NZ Level 7 (NZC) Level 2 (NCEA) Transformations of sine and cosine curves and equations
Lesson

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 factorise 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

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.

### 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(xc))+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.

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 vertically dilates the graph of $y=f\left(x\right)$y=f(x).
• $b$b horizontally dilates the graph of $y=f\left(x\right)$y=f(x).
• $c$c horizontally translates the graph of $y=f\left(x\right)$y=f(x).
• $d$d vertically translates the graph of $y=f\left(x\right)$y=f(x).

In the case that $a$a is negative, it has the additional property of reflecting the graph of $y=f\left(x\right)$y=f(x) about the horizontal axis.

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.

Careful!

If $c$c is negative, it may be convenient to represent the equation in the form $y=af\left(b\left(x+c\right)\right)+d$y=af(b(x+c))+d instead, where we've redefined $c$c using its absolute value. In this case, the value of $c$c represents translation to the left.

Similarly, if $d$d is negative, it may be convenient to represent the equation in the form $y=af\left(b\left(x-c\right)\right)-d$y=af(b(xc))d, where we've redefined $d$d using its absolute value. In this case, the value of $d$d represents 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

Consider the graphs of $y=\sin x$y=sinx and $y=5\sin\left(x+\left(\left(-60\right)\right)\right)$y=5sin(x+((60))).

1. What transformations have occurred?

Select all that apply.

Vertical translation

A

Horizontal translation

B

Vertical dilation

C

Horizontal dilation

D

Vertical translation

A

Horizontal translation

B

Vertical dilation

C

Horizontal dilation

D
2. Complete the following statement.

The graph of $y=\sin x$y=sinx has increased its amplitude by a factor of $\editable{}$ units and has undergone a phase shift of $\editable{}$ to the right.

##### question 2

The graph of $y=\cos x$y=cosx has been transformed into the graph of $y=\cos\left(2x+\left(\left(-60\right)\right)\right)$y=cos(2x+((60))).

1. What transformations have occurred?

Select all that apply.

Vertical translation

A

Horizontal translation

B

Horizontal dilation

C

Vertical dilation

D

Vertical translation

A

Horizontal translation

B

Horizontal dilation

C

Vertical dilation

D
2. Complete the following statement.

The graph of $y=\cos x$y=cosx has decreased its period by a factor of $\editable{}$ and then has undergone a phase shift of $\editable{}$ to the right.

3. Draw the graph of $y=\cos\left(2x+\left(\left(-60\right)\right)\right)$y=cos(2x+((60))).

##### question 3

The graph of $y=\sin x$y=sinx undergoes the series of transformations below.

What is the equation of the transformed graph in the form $y=-\sin\left(x+c\right)+d$y=sin(x+c)+d where $c$c is the lowest positive value in degree?

• The graph is reflected about the $x$x-axis.
• The graph is horizontally translated to the left by $60^\circ$60°.
• The graph is vertically translated downwards by $3$3 units.

### Outcomes

#### M7-2

Display the graphs of linear and non-linear functions and connect the structure of the functions with their graphs

#### 91257

Apply graphical methods in solving problems