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. 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 factor 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 |
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.
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.
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):
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.
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!
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.
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.
Horizontal translation $4$4 units to the left.
Vertical translation $4$4 units down.
Vertical translation $4$4 units up.
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?
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.
Reflection about the $x$x-axis, then translation $3$3 units up.
Translation $3$3 units up, then reflection about the $x$x-axis.
Translation $3$3 units down, then reflection about the $x$x-axis.
Consider the given graph of $y=\sin x$y=sinx.
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.
Move the graph to the right by $\frac{\pi}{2}$π2 radians and up by $3$3 units.
Move the graph to the right by $\frac{\pi}{2}$π2 radians and down by $3$3 units.
Move the graph to the left by $\frac{\pi}{2}$π2 radians and down by $3$3 units.
Hence graph $y=\sin\left(x-\frac{\pi}{2}\right)+3$y=sin(x−π2)+3 on the same set of axes.
What is the period of the curve $y=\sin\left(x-\frac{\pi}{2}\right)+3$y=sin(x−π2)+3 in radians?