# Transformations of Trig Functions

The Transformations of Trig Functions section covers:

We learned how to transform Basic Parent Functions here in the Parent Functions and Transformations section.
Now we will transform the six Trigonometric Functions.  Later we’ll be transforming the Inverse Trig Functions here.

Some prefer to do all the transformations with t-charts like we did earlier, and some prefer it without t-charts (see here); most of the examples will show t-charts.

# T-Charts for the Six Trigonometric Functions

Here are the trig parent function t-charts I like to use.   Note that each covers one period (one complete cycle of the graph before it starts repeating itself) for each function.  Also note that “undef” means the function is undefined for that x value; there is a vertical asymptote there. Note also that when the original functions (like sin, cos, and tan) have 0’s as y values, their respective reciprocal functions are undefined at those points (because of division of 0).

# Sin and Cos Transformations

Here’s one example of a general formula in order to transform a sin or cos function.  (I also include the remaining four trig functions in the chart).   Note that sometimes you’ll see the formula arranged differently; for example, with “a” being the vertical shift at the beginning.

## Drawing Transformed Graphs for Sin and Cos

Here are some examples, first with the sin function, and then the cos (the rest of the trig functions will be addressed later).  You will probably be asked to sketch one complete cycle for each graph, label significant points, and list the Domain, Range, Period and Amplitude for each graph.

## Transforming Without Using t-charts (more here)

Since we get can get the new period of the graph (how long it goes before repeating itself), by using  $$\frac{2\pi }{b}$$, and we know how to shift the graphs horizontally and vertically (including flips), some prefer not to use t-charts to draw these.   To do this, just graph the outside points using the new period and the shifts, the middle point, and then two more points in between, for a total of the five points.

Let’s use the example  $$y=-2\cos \left( \frac{1}{3}\left( x-\frac{\pi }{2} \right) \right)+1$$  that we did above, without using a t-chart.

We see that the graph is centered at y = 1, because of the vertical shift.  From here we go up 2 and down 2 so the y part of the graph goes from  –1  to  3.

The first (leftmost) point is  $$\frac{\pi }{2}$$  to the right, and it is “upside down” for cosine (because of the negative sign before the 2), so it will be at  $$\left( \frac{\pi }{2},-1 \right)$$.

Since the period is 6π, the middle point is 3π  farther than the start at  $$\frac{\pi }{2}$$, so the next (highest) point is at  $$\left( \frac{\pi }{2}+3\pi ,\,\,3 \right)=\left( \frac{7\pi }{2},\,\,3 \right)$$.   The last (rightmost) point is again at the lowest y point at  $$\left( \frac{7\pi }{2}+3\pi ,\,\,-1 \right)=\left( \frac{13\pi }{2},\,\,-1 \right)$$.

To get the point between  $$\left( \frac{\pi }{2},-1 \right)$$  and  $$\left( \frac{7\pi }{2},\,\,3 \right)$$, we take the average of the x values and the y again to get  $$\left( 2\pi ,\,\,1 \right)$$.  We do the same thing for the point between  $$\left( \frac{7\pi }{2},\,\,3 \right)$$  and  $$\left( \frac{13\pi }{2},\,\,-1 \right)$$  to get  $$\left( 5\pi ,\,\,1 \right)$$.   Now we have the 5 points and we can draw the graph!

## Writing Equations from Transformed Graphs for Sin and Cos

You may be asked to write trig function equations, given transformed graphs.  Here are the steps to do this; examples will follow:

