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.