Exploring Sinusoidal Functions

Task

In this task we will explore the effect that changing the parameters in a sinusoidal function has on the graph of the function. A general sinusoidal function is of the form $$y=A\sin(B(x-h))+k$$ or $$y=A\cos(B(x-h))+k.$$

1. Use the sliders in the applet to change the values of $A,\ k,\ h,$ and $B$ to create the functions in the table. Then describe the effect that changing each parameter has on the shape of the graph. Add more rows to the table, if necessary.

   Function	Effect on $y = \sin x$
   $y = 2\sin x$
   $y = -2\sin x$
   $y = \sin x + 2$
   $y = \sin x - 2$
   $y = \sin(x + 2)$
   $y = \sin(x - 2)$
   $y = \sin(2x)$
   $y = \sin(\frac12x)$
   $y =$
   $y =$

2. Describe how changing $A$, $k$, and $h$ changes the graph of the function.
3. There seems to be a relationship between $B$ and the period of the function but it is harder to describe than the other parameters. Experiment with different values of $B$ and fill in the corresponding period in the table below. In the last row of the table, use the data you have collected to infer a general relationship between $B$ and the period.

   $B$	$y=\sin(Bx)$	Period
   1	$y=\sin(x)$
   2	$y=\sin(2x)$
   4	$y=\sin(4x)$
   $1/2$	$y=\sin(\frac12x)$
   -2	$y=\sin(-2x)$
   -4	$y=\sin(-4x)$
   $-1/2$	$y=\sin(-\frac12x)$
   $B$	$y=\sin(Bx)$

IM Commentary

This task serves as an introduction to the family of sinusoidal functions. It uses a desmos applet to let students explore the effect of changing the parameters in $y=A\sin(B(x-h))+k$ on the graph of the function. Sinusoidal functions are the perfect type of function to illustrate how new functions can be built from already known functions by shifting and scaling. Even though the task statement only considers transformation of $y=\sin x$ it would be easy to change the applet and the problem statement to also explore transformations of $y=\cos x$.

The task can be used in a variety of ways. Students can work in class, possibly in groups, to experiment with the applet and then discuss their observations. Alternatively, students can do the exploration as homework and bring their observations to class for a whole class discussion and to pull together all the pieces.

In this task students engage in SMP 7 - Look For and Make Use of Structure and SMP 8 - Look For and Express Regularity in Repeated Reasoning. Since students are asked to verbalize their observations, the instructor can also use the opportunity to work on SMP 6 - Attend to Precision, in this case precision of language and notation is important.

Solution

1. Function	Effect on $y = \sin x$
   $y = 2\sin x$	Stretch graph vertically by a factor of two.
   $y = -2\sin x$	Stretch graph vertically by a factor of two and reflect about the $x$ axis.
   $y = \sin x + 2$	Shift graph up by two units.
   $y = \sin x - 2$	Shift down by two units.
   $y = \sin(x + 2)$	Shift left by two units.
   $y = \sin(x - 2)$	Shift right by two units.
   $y = \sin(2x)$	Compress horizontally by a factor of two.
   $y = \sin(\frac12x)$	Stretch horizontally by a factor of two.

2. Changing $A$ changes the amplitude of the function. In fact, the function has amplitude equal to $|A|$. For $k>0$, the graph is shifted up by $k$ units. For $k\lt0$, the graph is shifted down by $k$ units. Therefore, the function has midline $y=k$. For $h>0$, the graph is shifted to the right by $h$ units. For $h\lt0$, the graph is shifted to the left by $h$ units.

3. $B$	$y=\sin(Bx)$	Period
   1	$y=\sin(x)$	$2\pi$
   2	$y=\sin(2x)$	$\pi$
   4	$y=\sin(4x)$	$\pi/2$
   $1/2$	$y=\sin(\frac12x)$	$4\pi$
   -2	$y=\sin(-2x)$	$\pi$
   -4	$y=\sin(-4x)$	$\pi/2$
   $-1/2$	$y=\sin(-\frac12x)$	$4\pi$
   $B$	$y=\sin(Bx)$	$2\pi/|B|$