1. Write out the generic transformed Equation for sin or cos:  $$y=a\sin b\left( x-c \right)+d$$  or  $$y=a\cos b\left( x-c \right)+d$$.  To determine if it is a sin graph or a cos graph, see if the graph seems to peak at the y axis (cos graph), or have the (horizontal) middle closest to the y axis (sin graph).   Note that the graph may appear flipped across the x axis, but, by shifting the graph to the left or right, you may not have to write the equation as a flipped function (with a negative).
2. Find the amplitude (a) of the transformed function by subtracting the bottom y value from the top y value, and then dividing by 2.  (Remember that for the csc, sec, tan, and cot graphs, this is just called a “stretch”.)
3. To get d, or the vertical shift of the function, we add the amplitude to the bottom y value.   Notice that this is the y value for the middle of the function (the “average” of the bottom y and the top y); we also can just observe this from the graph.
4. To get b:   First find the period of the graph: see how long it goes before repeating itself, then subtract the two x values to get this new period.   For example, if it “starts” at π and then starts repeating itself again at 5π (a complete revolution), its period is 4π.   Once you get this period,   $$b=\frac{2\pi }{\text{new period}}$$ .    In our example, b would be   $$\frac{2\pi }{4\pi }=\frac{1}{2}$$.   (Remember that for the tan and cot graphs,  $$b=\frac{\pi }{\text{new period}}$$).
5. Find the phase shift of the graph by seeing how close it is to the y axis (x = 0); this value is  c.   You can actually write the graph as a cos graph or a sin graph; both will be correct, depending on what c is, but you typically want c to be the smallest it can be.   Look at the x value of the y high point for cos (since cos typically starts at y = 1), or middle point of graph for sin (since sin typically starts at y = 0), and compare to x = 0.  Since the formula has “xc” in it, if the graph is to the right of where it should be, c is positive; if it is to the left of where it should be, c is negative.
6. Check your equation in the graphing calculator by setting the x window exactly to the leftmost point of the graph (Xmin) and to the rightmost point of the graph (Xmax).   Similarly, set the y window to the lowest point on the graph (Ymin) and to the highest point on the graph (Ymax).   When you graph, you should see the exact graph for that problem.

Here are some examples:

# Sinusoidal Applications

Uh oh – more word problems!   These aren’t too bad, once you get the hang of them.  And now that you know how to transform sin and cos functions, that’s really all we’re doing here.

A sinusoidal function, or sinusoid is a fancy name for the sin (or cos) waves that  we’ve been working with.  Sinusoids are quite useful in many scientific fields; sine waves are everywhere!

With sinusoidal applications, you’ll typically have to decide between using a sin graph or a cos graph.  Sometimes it helps to remember that the sin graphs start in the middle of the graph, and the cos graphs start at the top of the graph.   Also, sometimes, the graphs will be “upside down” which means you might need to reflect the sin or cos (using a negative coefficient).

Let’s just start with an example, and see the steps:

Roller Coaster Problem:

A part of the track of a roller coaster has the shape of a sinusoidal function.

The highest and lowest points on the roller coaster are 150 feet apart horizontally and 100 feet apart vertically.    The lowest point of the roller coaster was actually built 10 feet below the ground.

Let y = the height of the track (with y = 0 as the ground), and x = the number of feet horizontally, with x = 0 at the highest point of the track.

###### (a)   Write the sinusoidal equation of this part of the track of the roller coaster.(b)   How high is the highest point of the roller coaster?   How high is the coaster at x =15 feet?  100 feet?(c)   How long (horizontally) is the roller coaster when the track is 75 feet above the ground?  15 feet above the ground?

Solution:

(a)   Let’s get the equation with these steps:

1. Graph the function’s high points and low points, and figure out the coordinates of these points, given what we know.  Since the track is 100 feet tall, and the lowest point is at y = –10, we know the highest point is at (0,  100 – 10) = (0,  90)  (since the highest point is when x = 0).  Since the horizontal distance between the highest point and lowest point is 150, the lowest point is at (150, –10).So here’s what we can graph:We’ll use a cos graph, since the highest point of the graph is on the y axis:   $$y=a\cos b\left( x-c \right)+d$$.
2. To get the amplitude, or a, we’ll subtract the lowest y point from the highest, and then divide by 2.  So  a  =   $$90-\left( -10 \right)=100\div 2=50$$.   So now we have  $$y=50\cos b\left( x-c \right)+d$$.
3. To get the middle of the function, or the vertical shift (d),  we add the amplitude to the lowest y value:  $$-10+50=40$$.   (We could also have just taken the average of 90 and –10).  So now we have  $$y=50\cos b\left( x-c \right)+40$$.
4. To get the period of the graph, we know that the horizontal distance between the highest point and the lowest point is one half of a period.  So the period of the graph is (2)(150) = 300 feet.  Now to get b, we have  $$b=\frac{2\pi }{\text{new period}}=\frac{2\pi }{300}=\frac{\pi }{150}$$.  So now we have  $$y=50\cos \frac{\pi }{150}\left( x-c \right)+40$$.
5. Since the highest point is on the y axis (= 0), there is no horizontal phase shift.  Thus the sinusoidal function is  $$y=50\cos \frac{\pi }{150}x+40$$.  We could put it in a graphing calculator to check it.

