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. 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 $y=-2\sin\left(3x+\frac{\pi}{4}\right)$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.

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.

Summary

To obtain the graph of $y=af\left(b\left(x-c\right)\right)+d$y=af(b(x−c))+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(x−c))−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.

Try experimenting with the value of each of these variables in the applet below!

Created with Geogebra

Practice questions

QUESTION 1

Consider the function $y=\sin x+4$y=sinx+4.

Complete the table of values.

$x$x $0$0 $\frac{\pi}{2}$π2 $\pi$π $\frac{3\pi}{2}$3π2 $2\pi$2π

$y$y $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$
Graph the function.

Loading Graph...
What transformation of the graph of $y=\sin x$y=sinx results in the graph of $y=\sin x+4$y=sinx+4?
Horizontal translation $4$4 units to the right.
A
Horizontal translation $4$4 units to the left.
B
Vertical translation $4$4 units down.
C
Vertical translation $4$4 units up.
D
What is the maximum value of $y=\sin x+4$y=sinx+4?
What is the minimum value of $y=\sin x+4$y=sinx+4?

$x$`x`	$0$0	$\frac{\pi}{2}$π2	$\pi$π	$\frac{3\pi}{2}$3π2	$2\pi$2π
$y$`y`	$\editable{}$	$\editable{}$	$\editable{}$	$\editable{}$	$\editable{}$

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?

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 $y=\sin x$y=sinx.

Loading Graph...

How can we transform the graph of $y=\sin x$y=sinx to create the graph of $y=\sin\left(x-\frac{\pi}{2}\right)+3$y=sin(x−π2)+3?
Move the graph to the left by $\frac{\pi}{2}$π2 radians and $up$ by $3$3 units.
A
Move the graph to the right by $\frac{\pi}{2}$π2 radians and $up$ by $3$3 units.
B
Move the graph to the right by $\frac{\pi}{2}$π2 radians and $down$ by $3$3 units.
C
Move the graph to the left by $\frac{\pi}{2}$π2 radians and $down$ by $3$3 units.
D
Hence graph $y=\sin\left(x-\frac{\pi}{2}\right)+3$y=sin(x−π2)+3 on the same set of axes.

Loading Graph...
What is the period of the curve $y=\sin\left(x-\frac{\pi}{2}\right)+3$y=sin(x−π2)+3 in radians?

Outcomes

11.SF.TF.1

Positive and negative angles. Measuring angles in radians and in degrees and conversion from one measure to another. Definition of trigonometric functions with the help of unit circle. Truth of the identity sin^2 x + cos^2 x = 1, for all x. Signs of trigonometric functions and sketch of their graphs. Expressing sin (x + y) and cos (x + y) in terms of sin x, sin y, cos x and cos y. Deducing the identities like following: cot(x + or - y), sin x + sin y, cos x + cos y, sin x - sin y, cos x - cos y (see syllabus document)

Transformations of sine and cosine curves and equations