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Sine and Cosine Graphs
Transcript of Sine and Cosine Graphs
GRAPH? WHAT ABOUT COSINE GRAPHS? This! Here is shown one cycle of a sine curve.
What does one cycle consist of, you may ask? Many things! To make this simpler, we will only focus on one cycle (two cycles are shown on the graph to indicate that the basic sine curve repeats indefinitely both negatively and positively). First to know is that the maximum value of the sine curve is 1, and the minimum value is -1 (See the yellow highlighted points on the graph). Involved with these points is the amplitude, which represents half the distance between the maximum and the minimum values of the function. This means that the amplitude of the basic sine curve is 1. Each function also has a period of 2 pi, meaning it takes a period of 2 pi to fully complete the cycle. The two highlighted points on the graph are the beginning and end of this function's cycle. The cosine function is similar to the sine function in more ways than one. Like the basic sine cycle, the basic cosine cycle also has an amplitude of 1, meaning it has a maximum of 1, and a minimum of -1 (See the highlighted points on the curve) The basic cosine curve also has a period of 2 pi per cycle, as shown in the graph above. SINE AND COSINE GRAPHS! ALEXX SMITH, JUSTIN CORBITT, AND WILL SOPER d Many things! Focus on the sine cycle that is on the positive side of the x axis (the cycle on the right just indicates that the basic sine curve repeats indefinitely both positively and negatively). In the basic sine function, the maximum value is always 1 and the minimum value is always -1, which indicates the graph's amplitude (See the hightlighted points). The amplitude represents half the distance between the maximum and minimum values of the function. WHAT IS A COSINE GRAPH? Many things! Focus on the complete sine cycle on the right side of the x and y axis (the cycle on the left just indicates that the curve repeats indefinitely). The maximum of a basic sine curve is always 1 and the minimum of a basic sine curve is always -1, which relates to the amplitude (See highlighted points). The amplitude represents half the distance between the maximum and the minimum values of the function. Because the distance between the maximum and minimum is 2, half of that is 1, making the amplitude of a basic sine curve always 1. QUICK! what have you learned so far? The basic sine curve's amplitude is 1. The basic sine curve's period is 2 pi. The basic cosine curve's amplitude is ALSO 1. The basic cosine curve's period is ALSO 2 pi. keep in mind, to draw the sine and cosine functions by hand, noting five key points in one period of each graph is helpful. These would be; MAXIMUM INTERCEPT INTERCEPT INTERCEPT MINIMUM MINIMUM INTERCEPT INTERCEPT MAXIMUM MAXIMUM NOW TRANSLATIONS OF SINE
AND COSINE CURVES! remember this formula:
y=asin(bx-c)+d (sine curve)
y=acos(bx-c)+d (cosine curve) a: the amplitude. when this is a number other than 1, the curve can shrink vertically (for example with an amplitude of 1/2) or stretch vertically (for example with an amplitude of 3). If you are given an amplitude, that number replaces the a.
-a: if the amplitude is negative, this means the function is reflecting over the x axis.
b: the period. b is replaced by the amount of cycles completed in the given period. For example, because one cycle is completed at 2pi, a 1 would replace the b. If you were given a period of 4pi, a 2 would replace the b, because two cycles have been completed.
c: a horizontal shift, which is not new material. if the translation has moved the function left or right on the graph, the amount of units it shifted over relaces the c. keep in mind that the x-axis counts by every pi/6.
d: a vertical shift, which is also not new material. if the translation has movede the function up or down on the graph, the amount of units it has shifted up or down relaces the d.