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## Graphs of Sine and Cosine

By Christine Sowa, Jennifer Walker, Lindsay Krakower, Kathy Teal, and Liz Butkusoddcast.com

The general form for all sin functions follows the format:

y= A sin (BX-C) + KIn this formula-

Ais theamplitudeof the graph, which represents half the distance between the maximum and minimum values of the function. Even if there is a negative sign in front ofA, theAvalue is always a positive number. TheAvalue represents how much the graph deviates from the line of equilibrium. However, if theAvalue is preceded by a negative sign, then the graph is reflected over the x-axis.A sin graph with an altered amplitude looks like this:

Bchanges theperiodof the graph. To find theperiodof the graph, use the formula2π/B; therefore, the larger theBvalue, the smaller the period of the graph will be.A sin graph with an altered period looks like this:

Cdetermines thephase shiftof the graph. To find thephase shiftof the graph, use the expressionC/B. IfCis a positive value, then the graph is shifted to the right; if it is negative, the graph is shifted to the left. (Be careful - since there is already a negative sign in front of C in the original equation, the horizontal shift ends up being in the opposite direction of the C value in the formula. Ex: If the value of C is negative 2, when you put it in the formula, the negatives will cancel out and become a positive 2, causing a positive phase shift to the right as compared to the parent graph.)A sin graph with an phase shift looks like this:

Kis thevertical shiftof the graph. The graph isshifted up(ifK is added to the equation) ordown(ifK is subtracted from the equation) the same number of units as the value of K.A sin graph with a vertical shift looks like this:

A basic sin graph looks like this:

y= sin x

The

amplitudeis one because theAvalue is one.The

periodof the graph is 2π becauseBis equal to one.There is no

phase shiftbecause there is noCvalue in the equation.The graph is not moved up or down because there is no

Kvalue.Notice the following:

Kis an integer (KEI).Example Problem Involving Sin:

A spring and mass system is pulled down from its line of equilibrium 5 inches. It takes 10 seconds for the spring to travel to 5 inches above the line of equilibrium and then back to the line of equilibrium. Make a sin function to represent this model.

First, since the graph starts out going below the x-axis, we know that the amplitude will have a negative sign in front of it. Next, the amplitude is 5 because the spring travels five inches below the x-axis as well as 5 inches above it. The period is ten seconds.

2π=10-->; 2π=10B -->; B= π/5B 1

Therefore, the equation is y= -5 sin πx/5

To better understand the origin of sine graphs, study the following image:

_

The general form for all cosine functions follows the format:

y= A cos (BX-C) + KIn this formula-

Ais theamplitudeof the graph, which represents half the distance between the maximum and minimum values of the function. Even if there is a negative sign in front ofA, theAvalue is always a positive number. TheAvalue represents how much the graph deviates from the line of equilibrium. However, if theAvalue is preceded by a negative sign, then the graph is reflected over the x-axis.A cos graph with an altered amplitude looks like this:

Bchanges theperiodof the graph. To find theperiodof the graph, use the formula2π/B; therefore, the larger theBvalue, the smaller the period of the graph will be.A cos graph with an altered period looks like this:

Cdetermines thephase shiftof the graph. To find thephase shiftof the graph, use the expressionC/B. IfCis a positive value, then the graph is shifted to the right; if it is negative, the graph is shifted to the left. (Be careful - since there is already a negative sign in front of C in the original equation, the horizontal shift ends up being in the opposite direction of the C value in the formula. Ex: If the value of C is negative 2, when you put it in the formula, the negatives will cancel out and become a positive 2, causing a positive phase shift to the right as compared to the parent graph.)Kis thevertical shiftof the graph. The graph isshifted up(ifK is added to the equation) ordown(ifK is subtracted from the equation) the same number of units as the value of K.A basic cos graph looks like this:

y= cos x

The

amplitudeis one because theAvalue is one.The

periodof the graph is 2π becauseBis equal to one.There is no

phase shiftbecause there is noCvalue in the equation.The graph is not moved up or down because there is no

Kvalue.Notice the following:

Create Your Own Sine or Cosine Graph!!!

Terms and Definitionssine curve - the graph of the sine function

one cycle - one period of the function

amplitude (absolute value of A) - half the distance between the maximum and minimum values of the function

period - distance to complete one cyle of the graph

phase shift - horizontal translation from parent graph

vertical translation - vertical shift from parent graph

Helpful Notesthe sine graph is symmetric with respect to the origin ----- sine function is odd

the cosine graph is symmetric with respect to the y-axis ----- cosine function is even

Graphing a Sine Function Programprogram simultaneously draws the unit circle and the corresponding points onthe sine curve

also connects points on unit circle with corresponding points on the sine curve

click program belowand scroll down to Graphing a Sine Function Program:

Graph a Sine Function

Example Sine Problemfind a, b, c, and k for the funtion f(x) = asine(bx-c)+k so the graph of f matches the graph below.

y-axis in increments of 1, x-axis in increments of π/2

Solution: a = 3, b = 1/2, c = 1, k = 2, graph is inverted because of negative sign before "a"

(from the equation f(x) = -3sin(x/2 -1)+2)

Example Cosine ProblemFind a, b, c, and k for the function f(x) = acos(bx-c)+k so the graph of f matches the graph below.

y-axis in increments of 1, x-axis in increments of π /8

Solution: a = 2, b = 4π, c = 2, k = 3, graph is inverted because of negative sign before "a"

(from the equation f(x) = -2cos(4πx-2)+3)