27.03 Vertical shifts and dilations of sine and cosine functions (Extended)

The graphs of trigonometric functions $y=\sin x$y=sinx and  $y=\cos x$y=cosxhave repeated sections. Functions that repeat themselves in fixed intervals are called periodic. The graphs of $y=\sin x$y=sinx and $y=\cos x$y=cosx are defined by their amplitude, phase and period.

In this lesson, we will define amplitude and consider the transformation of trigonometric graphs through vertical shifts and dilations. In subsequent lessons we will look at the definition of the period and phase shifts of trigonometric functions.

Amplitude

The sine and cosine functions have a minimum value, a maximum value, and an average value about which they oscillate. Both of the basic functions $y=\sin x$y=sinx and $y=\cos x$y=cosx have a maximum value of $1$1, a minimum value of $-1$−1, and an average value of $0$0. The distance of the maximum (or minimum) value from the average value is called the amplitude.

The range of sine and cosine functions is given by the difference between their maximum and minimum values. The amplitude is then half of the range (that is, $\frac{\text{max}-\text{min}}{2}$max−min2​). Note that the amplitude is always a positive number.

 

Vertical translations

In general, we say that the graph of a function $y=f\left(x\right)$y=f(x) is vertically translated when the resulting graph has the form $y=f\left(x\right)+c$y=f(x)+c, where $c$c is a non-zero constant.

Graphically speaking, a vertical translation takes each point on the graph of $y=f\left(x\right)$y=f(x) and adds (or subtracts) a constant to the $y$y-value of each point. For instance, all of the points on the graph of $y=\cos x$y=cosx shown below have been shifted upwards according to the constant term.

$y=\cos x$y=cosx vertically translated upwards by a positive constant $c$c.

 

The constant term does not necessarily have to be positive. In the case that $c$c is negative (that is $c<0$c<0), the resulting graph will be translated vertically downwards.

$y=\sin x$y=sinx vertically translated downwards by a negative constant $c$c.

 

Note that for the sine, cosine, and tangent functions, vertical translations do not change the domain. For sine and cosine functions the limits of the range either increase or decrease by the amount of the vertical translation, while the range of the tangent function remains unchanged when vertically translated.

Worked Examples

Example 1

Sketch the graph of $y=\sin x+4$y=sinx+4 for $0\le x\le360$0≤x≤360.

This requires us to vertically translate the graph of $\sin x$sinx upwards by $4$4 units:

 

Vertical dilations of sine and cosine functions

A vertical dilation of a function occurs when we multiply the function by a scale factor. That is, a function $kf\left(x\right)$kf(x) will be vertically dilated by a factor of $k$k. Equations of the form $y=k\sin x$y=ksinx or $y=k\cos x$y=kcosx undergo vertical dilation by a factor $k$k and have an amplitude of $k$k units.

Graphically this transformation corresponds to stretching the graph of $\sin x$sinx or $\cos x$cosxin the vertical direction when $k>1$k>1, and compressing the graph of $\sin x$sinx or $\cos x$cosx in the vertical direction when $00<k<1. This action of stretching and compressing the graph of a function is known as dilation. We can also reflect the graph across the $x$x-axis by multiplying the whole function by $-1$−1. The diagram below shows sine graphs with different amplitudes.

When the graphs of $\sin x$sinx or $\cos x$cosx are vertically dilated by a factor $k$k, the domain remains constant but the range of the function changes by the factor of the dilation.

We can summarise the effects of vertical dilations on the graphs of $y=\sin x$y=sinx or $y=\cos x$y=cosx and their amplitudes as follows:

Function	Dilation	Amplitude
$y=k\sin x$y=ksinx	vertical stretch $|k|>1$|k|>1 vertical compression $0<|k|<1$0<|k|<1 horizontal reflection $k<0$k<0	$|k|$|k|
$y=k\cos x$y=kcosx	vertical stretch $|k|>1$|k|>1 vertical compression $0<|k|<1$0<|k|<1 horizontal reflection $k<0$k<0	$|k|$|k|

Worked examples

Example 2

State the amplitude of the function $f\left(x\right)=5\sin x$f(x)=5sinx. Then sketch the curve for $-180\le x\le180$−180≤x≤180.

Think: When we compare $f\left(x\right)$f(x) to the standard function $\sin x$sinx we can see that all the function values of $5\sin x$5sinx will be five times larger than all the function values of $\sin x$sinx. This means that the amplitude of $5\sin x$5sinx is also five times larger than the amplitude of $\sin x$sinx.

Do: The amplitude of $\sin x$sinx is $1$1, so the amplitude of $f\left(x\right)=5\sin x$f(x)=5sinx is $5\times1=5$5×1=5.

Reflect: We can obtain the graph of $f\left(x\right)=5\sin x$f(x)=5sinx by starting with the graph of $y=\sin x$y=sinx and applying a vertical dilation by a factor of $5$5.

Below is the graph of the curve:
 

Example 3

The graph of $y=\cos x$y=cosx is reflected across the $x$x-axis, then compressed in the vertical direction so that its minimum value is $\frac{3}{4}$34​. What is the equation of the resulting function? What is the amplitude?

Think: Recall that a reflection across the $x$x-axis corresponds to multiplying the function by $-1$−1. This will "switch" the location of the maximum and minimum values of the graph, but the amplitude will still be a positive value.

Do: Let's keep track of how the equation of the graph changes at each stage of the transformation.

$y=\cos x$y=cosx	$\rightarrow$→	$y=-\cos x$y=−cosx	(Reflection across $x$x-axis)
$y=-\cos x$y=−cosx	$\rightarrow$→	$y=-\frac{3}{4}\cos x$y=−34​cosx	(Vertical compression)

 

The final equation of the resulting graph is $y=-\frac{3}{4}\cos x$y=−34​cosx. The amplitude of this equation is $\frac{3}{4}$34​.

Reflect: Compare the resulting equation with the original equation. The only difference is the constant multiple of $-\frac{3}{4}$−34​. Notice that although this number is negative, the amplitude of the resulting equation is positive. In general, the amplitude of the equation $y=k\cos x$y=kcosx is $\left|k\right|$|k|.