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1. Vertical dilations of trigonometric functions

By Dan Chicos. Last modified: 16 Jan 2023.

Terminology

dilation: The process of stretching or compressing the graph of a function horizontally or vertically.

scale factor: The value of `k` by which the graph of a function is dilated.

transformation: A general name for the process of changing the graph of a function by moving, reflecting or stretching it.

amplitude: The hight from the centre of a sine or cosine function to the maximum or minimum values (peaks and troughs of its graph respectivily). For `y=k sin ax` and `y=k cos ax`, the amplitude is `k`.

centre: The mean value of a sine or cosine function that is equidistant from the maximum and minimum values. For `y=k sin ax + c` and `y=k cos ax + c`, the centre is `c`.

period: The length of one cycle of a periodic function on the `x`-axis, before the function repeats itself.

phase: A horizontal shift (translation). For `y=k sin [a(x+b)]` and `y=k sin [a(x+b)]`, the phase is `b`; that is, the graph of `y=k sin ax + c` and `y=k cos ax + c` respectively are shifted `b` units to the left.

Vertical dilations

A vertical dilation of any function `y=f(x)` is `y=k f(x)` with scale factor `k`. In the case of sine and cosine functions, the vertical dilations are `y=k sinx` and `y=k cosx`.

Use the applet below to investigate various vertical dilations of `y=sinx` and `y=cosx`.

Instructions to use the applet

Choose one of the 3 functions.

On the left of the graph you'll see a vertical slider. Drag the slider to change the value of `k` and observe what effect it has on the vertical dilation of the graph.

Choose function:

 

Conclusions:

Vertical dilations `y=k sinx` and `y=k cosx` have amplitude `k`.

If `k>1`, the function is stretched.

If `0 < k < 1`, the function is compressed.

If `k < 0`, the vertical dilations are reflected in the x-axis.

The domain remains unchanged.

The range is `[-k,k]`  for the `sin` and `cos` cases.

Note: `y=k tan x` does not have an amplitude since `y=tanx` does not have an amplitude.

 

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