Transformations to Sinusoidal Functions
Previously we have studied transformations from y = f(x) to y = a f[b(x – c)] + d . These
transformations are also applied to the sinusoidal functions, y = sin x and y = cos x . The
transformations that parameters a, b, c, d represent are still the same, however in addition to noting
changes in the range, domain, x- and y- intercepts, we also need to address the changes to the
properties specific to sinusoidal functions – period, amplitude, maximum and minimum values, and
where the maximum and minimum values occur.
Notes: For the basic sinusoidal functions, a =1, b =1, c =0, d = 0. That is, no horizontal or vertical
stretch factor and no horizontal or vertical translation have been applied.
The software program uses the general form y = a f[bx + c] + d ; to avoid issues at the
moment, when c is not zero, b will be set to equal 1. Also, realize if we want to show a
horizontal translation of 6 right, for example, then the value of c must be entered as -6.
You will be working from two perspectives today. You will use a demo version (it will interrupt
every once in a while to remind you of this) of a software program called zap-a-graph to:
* enter a basic sinusoidal function and then have transformations applied to that function,
after which you will look at the effect on the period, amplitude, max/min values
 enter a sinusoidal function for given transformations by changing the parameters in the
function, and then seeing if the resulting graph shows those transformations
The program itself is very easy to use. Log on using your user name and password. Click on
My Computer, then Vol 1 on esbchs-server, open up the ICT Data folder, then the Zap Demo folder,
lastly click on zap-a-graph.exe (or the shortcut).
A grid and a menu across the top of the screen will appear. First, go to the menu choice GRID,
click on it and go down to All of the Above. For major x … insert shift p / 2 (shift p equals pi),
for major y… insert 1, for pixels insert 24, for minor x insert 2 (this will scale the x axis in
increments of quarter pi), and for minor y insert 2 (this will scale the y axis in increments of 0.5).
These settings should be appropriate for most of your work today, but feel free to adjust them as
you go along.
Next, you need to go the menu choice DEFINE. Here you will be working either with:
Sine > y =a sin (bx + c) + d
or Cosine > y =a cos (bx + c) + d
To graph the basic sin x or cos x function, use a = 1, b = 1, c = 0, d = 0. Click on PLOT, the graph
will be drawn. You may then choose TRANSFORM from the menu, and go to the appropriate
type of transformation. After the transformation has been drawn, at the bottom of the screen will be
the new equation – check that the parameter that has changed in the equation is the one that you had
anticipated. You can analyze the properties from the graph or you can also take a look at the
OPTIONS menu and select various items (table of values etc.) from there.
The ERASE menu choice allows you to get rid of the lst graph drawn, or the last graph drawn or all
of the graphs on the screen.
To graph sinusoidal functions with transformations applied, simply change the value(s) of a, b, c, d
as required. Once the graph has been plotted, make sure that the transformation that was described
is indeed shown by the graph that results from the equation you entered.
Log on and start up the program zap-a-graph using the instructions on page 1. Set up the grid as
described in the instructions and then complete the following:
Part I
1. Plot the graph of y = sin x . Go to the transform menu and apply a vertical stretch by a factor
of 3. Record the new equation and compare and contrast the properties of the two graphs and
then erase all.
2. Plot the graph of y = cos x . Apply a vertical stretch by a factor of ½ . Record the equation and
then compare and contrast the properties of the two graphs. Erase all.
3. Plot the graph of y = sin x . Apply a reflection in the x-axis. Compare and contrast the
properties of the two graphs.
4. Plot the graph of y = cos x and its reflection in the x-axis. Compare and contrast….
5. State how the value of the parameter a affects the properties of the basic functions y = sin x
and y = cos x. What could you state that the absolute value of a represents in terms of the
graphs of the basic sinusoidal functions?
Part II
1. Plot the graph of y = sin x . Go to the transform menu and apply a horizontal stretch by a factor
of 2. Record the new equation and compare and contrast the properties of the two graphs and
then erase all.
2. Plot the graph of y = cos x . Apply a horizontal stretch by a factor of ½ . Record the equation
and then compare and contrast the properties of the two graphs. Erase all.
3. Plot the graph of y = sin x . Apply a reflection in the y-axis. Compare and contrast the
properties of the two graphs.
4. Plot the graph of y = cos x and its reflection in the y-axis. Compare and contrast….
5. State how the value of the parameter b affects the properties of the basic functions y = sin x
and y = cos x. What operation between the original period of 2 rads for y = sin x and
y = cos x and the value of b would result in the new period of the transformed graph?
Part III
1. Plot the graph of y = sin x . Go to the transform menu and apply a horizontal translation of

units right (note the units here are radians). Record the new equation and compare and
4
contrast the properties of the two graphs. Erase the current graph and apply a horizontal
translation of  rads left. Record the equation. What other equation could be used to represent
the graph of this transformed function?

units right. Record the
2
equation and then compare and contrast the properties of the two graphs. The graph of
the transformed equation could also be represented by another function – what is that function.
Erase all.
2. Plot the graph of y = cos x . Apply a horizontal translation of
3. State how the value of the parameter c affects the properties of the basic functions y = sin x and
y = cos x .
Part IV
1. Plot the graph of y = sin x . Go to the transform menu and apply a vertical translation of 2 units
up. Record the new equation and compare and contrast the properties of the two graphs and
then erase all.
2. Plot the graph of y = cos x . Apply a vertical translation of 1 unit down. Record the equation
and then compare and contrast the properties of the two graphs. Erase all.
3. State how the value of the parameter d affects the properties of the basic functions y = sin x and
y = cos x . What would the value of d locate on the graph of either sinusoidal function?
Part V
1. For each of the transformations described below, write an equation and then enter that
with appropriate values for a, b, c, d . Plot the graph to verify whether or not your
equation is correct.
a. the graph of y = sin x undergoes a vertical stretch with respect to the x-axis by a factor
of 2 and a horizontal stretch with respect to the y-axis by a factor of 2
b. the graph of y = cos x undergoes a vertical stretch with respect to the x-axis by a factor
of 3 and a vertical translation of 2 units down
c. the graph of y = sin x undergoes a horizontal translation of

rad right
3
2. For each of the following, predict the amplitude, period and range. Then either apply the
transformations using the transform menu choice (after you have entered the basic equation) or
write an equation that would represent the function. Plot and check your predictions.
a. the graph of y = sin x undergoes a vertical stretch with respect to the x-axis by a factor of 3
amplitude:
period:
range:
b. the graph of y = cos x undergoes a horizontal stretch with respect to the y-axis by a factor
of 2 and a vertical translation of 2 units up
amplitude:
period:
range:
c. the graph of y = sin x undergoes a vertical stretch with respect to the x-axis by a factor of 2
and a vertical translation of 1 unit down
amplitude:
period:
range:
3. For each equation, describe the transformation then state the amplitude, period and range for the
graph of the function. Check using zap-a-graph.
a.
f ( x)  2.5 cos( x   )
b. g ( x)  sin 4 x
c. h( x) 
1 1 
sin  x   2
2 2 