**GRAPHING TRIGONOMETRIC FUNCTIONS**

**GRAPHING TRIG FUNCTIONS**

We will review:

(1) equations of trig functions and their different parts.

(2)how different parts change or transform the graph.

(3)the basic graphs: y=sin(x), y=cos(x), y=tan(x),

y=cot(x), y=sec(x), and y=csc(x).

(4)going from equations to graphs and vice versa

and how to graph.

(5)word problems

(6) We will learn about Inverse graphs more in depth.

**The General Equation**

**y =A sinB(X+C)+D**

phase

shift

period

2pi/B

IAI =

amplitude

vertical

shift

http://www.regentsprep.org/Regents/math/algtrig/ATT7/sinuso39.gi f

Find the sinusodial axis (middle height of graph).

Find the highest or lowest point of the graph (0,2) or (0,-2) .

This distance fromt these points to sinusodial axis is the amplitude (2 in this case).

Red

arrows show amplitude.

Amp is always positive because it's a distance.

-Period: Take take 2pi and divide B by it.

http://images.tutorcircle.com/cms/images/tcimages/sinegraph.JPG

-On a graph where B is not given, the period is the "tip-to-tip" distance on a graph. In this case, (5pi/2 - pi/2= 2pi.) To find B on the equation, do 2pi/B= 2pi. B=1.

period

3pi/2

pi/2

5pi/2

Phase Shift (c)

This is a graph of y=sin(x). To create a phase shift, c is added to x.

Phase shift moves the graph left or right.

http://math.ucsd.edu/~wgarner/math4c/textbook/chapter5/graph_trig_func_files/image003.gi f

y=sin(x)

<-- Since pi/4 is subtracted from

x, the graph moves

right

.

(x+pi/4)

added to X

means graph moves left

Vertical Shift (d)

http://dwb4.unl.edu/chem/chem869m/chem869mmats/C07MatsFigs/Fig16.jp g

-Vertical shift moves a graph up or down.

-If a positive shift, (+2 in this case) is added to sin(x), then the graph moves

up

two units.

If the graph had a (-2) instead of a (+2), the graph would have

gone down

two units.

vertical shift

Going from Equation to Graph

**GRAPH: y= 2sin(X+ (pi/2)) +5**

Doesn't have to be sine; this could be cosine , tan, sec etc.

BASIC GRAPHS OF TRIG FUNCTIONS

To graph these basic functions, plugging points is the best way.

http://ramanujan.math.trinity.edu/rdaileda/teach/m1312f08/invtrig/5.jp g

http://math.ucsd.edu/~wgarner/math4c/textbook/chapter5/graph_trig_func_files/image003.gi f

y= tan (X)

-these dashed lines represent aymptotes. Since tan(X)= sin(X)/cos(x), cos(x) cannot be zero.

-In places where cosine does end up being zero (x= 90, 270), the function is undefined.

y= cot(X)

sin(X) cannot be zero and hence, the asymptotes are drawn where sin(X) = 0

y=sin (X)

y= cos (X)

y= sec (X)

cos(x) cannot equal zero.

y= csc (X)

sin(X) cannot equal zero.

Graphing cosine

Word Problems

All around us, there are examples of sinusoidal functions and the graphs and equation for sinusoidal functions can also be used with real-world applications such as word problems.

-same method as sine graph.

GRAPH: y=3cos 2(X-(pi/2))+2

(1)&(2) Label and list key details:

3=amplitude,

y=2 sinusodial axis,

2=vertical shift,

period= 2pi/2= pi,

(X-(pi/2))= phase shift

Steps to solving word problems:

1) Convert word problem to general equation

2) Graph general equation

(3) Graph

Word Problem Example

The average depth of water at the end of a dock is 6 feet with an 8 feet high tide at 4AM. If the tide goes from low to high every 6 hours, find the sinusoidal function describing the depth of the water corresponding with time.

1) First, we choose to represent this word problem as either sine or cosine. In this case, we can choose cosine.

2) We need to find the value of a, b, c, and d from the general equation. We know d, the sinusoidal axis right away is 6 feet.

Word Problem Example (cont.)

Tangent and Cotangent

Same procedure again except

remember to include:

-aymptotes

-in the equation of tangent and cotangent, B must be divided by pi, NOT 2pi to get period.

y =A tan B(X+C)+D

3) We also know that A is 2 feet because the variation between the sinusoidal axis and the high tide is 2.

4) B and C are harder to find. However, we know that the high point of the cosine function is at 4AM so C must be adjusted four hours to the right.

5) The tide goes from high to low in 6 hours so one radian must be equivalent to 6 hours. B essentially must be pi/6 because the period is 6 hours.

divide by pi not 2pi.

When a value is subtracted from X, graph goes right and when something is added, it moves to the left.

sinusodial axis

Once you find the point on the sinusodial axis (by plugging in values), you can "eyeball" the graph going from left to right (dont touch the asymptote).

TANGENT GRAPH

COTANGENT GRAPH- goes from right to left

GRAPHING COSECANT

-Secant and cosecant also have asymptotes because their denominators cannot be 0.

The red graph

is y=sin(x)+1. When sine is graphed, the tips of the sine graphs also mark the beginning of the cosecant graph (in green).

GRAPHING SECANT

Word Problem Example (Cont.)

We now have all the components needed for the equation!! Replace the variables with the numbers and we get: y= 2 cos pi/6 (x-4) + 6

Test Tip: If you have time, plug in a few numbers for x to see if it makes sense and works

understand the relations in graphs between each function

know how to convert into the standard equation form

use substitution to your advantage when checking work

be wary of negative signs that can dramatically change a graph

Study Tips (graphing)

Inverse Functions

The below are the inverse functions. Important things to keep in mind are that csc^-1, sec^-1, cot^-1, and tan^-1 all have asymptotes in the x-axis direction whil cos^-1and sin^-1 do not have asymptotes.

-they find the angle behind each ratio-

General Tips

Always check your work (since you will have your calculator, doesn't hurt to check basic trig problems on it.)

Time management is key. SKIP what you don't understand and come back to it.

Take it step by step- don't rush too much.

PRACTICE PROBLEMS

(1) Label all parts of this equation: y= 4cot 6(X-pi/3) + 2.

(2) Graph: y= 4 cos 3X (10)

(3) Graph: y=(1/2)secX

(4) Graph: y=2 csc 2X

(5)Graph: y=sin3X

(6)Graph: (1/2) tan(X/3)

(7)Graph: 2cot2X

(8)&(9) Write an equation of these graphs (zoom in):

(2)

(3)

(4)

(5)

(6)

(7)

(1) y= 4cot 6(X-pi/3) + 2 , 4=amp, pi/6= period, (X-pi/3)

phase shift, 2 vertical shift, y=2 sinusodial axis.

file:///C:/Users/Sami/Downloads/writing_equations_from_graphs.pdf

(8) y =10cos(0.5X)

(9) y= 2sin(x-pi/4)

(10)y= -2cos(3piX)

Inverse functions

Inverse functions are the same as the regular function, but the x and y coordinates are switched

in addition, generally a half cycle of the y values starting from the minmum x value to the maximum x valueclosest to 0 are used