delete -- Harmonic oscillations (update see german version)

Before we start, recall the geometric transformations "stretching" and "shifting". Here are some exercises.

Harmonic oscillations are simply stretched and shifted sine, cosine and tangent functions. Recall that we have discussed how to transform quadratic functions in this way.

With the additional parameters we can change the shape of the sine wave. We can make the waves larger, wider, flip them or shift them left or right or up and down. Each parameter is responsible for one of these transformations. Let's explore this further in the following exercise.

Exercise 4

Use a table of values to sketch the graphs $f$ and $g$ into the same coordinate system. Also draw the graph $h(x)=\sin(x)$ into the same coordinate system and try to figure out with what geometric transformation we get from $f$ to $h$, and from $g$ to $h$.

Also determine for each $f$ and $g$ the value of the parameters $A, u,v$ und $b$.

1. $f(x)=2\sin(x)$ und $g(x)=0.5\sin(x)$

2. $f(x)=\sin(2x)$ und $g(x)=\sin(0.5x)$

3. $f(x)=\sin(x+\frac{\pi}{4})$ und $g(x)=\sin(x-\frac{\pi}{4})$

4. $f(x)=\sin(x)+0.5$ und $g(x)=\sin(x)-0.5$

Solution

$A=2$ and $A=0.5$, $u=1, v=0, b=0$

$A=1$, $u=2$ and $u=0.5$, $v=0, b=0$

$A=1$, $u=1$ and $v=-\frac{\pi}{4}$ and $v=\frac{\pi}{4}$, $b=0$

$A=1, u=1, v=0$, $b=0.5$ and $b=-0.5$

The following is evident from the exercise above. Let $h(x)=\sin(x)$ and $f(x)=A\sin(u(x-v))+b$. The parameters $A, u, v$ and $b$ describe geometric transformations to get from the graph of $h$ to the graph of $f$:

1. $A$ stretches the graph of $h$ by a factor $A$ in the $y$-direction.
   1. $\sin(x) \rightarrow 2\sin(x)$: Graph is stretched by the factor of $2$ in $y$-direction (elongation).
   2. $\sin(x) \rightarrow \frac{1}{2}\sin(x)$: Graph is stretched by the factor of $\frac{1}{2}$ in $y$-direction (compression).
2. $u$ stretches the graph of $h$ by the factor $\frac{1}{u}$ in $x$-direction.
   1. $\sin(x) \rightarrow \sin(2x)$: Graph is stretched by the factor of $\frac{1}{2}$ in $x$-direction (compression).
   2. $\sin(x) \rightarrow \sin(\frac{1}{2}x)$: Graph is stretched by the factor of $2$ in $x$-direction (elongation).
3. $v$ shifts the graph of $h$ by $v$ in the $x$ direction.
   1. $\sin(x) \rightarrow \sin(x-\frac{\pi}{4})$: Graph is shifted to the right by $\frac{\pi}{4}$.
   2. $\sin(x) \rightarrow \sin(x+\frac{\pi}{4})$: Graph is shifted to the left by $\frac{\pi}{4}$.
4. $b$ shifts the graph of $h$ by $b$ in the $y$ direction.
   1. $\sin(x) \rightarrow \sin(x)+\frac{1}{2}$: Graph is shifted upwards by $\frac{1}{2}$ .
   2. $\sin(x) \rightarrow \sin(x)-\frac{1}{2}$: Graph is shifted downwards by $\frac{1}{2}$.

The same is true for the graphs of the functions $\cos(x)$ and $\tan(x)$. Let's generalise:

Theorem 1: Harmonic oscillations

Consider the function

$$f(x)=A\sin(u(x-v))+b$$

We obtain the graph of $f$ by applying the following transformations to the graph of $\sin(x)$:

1. $A$: stretch $\sin(x)$ in $y$ direction by factor $A$
2. $u$: stretch the new graph in $x$ direction by factor $\frac{1}{u}$
3. $v$: shift the new graph to the right ($v>0$) or the left ($v<0$) by $v$.
4. $b$: shift the new graph up ($b>0$) or down ($b<0$) by $b$.

The order you execute these transformations is important, you have to start with the stretching in $y$- or $x$-direction, followed by shifting in $x$- or $y$-direction. However, the order of the two stretching transformations could be changed, and also the order of the two shifting transformations. But it is probably best to follow the order shown above, which corresponds to the order of the parameters in the formula!

The effect that parameters $A$ and $b$ have on the graph should be clear. Both modify the output of the function $\sin(x)$ by either multiplying the output by $A$ or adding to the output $b$ (see figure below). Because the output is the $y$-coordinate of the points on the graph, we immediately see that the graph of $\sin$ is stretched in $y$-direction by $A$ or shifted up or down by $b$.

Less clear is the effect of $u$ and $v$, both of which modify the input $x$ before it is fed into the $\sin$ function (see figure below).

