topic badge
AustraliaVIC
VCE 11 Methods 2023

8.06 Transformations of sine and cosine

Lesson

Recall from the lesson on transformation of functions, the impact of the parameters $a$a$b$b$c$c and $d$d on the graph of $y=af\left(b\left(x-c\right)\right)+d$y=af(b(xc))+d was summarised as follows:

Summary

To obtain the graph of $y=af\left(b\left(x-c\right)\right)+d$y=af(b(xc))+d from the graph of $y=f\left(x\right)$y=f(x):

  • $a$a dilates(stretches) the graph by a factor of $a$a from the $x$x-axis 
  • When $a<0$a<0 the graph was reflected about the $x$x-axis
  • $b$b dilates(stretches) the graph by a factor of $\frac{1}{b}$1b from the $y$y-axis 
  • When $b<0$b<0 the graph was reflected about the $y$y-axis
  • $c$c translates the graph $c$c units horizontally
  • $d$d translates the graph $d$d units vertically

Now look at these more closely in relation to the graphs $f\left(x\right)=\sin\left(x\right)$f(x)=sin(x) and $f\left(x\right)=\cos\left(x\right)$f(x)=cos(x)and the impact on key features of these graphs. Use the applet below to observe the impact of $a$a$b$b$c$c and $d$d on $f\left(x\right)=a\sin\left(b\left(x-c\right)\right)+d$f(x)=asin(b(xc))+d and $f\left(x\right)=a\cos\left(b\left(x-c\right)\right)+d$f(x)=acos(b(xc))+d:

Notice that the parameters had the expected impact on the graph and the key features can be described in relation to these.

Key features

Key features of the graph of $y=a\sin\left(b\left(x-c\right)\right)+d$y=asin(b(xc))+dor $y=a\cos\left(b\left(x-c\right)\right)+d$y=acos(b(xc))+d are:

  • Principal axis: This is the line about which the graph oscillates. For the base graph of $y=\sin x$y=sinxor $y=\cos x$y=cosx this is the $x$x-axis(the line $y=0$y=0). Since the graph has been translated vertically by $d$d units, the principal axis has the equation $y=d$y=d.
  • Amplitude: The vertical distance from the principal axis to the maximum point is the amplitude. Hence, as the base graph has an amplitude of $1$1 and graph has been dilated by a factor of $a$a the amplitude$=|a|$=|a|.
  • Period: The period is the length of a cycle. This can be found as the horizontal distance between two successive maximums or minimums. As $b$b dilates the graph horizontally by a factor of $\frac{1}{b}$1b and the base graph has a period of $2\pi$2π, the transformed graph has period$=\frac{2\pi}{b}$=2πb.
  • Phase shift: $c$c translates the graph $c$c units horizontally. 
  • Maximum: The maximum value of the function is $y=d+|a|$y=d+|a|
  • Minimum: The minimum value of the function is $y=d-|a|$y=d|a|

Worked example

Consider the graphs of $y=\sin x$y=sinx and $y=-2\sin\left(3x+\frac{\pi}{4}\right)+2$y=2sin(3x+π4)+2 which are drawn below.

The graphs of $y=\sin x$y=sinx and $y=-2\sin\left(3x+\frac{\pi}{4}\right)+2$y=2sin(3x+π4)+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 $y=-2\sin\left(3x+\frac{\pi}{4}\right)+2$y=2sin(3x+π4)+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. The result is a horizontal dilation by a factor of $\frac{1}{3}$13, note the new graph will now cycle every $\frac{2\pi}{3}$2π3.

The graph of $y=-2\sin3x$y=2sin3x

 

Our next step will be to obtain the graph of $y=-2\sin\left(3x+\frac{\pi}{4}\right)$y=2sin(3x+π4), and we can do so by applying a horizontal translation. In order to see what translation to apply, however, we first factorise the function into the form $y=-2\sin\left(3\left(x+\frac{\pi}{12}\right)\right)$y=2sin(3(x+π12)).

In this form, we can see that the $x$x-values are increased by $\frac{\pi}{12}$π12 inside the function. This means that to get a particular $y$y-value, we can put in an $x$x-value that is $\frac{\pi}{12}$π12 smaller than before. Graphically, this corresponds to shifting the entire function to the left by $\frac{\pi}{12}$π12 units.

