Transformations of sine and cosine
The impact of the parameters $a$a, $b$b, $c$c and $d$d, have on the graph of $y=af\left(b\left(x-c\right)\right)+d$y=af(b(x−c))+d, can be summarised as follows:
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 dilates (stretches) the graph by a factor of $a$a from the $x$x-axis, parallel to the $y$y-axis
- When $a<0$a<0 the graph is reflected about the $x$x-axis, parallel to the $x$x-axis
- $b$b dilates (stretches) the graph by a factor of $\frac{1}{b}$1b from the $y$y-axis, parallel to the $x$x-axis
- When $b<0$b<0 the graph is reflected about the $y$y-axis
- $c$c translates the graph $c$c units horizontally, when $c>0$c>0 the graph shifts $c$c units to the right and when $c<0$c<0 the graph shifts $|c|$|c| units to the left
- $d$d translates the graph $d$d units vertically, when $d>0$d>0 the graph shifts $d$d units upwards and when $d<0$d<0 the graph shifts $|d|$|d| units downwards
Let's 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(x−c))+d and $f\left(x\right)=a\cos\left(b\left(x-c\right)\right)+d$f(x)=acos(b(x−c))+d:
We can see the parameters had the expected impact on the graph and we can describe key features in relation to these.
Key features of the graph of $y=a\sin\left(b\left(x-c\right)\right)+d$y=asin(b(x−c))+d or $y=a\cos\left(b\left(x-c\right)\right)+d$y=acos(b(x−c))+d are:
- Principal axis: This is line about which the graph oscillates. For the base graph of $y=\sin x$y=sinx or $y=\cos x$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 is $|a|$|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|
Exploration
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.
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| 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).
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| 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.
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| 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.
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| 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)). Remember we do this because usually, in the form presented, we need to translate horizontally before we can horizontally dilate. But for trigonometry graphs, performing the horizontal dilation first is far easier when drawing.
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.
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| 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.
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| The graph of $y=-2\sin\left(3x+\frac{\pi}{4}\right)+2$y=−2sin(3x+π4)+2 |
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.
What is the maximum value of the function?
What is the minimum value of the function?
What is the amplitude of the function?
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.
AHorizontal translation.
BReflection across the $x$x-axis.
CVertical translation.
D
Question 2
Determine the equation of the graphed function given that it is of the form $y=\cos\left(x-c\right)$y=cos(x−c), where $c$c is the least positive value.
Question 3
The functions $f\left(x\right)$f(x) and $g\left(x\right)=jf\left(kx\right)$g(x)=jf(kx) have been graphed on the same set of axes in grey and black respectively.
What transformations have occurred from $f\left(x\right)$f(x) to $g\left(x\right)$g(x)?
Select all that apply.
Vertical stretching by a factor of $4$4.
AHorizontal compression by factor of $3$3.
BVertical compression by a factor of $3$3.
CHorizontal stretching by factor of $4$4.
DDetermine the value of $j$j.
Determine the value of $k$k.
Question 4
Consider the function $y=3\sin\left(x-\frac{\pi}{3}\right)+2$y=3sin(x−π3)+2.
Determine the period of the function, giving your answer in radians.
Determine the amplitude of the function.
Determine the maximum value of the function.
Determine the minimum value of the function.
Graph the function.
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