Final answer:
The equation g(x) = f(2x) represents a horizontal shrink of the function f, compressing its graph horizontally by a factor of 2. A translation occurs when every point on a graph (representing a function) moves by the same amount in the same direction. There are two types of translations of functions. Horizontal translations,Vertical translations.
Step-by-step explanation:
When discussing transformations of functions, the equation g(x) = f(2x) indicates a horizontal shrink of the function f to obtain the function g. This is because each input value of x in the function g is being multiplied by 2 before applying the original function f, effectively compressing the graph of f horizontally by a factor of 2. This means that what was stretched over a certain horizontal distance in f will now take up half of that distance in g. This is different from a vertical shrink or stretch, which would affect the output values Y, and different from a horizontal stretch, which would make the graph wider rather than narrower.
Transformation of functions means that the curve representing the graph either "moves to left/right/up/down" or "it expands or compresses" or "it reflects". For example, the graph of the function f(x) = x2 + 3 is obtained by just moving the graph of g(x) = x2 by 3 units up. Function transformations are very helpful in graphing the functions just by moving/expanding/compressing/reflecting the curve without actually needing to graph it from scratch.
The horizontal dilation (also known as horizontal scaling) of a function either stretches/shrinks the curve horizontally. It changes a function y = f(x) into the form y = f(kx), with a scale factor '1/k', parallel to the x-axis. Here,
- If k > 1, then the graph shrinks.
- If 0 < k < 1, then the graph stretches.
In this dilation, there will be changes only in the x-coordinates but there won't be any changes in the y-coordinates. Every old x-coordinate is multiplied by 1/k to find the new x-coordinate.