can you explain to me the transformations in sinusoidal functions especially the vertical compression/stretch and horizontal compression/stretch?
for example
y = a sin k x
where a and k are constants that we can fool with?
If you increase a, you increase the height of the function. If a is 1, then the function all happens between y = -1 and y = +1
If a is 2, then y swings back and forth between -2 and +2
etc
In the horizontal or x direction, the function repeats every time kx changes by 2pi
so if kx = 0, y = 0 and increasing
then when kx = 2pi, y = 0 and increasing again
that is when x = 2 pi/k
so the "wavelength", the x distance over which the function repeats, is 2 pi/k
so it
Of course! Sinusoidal functions, like the sine and cosine functions, can be transformed by manipulating their amplitude, period, and phase shift. Let's focus on the vertical compression/stretch and horizontal compression/stretch.
1. Vertical Compression/Stretch:
The vertical compression or stretch of a sinusoidal function refers to changing the amplitude of the graph. Amplitude determines the maximum distance the graph reaches from its midline. A compression makes the graph narrower, while a stretch makes it wider.
To compress or stretch the graph vertically, you need to multiply the function by a constant inside the function. Let's use the general form of a sine function, f(x) = a * sin(bx + c), as an example:
- To compress the graph vertically, multiply the function by a constant less than 1. For example, if you multiply it by 0.5, the amplitude will be halved, resulting in a compressed graph.
- To stretch the graph vertically, multiply the function by a constant greater than 1. For instance, if you multiply it by 2, the amplitude will be doubled, leading to a stretched graph.
2. Horizontal Compression/Stretch:
The horizontal compression or stretch of a sinusoidal function refers to the change in its period. The period represents the length of one complete cycle of the function.
To compress or stretch the graph horizontally, you need to divide or multiply the variable x inside the function. Using the same general form of a sine function:
- To compress the graph horizontally, divide x by a constant greater than 1. For example, if you divide it by 2, the period will be halved, resulting in a compressed graph.
- To stretch the graph horizontally, multiply x by a constant greater than 1. For instance, if you multiply it by 2, the period will be doubled, leading to a stretched graph.
Remember that changing the period also affects the frequency of the function. The frequency is the reciprocal of the period, indicating the number of cycles per unit.
By understanding how to compress or stretch sinusoidal functions both vertically and horizontally, you can manipulate the amplitude and period to match specific conditions or graphical requirements.