19)+Trigonometric+Graphing

__Given a trigonometric graph, state the equation of the graph __ I decided to explain how to state equations from trigonometric graphs due to the fact that this was one of my weaker subjects throughout the year. Although I found this section to be a little challenging, I enjoyed the fact that I was able to see how a relatively simple graph could be written using numerous different equations. I was able to fully master the skill of writing equations from trigonometric graphs after this project. In order to state the equation of a trigonometric graph, one must first understand what all of the parts of the equation mean. Our most basic example of trigonometric equation will look like this: Y=asin(bx+c)+d  Or  Y=acos(bx+c)+d  Or  Y=atan(bx+c)+d  Or  Y=acsc(bx+c)+d  Or  Y= asec(bx+c)+d  Or  Y=acot(bx+c)+d

•A= Amplitude (maximum-minimum)/2
====•B=B helps us to calculate the period which is found by 2 p /B ====

<span style="color: #3e00ff; font-family: 'Comic Sans MS',cursive; font-size: 110%;">•D=Vertical shift
Now that we have a general equation to work with we can dissect each peace to understand what they mean. We must begin our equation writing by deciding which type of graph is drawn. There are three different sinusoidal functions; therefore there are three different looking graphs. Sine: Any graph that is not translated that is a sine equation goes through the point (0,0). Ex.

Cosine: Any graph that is not translated that is a cosine equation goes through the point (0,1). Ex. Tangent: The tangent graph looks very different than both the cosine and sine graphs. When graphing the tangent function, one must switch the x and y values, which makes the graph vertical and also contain asymptotes Ex.

Cotangent= Any equation with cotangent in it looks very similar to the tangent graph except the asymptotes are translated to the left 90 degrees or if the graph is in radians, p /2. The graph is also changing direction from a steady increase to a steady decrease. Ex.

Cosecant= The graph of cosecant is found from 1/sin and the parent graph has asymptotes at 0 and 180 or 0 and p. The graph itself looks like a parabola. Ex. Secant= Secant is derived from the equation 1/cos and is very similar to csc, however, the asymptotes are at –90 and 90 or - p /2 and p /2. Ex.

Now that we know what each parent graph looks like we can begin writing equations.

A= The first piece of the equation which we can find is the amplitude. The amplitude, in other words, is the maximum of the graph subtracted by the minimum of the graph divided by two. Ex. http://www.intmath.com/trigonometric-graphs/trigo-graph-intro.php

The maximum of this graph is 10 while the minimum is –10. In order to find the amplitude you do the simple calculation: 10-(-10)/2=10  A=10

<span style="color: blue; display: block; font-family: 'comic sans ms'; font-size: 14pt; text-align: center;">Our equation so far would be y=10sinB(x)

B= The next step of writing and equation is to find the B value. In order to find this value we must us the equation:

2 <span style="font-family: Symbol,sans-serif;">p/ <span style="font-family: Arial,Helvetica,sans-serif;">b= period/1  <span style="display: block; font-family: 'Comic Sans MS'; font-size: 130%; line-height: 27px; text-align: center;">or <span style="display: block; font-family: 'Comic Sans MS'; font-size: 130%; line-height: 27px; text-align: center;">360/b=period/1

<span style="display: block; font-family: Symbol,sans-serif; font-size: 130%; text-align: center;">B=2

The period for any graph is the time it takes for it to repeat itself. For most graphs that do not have a stretch or shrink have a period of either 2p or 360 degrees.

= Now that we know how to find the //b// value we can begin to practice with a few different graphs. = = = = Example 1: =



Notice that with this graph the period that it takes for the peak of the cosine graph to repeat itself is p /4. In order to calculate the b value for our equation we must do the simple calculation: = 2 <span style="font-family: Symbol,sans-serif;">p/B=p/4  = =<span style="font-family: Symbol,sans-serif; font-size: 140%;">B=8 =

Our equation for this cosine graph would then be y=3cos(8x)

Example 2: We use the same process as with previous graphs in order to find the equation of this sine graph.

= 2 <span style="font-family: Symbol,sans-serif;">p/B=6p  =

This graph is very similar to the other graphs that we have seen. We can see that the origin of the graph begins at (0,0) therefore we know that it is a sin equation. We now can see that the equation is =Y=4sin1/3x=

We have now covered the basics of writing equations from graphs with either stretches in the period or the amplitude. Now we will learn how to write equations of graphs that have phase shifts. ** Writing phase shifts are very easy. **  ** Example 1 **

http://www.mathamazement.com/images/Pre-Calculus/04_Trigonometric-Functions/04_05_Graphs-of-Sine-and-Cosine/sine-graph.JPG

Take for example this parent cosine graph.
Y=sin(x+ p /2 )
 * Assume that the numbers on the x-axis are in terms of p so the minimum amplitude of the graph would be at (3.14,-1).
 * We can write an equation in as if they graph was a sine graph. The way we do this is through a phase shift. In this case the equation for the sine graph would by
 * == We see that the period is 6.28 because the graph is a parent graph and 2 p = 6.28. By moving the graph to the left by p /2 we are able to get the sine version of the graph. ==

Example 2
==In this example we are able to see the original graph, the dotted line, which has no phase shifts. You can then clearly see the equation that is graphed which is a solid line with a phase shift to the right a distance of p/8. ==  y=2cos(x- p/8) Example 3
 * ==Using the knowledge that we have learned we can write the entire equation for this line which is:==

This graph may appear to be very tricky, however, in order to write the equation if you take each step at a time, the final writing of the equation will be much easier.

 * == To begin we must decide what time of graph it is and what the period of the graph is. For this equation we can conclude that it is a cosine graph because the origin is at (0,1) and we can conclude that the period is p, <span style="font-family: 'Comic Sans MS',cursive; font-size: 12pt;">therefore, the b value in our equation is 2. ==
 * ===The second part of writing out equation is finding the amplitude of the graph. The maximum of the graph is 1 and the minimum is -1, therefore by using the equation: maximum-minimum divided by 2= the amplitude of the equation. One minus negative one equals two, which when divided by two leaves you with 1 as the amplitude.===
 * The final step of writing the equation is to recognize the phase shift.

== The same strategy applies for all types of graphs with phase shifts. The easiest way to master this skill is by first drawing in the parent graph with the change in amplitude and period and then move the graph either left or right depending on the phase shift. ==
 * When writing equations from tangent graphs you use the same process for finding the b value as well as the phase shift.
 * However, in order to find the change in amplitude you must insert two points from the graph into the x and y of the equation. This will allow you to see the change in amplitude.
 * Also, for cosecant and secant graphs, the amplitude is the differences from the maximum and minimum of the two different graphs.

We have covered every part of writing equations from graphs from regular cosine, sine, and tangent graphs. The same process is preformed for the other three graphs, cosecant, secant, and cotangent.

When there is any change from the parent graph you must follow the same simple steps to find the b value, phase shift, and amplitude.
 * The secant and cosecant graphs follow the same rules as the sine and cosine graphs
 * Tangent and Cotangent graphs also follow the same rules when writing equations for the graphs.

We now have completely learned how to write trigonometric equations from a graph! =Class set problem!= =Write the equation for this graph:=

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http://www.purplemath.com/modules/grphtrig3.htm Graphs from: http://www.intmath.com/

Videooo! http://www.youtube.com/watch?v=yOjDutd9e00