Graphs of Sine and Cosine

By Christine Sowa, Jennifer Walker, Lindsay Krakower, Kathy Teal, and Liz Butkus

oddcast.com



The general form for all sin functions follows the format:
y= A sin (BX-C) + K

In this formula-

A is the amplitude of 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 of A, the A value is always a positive number. The A value represents how much the graph deviates from the line of equilibrium. However, if the A value 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:
y=3sinx.jpg

B changes the period of the graph. To find the period of the graph, use the formula 2π/B; therefore, the larger the B value, the smaller the period of the graph will be.
A sin graph with an altered period looks like this:
y=sin(2x).jpg

C determines the phase shift of the graph. To find the phase shift of the graph, use the expression C/B. If C is 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:
y=sin(x

K is the vertical shift of the graph. The graph is shifted up (if K is added to the equation) or down (if K 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:
y=sinx+1.jpg
A basic sin graph looks like this:
y= sin x
y=sinx.jpg
The amplitude is one because the A value is one.

The period of the graph is 2π because B is equal to one.

There is no phase shift because there is no C value in the equation.
The graph is not moved up or down because there is no K value.

Notice the following:
  • The domain goes from negative infinity to positive infinity.
  • The range goes from negative 1 to positive 1.
  • The x-intercepts are Kπ where K is an integer (KEI).
  • The y-intercept is at 0.




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.

external image Simple_harmonic_oscillator.gif
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.
= 10 -->; 2π=10B -->; B= π/5
B 1

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

To better understand the origin of sine graphs, study the following image:
external image learn3.gif
_

The general form for all cosine functions follows the format:
y= A cos (BX-C) + K

In this formula-
A is the amplitude of 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 of A, the A value is always a positive number. The A value represents how much the graph deviates from the line of equilibrium. However, if the A value 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:
y=cos(2x).jpg

B changes the period of the graph. To find the period of the graph, use the formula 2π/B; therefore, the larger the B value, the smaller the period of the graph will be.
A cos graph with an altered period looks like this:
y=cos(2x).jpg

C determines the phase shift of the graph. To find the phase shift of the graph, use the expression C/B. If C is 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.)

K is the vertical shift of the graph. The graph is shifted up (if K is added to the equation) or down (if K 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

external image image030.jpgy=cosx.jpg

The amplitude is one because the A value is one.
The period of the graph is 2π because B is equal to one.
There is no phase shift because there is no C value in the equation.
The graph is not moved up or down because there is no K value.

Notice the following:
  • The domain goes from negative infinity to positive infinity.
  • The range goes from negative 1 to positive 1.
  • The x-intercepts are (2K+1)π/2 where KEI.
  • The y-intercept is at 1.

Create Your Own Sine or Cosine Graph!!!


Terms and Definitions
sine 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 Notes
the 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 Program
program 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 Problem
find 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
graph.jpg

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 Problem
Find 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
cos_graph.jpg

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)