(t), and the ripples doubled the wave's height, the new function would be h(t) = 2sin(t).
FAQ
In economics, function transformations can be used to model the impact of real-world events like inflation. For example, if a country experiences inflation, the purchasing power of its currency decreases. This can be represented by a vertical compression of the function that models the value of the currency over time. Similarly, economic growth or a surge in demand for a product can be modelled as a vertical stretch of the respective economic function. Thus, understanding function transformations provides valuable insights into economic trends and behaviours.
Transformations can alter the appearance of a function's graph, but they do not change its fundamental shape. For instance, stretches, compressions, translations, and reflections can modify the size, position, or orientation of the graph, but the basic form remains consistent. A parabola remains a parabola, and a sine wave remains a sine wave, regardless of the transformations applied. However, the specific details, such as width, height, or direction, can vary based on the transformation.
Transformations can influence the position of the zeros or roots of a function. Specifically, horizontal translations will shift the zeros left or right, while vertical translations will move the entire graph up or down but won't change the x-values of the zeros. Horizontal stretches and compressions can spread out or squeeze the zeros, respectively. However, vertical stretches or compressions do not change the x-values of the zeros; they only alter the y-values of the function at those points.
Yes, reflections are transformations that can invert a function. Reflecting a function over the x-axis will invert its y-values, essentially flipping the graph upside down. Similarly, reflecting over the y-axis will invert the x-values, mirroring the graph. These reflections can be particularly useful in understanding the symmetries of functions and their inverses. For instance, the inverse of a function can be visualised as a reflection of the original function over the line y = x.
Transformations can significantly influence the periodicity of trigonometric functions. Specifically, horizontal stretches and compressions directly impact the period of functions like sine and cosine. For instance, if the function is h(t) = sin(kt), the period of the function becomes 2π/k. If k is greater than 1, the function compresses, resulting in a shorter period. Conversely, if 0 < k < 1, the function stretches, leading to a longer period. However, vertical transformations, such as stretches and compressions, do not affect the periodicity; they only alter the amplitude.
Practice Questions
To represent the doubling of the profit, we multiply the function by 2, resulting in a vertical stretch. The growth starting 2 months earlier can be represented by a horizontal translation to the left by 2 units. Therefore, the new profit function would be P(t) = 4(t + 2)2. This function indicates that the company expects a significant increase in profit due to the new marketing strategy, with the growth being anticipated earlier in the year.
The introduction of the loudspeaker, which triples the amplitude of the sound waves, can be represented as a vertical stretch in the cosine function. If the original function was h(t) = cos(t), and the loudspeaker triples the wave's height, the new function would be h(t) = 3cos(t). This function suggests that the loudspeaker has a significant impact on the height of the sound waves in the room, making them much more pronounced while maintaining the same frequency.
