Transformations of sine and cosine curves and equations

Consider the graphs of $y=\sin x$y=sinx and $y=-2\sin\left(3x+45^\circ\right)+2$y=−2sin(3x+45°)+2 which are drawn below.

The graphs of $y=\sin x$y=sinx and $y=-2\sin\left(3x+45^\circ\right)+2$y=−2sin(3x+45°)+2

Starting with the graph of $y=\sin x$y=sinx, we can work through a series of transformations so that it coincides with the graph of $y=-2\sin\left(3x+45^\circ\right)+2$y=−2sin(3x+45°)+2.

We can first reflect the graph of $y=\sin x$y=sinx about the $x$x-axis. This is represented by applying a negative sign to the function (multiplying the function by $-1$−1).

The graph of $y=-\sin x$y=−sinx

Then we can increase the amplitude of the function to match. This is represented by multiplying the $y$y-value of every point on $y=-\sin x$y=−sinx by $2$2.

The graph of $y=-2\sin x$y=−2sinx

Next we can apply the period change that is the result of multiplying the $x$x-value inside the function by $3$3. This means that to get a particular $y$y-value, we can put in an $x$x-value that is $3$3 times smaller than before. Notice that the points on the graph of $y=-2\sin x$y=−2sinx move towards the vertical axis by a factor of $3$3 as a result.

The graph of $y=-2\sin3x$y=−2sin3x

 

Our next step will be to obtain the graph of $y=-2\sin\left(3x+45^\circ\right)$y=−2sin(3x+45°), and we can do so by applying a horizontal translation. In order to see what translation to apply, however, we first factor the function into the form $y=-2\sin\left(3\left(x+15^\circ\right)\right)$y=−2sin(3(x+15°)).

In this form, we can see that the $x$x-values are increased by $15^\circ$15° inside the function. This means that to get a particular $y$y-value, we can put in an $x$x-value that is $15^\circ$15° smaller than before. Graphically, this corresponds to shifting the entire function to the left by $15^\circ$15°.

The graph of $y=-2\sin\left(3x+45^\circ\right)$y=−2sin(3x+45°)

 

Lastly, we translate the graph of $y=-2\sin\left(3x+45^\circ\right)$y=−2sin(3x+45°) upwards by $2$2 units, to obtain the final graph of $y=-2\sin\left(3x+45^\circ\right)+2$y=−2sin(3x+45°)+2.

The graph of $y=-2\sin\left(3x+45^\circ\right)+2$y=−2sin(3x+45°)+2

 

 

The general case

In the example above we were transforming the graph of $y=\sin x$y=sinx. The particular function $y=\sin x$y=sinx was not important, however. We could have just as easily transformed the graph of $y=\cos x$y=cosx, or even a non-trigonometric function, using the same method!

Consider a function $y=f\left(x\right)$y=f(x). Then we can obtain the graph of $y=af\left(b\left(x-c\right)\right)+d$y=af(b(x−c))+d, where $a,b,c,d$a,b,c,d are constants, by applying a series of transformations to the graph of $y=f\left(x\right)$y=f(x). These transformations are summarised below.

 

 

If $y=f\left(x\right)$y=f(x) is the equation of a trigonometric function, then a vertical dilation corresponds to an amplitude change, a horizontal dilation corresponds to a period change and a horizontal translation corresponds to a phase shift.

The signs of $c$c and $d$d determine the direction of the horizontal and vertical translations respectively. If $c$c is positive the transformation describes a translation to the right, and if $c$c is negative the transformation describes a translation to the left. If $d$d is positive the transformation describes a translation upwards, and if $d$d is negative the transformation describes a translation downwards.

 

 

Lastly, the magnitude of $a$a and $b$b determine whether the vertical and horizontal dilations each describe a compression or an expansion.

For a value of $a$a where $\left|a\right|>1$|a|>1, the graph of $y=f\left(x\right)$y=f(x) vertically expands or stretches. For a trigonometric function, we say that the amplitude increases. If $\left|a\right|<1$|a|<1, the graph of $y=f\left(x\right)$y=f(x) vertically compresses. For a trigonometric function, we say that the amplitude decreases.

For a value of $b$b where $\left|b\right|>1$|b|>1, the graph of $y=f\left(x\right)$y=f(x) horizontally compresses. If $\left|b\right|<1$|b|<1, then the graph horizontally expands or stretches. In the case that the graph describes a trigonometric function, a horizontal compression means the period decreases and a horizontal expansion means the period increases.

 

Practice questions

question 1

question 2

question 3