y = cos x

To examine the graph of y = cos x, I will examine y = A cos (Bx +C) for different values of A, B, and C. This will allow me to make a generalization for the values of A, B, and C and thus will know how to graph a function of y = cos x quickly.

Let's us first look at the graph y = cos x. This is were A and B equal 1 and C equals 0. This is the graph that we will compare other graphs to.

y = cos x |

y = 2cos x |

y = 3cos x |

y = -cos x |

y = -2cos x |

y = -3cos x |

We can see from the graph above what the differences are when we change
the value of "A" in y =A cosx. Notice that the magnitude of the
curves is what is affected. This is called the __amplitude__ of the curve.
So A affects amplitude. To figure out the amplitude of a curve we can use
this easy formula.

Now that we know what the magnitude of the amplitude is, we need to decide if the sign of A is of any importance. Notice that the last three graphs are negative values for A. Does this make a difference? We can see that they reflect the graphs with the same numerical values (only positive) about the x-axis.

y = cos x |
y = cos ( -x) |

y = cos 2x |
y = cos ( -2x) |

y = cos 3x |
y = cos ( -3x) |

It appears that B affects the period of the curve. To see if this is true, lets graph some curves where the value of B is less than zero.

y = cos x |

y
= cos ( 1/2 x) |

y
= cos ( 1/3 x) |

We can see that in fact, B does affect the period of the curve. It takes 1/B times to complete a period of a curve. If B is equal to 1, then it takes 2pi to complete a period. If B is equal to 2, then it takes only pi to complete a period. If B is equal to 1/2, then it takes only 4pi to complete a period, twice as long as a normal period. Once again we also see that a negative only reflects the curve about the x-axis.

y = cos x |

y = cos (x + 1) |

y = cos (x + 2) |

y = cos (x - 1) |

y = cos (x - 2) |

Automatically we can see that the actual picture of the graph does not change, it only shifts. If C is positive it shifts to the right, if C is negative it shifts to the left. Thus C affects the horizontal displacement (or shift) of the graph.

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assignment one |
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