(b)   At the highest point, the roller coaster is 90 feet above the ground ($$y=50\cos \left( \frac{\pi }{150}*0 \right)+40=90$$) .   At x = 15 feet, the roller coaster is   $$y=50\cos \left( \frac{\pi }{150}*15 \right)+40$$, or 55  feet tall.  At x = 100 feet, the roller coaster is  $$y=50\cos \left( \frac{\pi }{150}*100 \right)+40$$, or 15 feet tall.   It’s easiest to put the function  in the graphing calculator, and use the 2nd TRACE (CALC) value function to get these values (see the WINDOW you can use below).

(c)   When the track is 75 feet above the ground, y = 75. The easiest way to get the x at that point is to use the Intersect feature in the graphing calculator:  (2nd Trace (Calc), 5, ENTER, ENTER, ENTER).   So when the track is 75 feet above the ground, the roller coaster is about 37.98 feet from the highest point at x = 0. Note the window I used to match the graph of the roller coaster.

You can use the same steps to see that when the roller coaster track is 15 feet above the ground, the roller coaster is 100 feet from the beginning point.

Bouncing Spring Problem:

The weight on a long spring bounces up and down sinusoidally with time.  You are looking at a second hand on a clock and notice that when the clock reads .2 seconds, the weight first reaches a high point that is 50 centimeters (cm) above the ground.  The next low point is at 30 cm above the ground, and this occurs at 1.5 seconds.

###### (d)   What would be the first positive value for the time when the weight is 45 cm above the ground?

Solution:

(a)   Let’s get the equation with these steps:

1. Graph the function’s high points and low points, and figure out the coordinates of these points, given what we know. Since the high point is at 50 cm at .2 sec, we know the high point is at (.2, 50).  We also know that the low point is at (1.5, 30).     So here’s what we can graph:   We’ll use a cos graph again, since the highest point of the graph is on the y axis:  $$y=a\cos b\left( x-c \right)+d$$.
2. To get the amplitude, or a, we’ll subtract the lowest y point from the highest, and then divide by 2. So a  = $$50-\left( 30 \right)=20\div 2=10$$.  So now we have  $$y=10\cos b\left( x-c \right)+d$$.
3. To get the middle of the function, or the vertical shift (d), we add the amplitude to the lowest y value:   $$30+10=40$$.   (We could also just taken the average of 30 and 50).  So now we have  $$y=10\cos b\left( x-c \right)+40$$.
4. To get the period of the graph, we know that the horizontal distance between the highest point and the lowest point is one half of a period. So the period of the graph is  (1.5 – .2) (2) = 2.6 cm.  Now to get b, we have  $$b=\frac{2\pi }{\text{new period}}=\frac{2\pi }{2.6}=\frac{10\pi }{13}$$.  So now we have  $$y=10\cos \frac{10\pi }{13}\left( x-c \right)+40$$.
5. Since the highest point is at x = .2, the horizontal phase shift is  .2 to the right. Thus  c = .2.  Thus the sinusoidal function is $$y=10\cos \frac{10\pi }{13}\left( x-.2 \right)+40$$.   We could put it in a graphing calculator to check it.

(b)  When the clock reads 18 seconds, we can plug in 18 for x to get y (the distance from the ground):   $$y=10\cos \frac{10\pi }{13}\left( 18-.2 \right)+40$$, which is about 45.68 cm.   This is after many bounces, as you could see if you graphed the function and made the window of x larger.

(c)  The approximate distance from the ground when the clock is at t = 0 seconds is  $$y=10\cos \frac{10\pi }{13}\left( 0-.2 \right)+40$$ = 48.85 cm.

(d)   Let’s use the graphing calculator to find the first positive value when the weight is 45 cm above the ground. After you hit GRAPH, you may have to use the TRACE button to get the cursor closer to the first point of intersection before you use intersect.     This value is approximately .63 seconds:

Tidal Problem:

Let’s do one more, where we’ll use a sin function:

A tsunami or tidal wave is an ocean wave caused by an earthquake.  The water first goes down from its normal level and then rises an equal distance above its normal level, and so on.    Let’s say the amplitude for this particular tsunami is 12 meters, it’s period is about 20 minutes, and it’s normal depth is 10 meters.    Assuming the depth of the water varies sinusoidally with time, find the sinusoidal function for this tsunami.

Solution:

Let’s first draw a graph, assuming that the water is at its regular depth when t = 0, and then goes down to its lowest point, and then up to its highest point.  (Notice that the water technically goes below the surface of the ocean; we won’t worry about the scientific consequences of this.)

We know the lowest point is at 5 minutes, and the period is 20 minutes, we can figure out that the highest point is at half the distance of the period (10 minutes) from that lowest point.  So we can plot the following points and draw the graph:

Now for this graph, we will use the sin function since the middle of the function goes through the y axis (x = 0).  But notice how the graph is flipped, so we will use –sin.

