NZ Level 8 (NZC) Level 3 (NCEA) [In development]
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Transformations of sine and cosine curves and equations
Lesson

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 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(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.

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

Practice questions

QUESTION 1

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

  1. 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 $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$
  2. Graph the function.

    Loading Graph...

  3. 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

    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
  4. What is the maximum value of $y=\sin x+4$y=sinx+4?

  5. What is the minimum value of $y=\sin x+4$y=sinx+4?

QUESTION 2

What two 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 all correct options.

  1. 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

    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...

  1. 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(xπ2)+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

    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
  2. Hence graph $y=\sin\left(x-\frac{\pi}{2}\right)+3$y=sin(xπ2)+3 on the same set of axes.

    Loading Graph...

  3. What is the period of the curve $y=\sin\left(x-\frac{\pi}{2}\right)+3$y=sin(xπ2)+3 in radians?

Outcomes

M8-2

Display and interpret the graphs of functions with the graphs of their inverse and/or reciprocal functions

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