In fact, the effect of these modifications are quite counter-intuitive. For example, $f(x)=\sin(2x)$ (that is, $u=2$) is a compressed version of the $\sin$-graph, and not an elongated version, and $f(x)=\sin(x+\frac{\pi}{4})$ (that is, $v=-\frac{\pi}{4}$) moves the $\sin$-graph to the left, and not to the right, which is quite unexpected. We can understand this by investigating the effect of the parameters $u$ and $v$ on the $x$-intercepts of $\sin$. This is shown in the example below:

Example 1

We already know that the $x$-intercepts of the function $\sin(x)$ are

$$...,-\pi, 0, \pi, 2\pi, 3\pi, ...$$

Determine the $x$-intercepts of the function $f$ below by solving the equation

$$f(x)=0$$

1. $f(x)=\sin(2x)$, that is, $u=2$
2. $f(x)=\sin(u\cdot x)$
3. $f(x)=\sin(x-2)$, that is, $v=2$
4. $f(x)=\sin(x+2)$, that is, $v=-2$

Determine in each case the geometric transformation that has to be applied to get from the $x$-intercepts of $\sin(x)$ to the $x$-intercept of $f$.

Solution

• $f(x)=\sin(2x)$: We have to find $x$ with

  $$\sin(2x)=0$$

  We know this is the case if $2x$ equals to $...,-\pi,0,\pi,2\pi,3\pi, ..., s,...$. Thus find $x$ with

  $$\begin{array}{llll} 2x &=& -\pi & \rightarrow x=-\frac{\pi}{2}\\ 2x &=& 0 & \rightarrow x=0\\ 2x &=& \pi & \rightarrow x=\frac{\pi}{2}\\ 2x &=& 2\pi & \rightarrow x=\pi\\ .. & & &\\ 2x &=& s & \rightarrow x=\frac{1}{2}s\\ \end{array}$$

  So if $s$ in an $x$-intercepts of $\sin(x)$, then $\frac{1}{2}s$ is an $s$-intercept of $f(x)=\sin(2x)$. Clearly, the new $x$-intercepts are obtains by stretched the old ones the factor $\frac{1}{2}$ in $x$-direction.

• $f(x)=\sin(u\cdot x)$: Similar, if $s$ is an $x$-intercept of $\sin(x)$, then $x$ is an $x$-intercept of $\sin(u\cdot x)$ if

$$u\cdot x= s$$

and thus

$$x=\frac{1}{u}s$$

So clearly, the $x$-intercepts of $f(x)=\sin(u\cdot x)$ are obtained by stretching the $x$-intercepts of $\sin(x)$ by the factor $\frac{1}{u}$.

• $f(x)=\sin(x-2)$: Similar, if $s$ is an $x$-intercept of $\sin(x)$, then $x$ is an $x$-intercept of $\sin(x-2)$ if

$$x-2= s$$

and thus

$$x=s+2$$

So clearly, the $x$-intercepts of $f(x)=\sin(x-2)$ are obtained by shifting the $x$-intercepts of $\sin(x)$ to the right by $2$.

• $f(x)=\sin(x+2)$: Similar, if $s$ is an $x$-intercept of $\sin(x)$, then $x$ is an $x$-intercept of $\sin(x+2)$ if

$$x+2= s$$

and thus

$$x=s-2$$

So clearly, the $x$-intercepts of $f(x)=\sin(x+2)$ are obtained by shifting the $x$-intercepts of $\sin(x)$ to the left by $2$.

Another "strange" thing that needs some explaining is that the shift parameter $v$ is in a bracket

However, arguing in the same way as in the example above, we see that only if this is the case is it true that $v$ is "shifting to the right (or left) by $v$. See the example below.

Note that for the function $f(x)=\sin(3x-1)$ we stretch the graph of $\sin(x)$ in $x$-direction by the factor $\frac{1}{3}$, and then shift the resulting graph to the right by $\frac{1}{3}$. Why? Because we can factor out the $3$ in the term $3x-1$ and get

And now some exercises ...

Exercise 5

Find the function equation of the harmonic oscillation $f$ shown below (blue stippled line). To do so, identify the correct geometric transformations to get from the graph of $\sin(x)$ (shown in gray) to the graph of $f$.

Solution

No stretching in $y$-direction, no shifting up or down.

1. Stretch $\sin(x)$ in $x$-direction by $\frac{1}{2}$

2. Shift to the right by $\frac{\pi}{4}$. Thus $A=1, u=\frac{1}{\frac{1}{2}}=2, v=\frac{\pi}{4}, b=0$. Thus,

   $$f(x)=\sin(2(x-\frac{\pi}{4}))$$

Exercise 6

Find the geometric transformations to get from the graph of $h(x)=\sin(x)$ to the graph of $f(x)=3\sin(0.5x-\frac{\pi}{4})+1$?

Draw the graph of $f$ by applying these transformations to the graph of $h$.

Solution

We first write

$$f(x)=3\sin(0.5x-\frac{\pi}{4})+1=3\sin(0.5(x-\frac{\pi}{2}))+1$$

so that we see that $A=3, u=0.5, v=\frac{\pi}{2}$, and $b=1$. Thus, the geometric transformations applied to the graph of $\sin(x)$ are

1. stretch in $y$-direction by the factor $3$, then
2. stretch in $x$-direction by the factor $2$, then
3. shift to the right by $\frac{\pi}{2}$, then
4. shift upwards by $1$.