We are already given the amplitude (12 meters), vertical shift (normal depth is at 10 meters), and period (20 minutes), so  $$b=\frac{2\pi }{\text{new period}}=\frac{2\pi }{20}=\frac{\pi }{10}$$.   There is no horizontal phase shift, so the sinusoidal function is   $$y=-12\sin \frac{\pi }{10}x+10$$.

# Secant and Cosecant Transformations

The reciprocal functions secant (sec) and cosecant (csc) are transformed the same way as the sin and cos, yet the “a” part of the transformation is not called an amplitude, but a stretch, as we are used to with “regular” transformed functions.

Remember, again, like the sin and cos transformations, the new period =  $$\frac{2\pi }{b}$$.

If asked for asymptotes of transformed functions,  you’ll perform the same transformations on them as you would the x values of the graph.   You can look at one of the new asymptotes of the transformed graph, and then add  $$\left( {\frac{{\text{new period}}}{2}} \right)k$$,  since there are two asymptotes per period for the csc and sec graphs.

We have the parent graph t-charts from above, so let’s go right to examples.

# Tangent and Cotangent Transformations

The reciprocal functions tangent (tan) and cotangent (cot) are transformed the same way as the csc and sec, (with “a” part of the transformation a stretch and not an amplitude).  The difference however is that, since the period of the tan and cot functions (how long the graph goes before repeating itself) is π instead of , we have  new period =  $$\frac{\pi }{b}$$.

If asked for asymptotes of transformed functions,  you’ll perform the same transformations on them as you would the x values of the graph.   You can look at one of the new asymptotes of the transformed graph, and then add (new period)k,  since there is one asymptote per period for the tan and cot graphs.

We have the parent graph t-charts from above, so let’s go right to examples.

# Transformation of all Trig Functions without T-Charts

Let’s go over once again how to transform trig functions without t-charts.  Note that in order to perform the transformations accurately and quickly, you must know your 6 trig functions graphs inside out!

Here are the steps for sin and cos graphs:

1. Put the trig function in the  $$y=a\sin b\left( {x-c} \right)+d\,\,\,\text{or}\,\,\,y=a\cos b\left( {x-c} \right)+d$$  format.
2. The new domain will still be  $$\left( {-\infty ,\,\infty } \right)$$.
3. The new range will be  $$\left[ {\text{vertical shift (}d\text{)}-\,\,\left| a \right|\,\text{,}\,\,\text{vertical shift (}d\text{)}+\,\,\left| a \right|} \right]$$.    ($$\left| a \right|$$   is amplitude)
4. New period will be  $$\frac{{2\pi }}{b}$$.
5. Phase shift will be c; start graph at this point, and if a is negative, make graph upside down.
6. First draw line where vertical shift is, and lines where lower and upper ranges are. Graph will be centered vertically at the vertical shift, go up to the top line, and down to the bottom line.
7. Then start graph (first point) where phase shift starts and count over (to right) an amount that is the new period (make complete revolution or cycle of the sin or cos). Draw the last point here.
8. Draw middle point halfway between; this will be at the halfway mark of the complete cycle. So we’ll have a total of five points.

Here are the steps for tan and cot graphs:

1. Put the trig function in the  $$y=a\tan b\left( {x-c} \right)+d\,\,\,\text{or}\,\,\,y=a\cot b\left( {x-c} \right)+d$$ format.
2. Remember that asymptotes for tan are at  $$\frac{\pi }{2}+\pi k$$  and for cot are  $$\pi k$$  (the c’s (cot and csc) have the “easier” ones).  To get the new asymptotes, set the trig argument to asymptotes, and solve for x.  For example, if we have  $$\tan \left( {3x+\pi } \right)\,\,(\text{which would be}\,\,\tan \left( {3\left( {x+\frac{\pi }{3}} \right)} \right))$$,  we would solve:  $$\displaystyle 3x+\pi =\frac{\pi }{2}+\pi k;\,\,\,\,\,\,3x=\left( {\frac{\pi }{2}-\pi } \right)+\pi k;$$   $$\displaystyle 3x=-\frac{\pi }{2}+\pi k;\,\,\,\,x=\frac{{-\frac{\pi }{2}}}{3}+\frac{{\pi k}}{3};\,\,\,\,x=-\frac{\pi }{6}+\frac{{\pi k}}{3}$$.
3. The new range will still be  $$\left( {-\infty ,\,\infty } \right)$$.
4. Draw asymptotes first; phase shift will take care of itself. (To get an asymptote starting point, you can set k = 0 in your new asymptote equation, k = –1 for one to the left, k = 1 for one to the right).  Right in between the asymptotes (you can take average of the x’s), draw the middle of the graph (but make sure it is shifted up or down according to the vertical shift, or d).
5. New period will be  $$\frac{\pi }{b}$$  (since tan and cot have periods of ).   This should be the distance between asymptotes.
6. If trig function is negative, make sure you flip the graph.
7. New domain will be all x values except for the asymptotes.
8. To get two more points on either side of the center point, take the center or average again of the middle point of the asymptotes for the x’s. To get the y value for these points (since these are normally at y = 1 and y = –1), shift the y according to the shifted function (for example, if the function is  $$3\tan \left( {x+\pi } \right)-4$$,  the y’s would be 3(1) – 4 = –1 for one of the points, and 3(–1) – 4 = –7 for the other one.