The graph of $y=-2\sin\left(3x+\frac{\pi}{4}\right)$y=2sin(3x+π4)

 

Lastly, we translate the graph of $y=-2\sin\left(3x+\frac{\pi}{4}\right)$y=2sin(3x+π4) upwards by $2$2 units, to obtain the final graph of $y=-2\sin\left(3x+\frac{\pi}{4}\right)+2$y=2sin(3x+π4)+2.

The graph of $y=-2\sin\left(3x+\frac{\pi}{4}\right)+2$y=2sin(3x+π4)+2

 

Careful!

When we geometrically apply each transformation to the graph of $y=\sin x$y=sinx, it's important to consider the order of operations. If we had wanted to vertically translate the graph before reflecting about the $x$x-axis, we would have needed to translate the graph downwards first. Alternatively, you can apply the reflection last by reflecting about the new principal axis.

 

Practice questions

Question 1

Consider the function $y=-3\cos x$y=3cosx.

  1. What is the maximum value of the function?

  2. What is the minimum value of the function?

  3. What is the amplitude of the function?

  4. Select the two transformations that are required to turn the graph of $y=\cos x$y=cosx into the graph of $y=-3\cos x$y=3cosx.

    Vertical dilation.

    A

    Horizontal translation.

    B

    Reflection across the $x$x-axis.

    C

    Vertical translation.

    D

Question 2

Which combinations of transformations could be used to turn the graph of $y=\cos x$y=cosx into the graph of $y=-\cos x+3$y=cosx+3?

  1. Select the two correct options.

    Reflection about the $x$x-axis, then translation $3$3 units down.

    A

    Reflection about the $x$x-axis, then translation $3$3 units up.

    B

    Translation $3$3 units up, then reflection about the $x$x-axis.

    C

    Translation $3$3 units down, then reflection about the $x$x-axis.

    D

Question 3

Consider the given graph of $y=\cos\left(x+\frac{\pi}{2}\right)$y=cos(x+π2).

Loading Graph...

  1. What is the amplitude of the function?

  2. How can the graph of $y=\cos x$y=cosx be transformed into the graph of $y=\cos\left(x+\frac{\pi}{2}\right)$y=cos(x+π2)?

    By reflecting it about the $x$x-axis, and then translating it horizontally $\frac{\pi}{2}$π2 units to the left.

    A

    By reflecting it about the $x$x-axis, and then translating it horizontally $\frac{\pi}{2}$π2 units to the right.

    B

    By translating it horizontally $\frac{\pi}{2}$π2 units to the right.

    C

    By changing the period of the function.

    D

    By translating it horizontally $\frac{\pi}{2}$π2 units to the left.

    E

Question 4

Determine the equation of the graphed function given that it is of the form $y=\cos\left(x-c\right)$y=cos(xc), where $c$c is the least positive value.

Loading Graph...

Question 5

Determine the equation of the graphed function given that it is of the form $y=\sin bx$y=sinbx or $y=\cos bx$y=cosbx, where $b$b is positive.

Loading Graph...

Outcomes

U1.AoS1.7

the effect of transformations of the plane, dilation, reflection in axes, translation and simple combinations of these transformations, on the graphs of functions

U1.AoS1.13

describe the effect of transformations on the graphs of relations and functions

U1.AoS2.9

representations of points and transformations

U1.AoS2.5

transformations of the plane and application to basic functions and relations by simple combinations of dilations (students should be familiar with both ‘parallel to an axis’ and ‘from an axis’ descriptions), reflections in an axis and translations (matrix representation may be used but is not required)

U2.AoS1.19

sketch by hand the unit circle, graphs of the sine, cosine and exponential functions, and simple transformations of these to the form Af(bx)+c , sketch by hand graphs of log_a(x) and the tangent function, and identify any vertical or horizontal asymptotes

U2.AoS1.5

circular functions of the form y=Af(nx)+c and their graphs, where f is the sine, cosine or tangent function

U2.AoS1.15

the effect of transformations of the plane on the graphs of sine, cosine, tangent and exponential functions

U2.AoS1.21

describe the effect of transformations of the plane on the graphs of the sine, cosine, tangent and exponential functions

What is Mathspace

About Mathspace