Here are the steps for csc and sec graphs:

1. Put the trig function in the  $$y=a\csc b\left( {x-c} \right)+d\,\,\,\text{or}\,\,\,y=a\sec b\left( {x-c} \right)+d$$  format.
2. Remember that asymptotes for sec are at  $$\frac{\pi }{2}+\pi k$$  and asymptotes for csc are  $$\pi k$$  (the c’s (cot and csc) have the “easier” ones).  To get the new asymptotes, set the trig argument to asymptotes, and solve for x.  For example, if we have  $$\csc \left( {3x-\frac{\pi }{6}} \right)$$,  we would solve:  $$3x-\frac{\pi }{6}=\pi k;\,\,\,3x=\frac{\pi }{6}+\pi k;\,\,\,x=\frac{\pi }{{18}}+\frac{{\pi k}}{3}=\frac{\pi }{{18}}+\frac{{6\pi k}}{{18}}$$
3. The new range will be  $$\left( {-\infty ,\,\,\text{vertical shift (d)}-\,\,\left| a \right|} \right]\cup \left[ {\text{vertical shift (d)}+\,\,\left| a \right|,\,\,\infty } \right)$$. ($$\left| a \right|$$   is amplitude)
4. Draw asymptotes first; the “cups” will fall into these areas.  (To get an asymptote starting point, you can set k = 0 in your new asymptote equation, k = –1 for one to the left, k = 1 for one to the right).  The center of the “cups” will fall halfway between the asymptotes.
5. New period will be  $$\frac{{2\pi }}{b}$$  (since csc and sec have periods of  $$2\pi$$).   (This should be the distance between every other asymptote).
6. To get where graphs start (“cup up” or “cup down”), shift the graph according to the phase shift (c) (using the y axis as the starting point), and draw the graph according to the original trig function (csc or sec) and whether or not it’s negative (if negative, you flip it vertically).
7. New domain will be all x values except for the asymptotes.

It’s a good idea to graph your answers on a graphing calculator (radians) with window of one period (with the Xmin and Xmax) and range (with Ymin and Ymax) to check your graphs.

Here are some examples:

Here’s one more with csc:

Understand these problems, and practice, practice, practice!

Click on Submit (the arrow to the right of the problem) to solve this problem. You can also type in more problems, or click on the 3 dots in the upper right hand corner to drill down for example problems.

If you click on “Tap to view steps”, you will go to the Mathway site, where you can register for the full version (steps included) of the software.  You can even get math worksheets.

You can also go to the Mathway site here, where you can register, or just use the software for free without the detailed solutions.  There is even a Mathway App for your mobile device.  Enjoy!

On to The Inverse Trigonometric Functions – you’re ready!

## 5 thoughts on “Transformations of Trig Functions”

1. I found wonderful, challenging problems with very clear explanations! This helped me soooo much. Thank you!

Instead of using t-charts, though, for trig transformations, I learned to draw guidelines onto my graph depending on amplitude, vertical/horizontal shifts, etc. Then I sketch the graph using my new guidelines instead of the y and x-axes. I find the period using: { angular frequency *period = 2pi } and use the period to number my x axis.

• Thanks so much for writing; comments like these make me want to keep writing! Please spread the word about She Loves Math 😉 Also, if you see any mistakes or have any suggestions, please let me know. OH thanks so much for your suggestion – I’ll put that on the site. Thanks! Lisa

• Thanks so much for your sweet note! Please spread the word about SheLovesMath.com 🙂 Lisa