G RAPHS OF S INE AND C OSINE FUNCTIONS Objectives: Sketch the graphs of basic sine and cosine functions Use amplitude, period and translations to help sketch the graphs of sine and cosine functions

G RAPHING S INE AND C OSINE F UNCTIONS In this unit, we will take the unit circle introduced in the previous unit, and graph it on the Cartesian plane. To do this, we are going to “unravel” the unit circle. Recall that for the unit circle the coordinates are where is the central angle. To graph, rewrite the coordinates as where is the central angle, in radians. We expanded the sine coordinates for.

S INE CURVE

Notice that the curve ranges from 1 to -1. The maximum value is 1, which is at. The minimum value is -1 at. This “height” of the sine function is called the amplitude. If we had continued the curve, it would repeat. This means that the sine curve is periodic. Look back at the unit circle, the sine value changes until it reaches. Therefore, the curve will repeat every units, making the period. The domain is all real numbers.

C OSINE CURVE Similarly, when we expand the cosine curve, from the unit circle, we have: Notice that the range is also and the domain is also. The period is also

B ASIC S INE AND C OSINE C URVES o Comparing, we see that the curves are almost identical, except that the sine curve starts at and the cosine curve starts at.

C HARACTERISTICS OF S INE AND C OSINE One Cycle: the period of each of these curves is from. The curves repeat indefinitely to the left and right. Domain: Range: Symmetry: Sine curve is symmetric with respect to the origin (odd function) and Cosine curve is symmetric with respect to the y-axis (even function)

C HARACTERISTICS OF S INE AND C OSINE To sketch the graphs of the basic sine and cosine functions by hand, it helps to note five key points in one period of each graph: the intercepts, maximum points and minimum points. Sine key points: Cosine key points:

T RANSFORMATIONS OF SINE AND COSINE You will study the graphic effect of each of the constants in equations of the forms: Sine : Cosine: Amplitude represents half the distance between the max and the min values of the function (multiply the y-values) If, the basic curve is stretched and if is shrunk Period the horizontal distance needed to complete one cycle of the function If, the period of the graph is greater than and if is less than

T RANSFORMATIONS OF SINE AND COSINE Phase shift : the c amount shifted right or left to determine the new endpoints of the one cycle If, the graph has a horizontal translation to the right and if to the left. Vertical translation: the d amount shifted up or down to determine the new max and min of the one cycle If, the graph has a vertical translation up and if the graph goes down Reflection: the sign in front of the determines if it is reflected over the x-axis

EX: F IND THE AMPLITUDE, THE PERIOD, THE REFLECTION, ANY VERTICAL TRANSLATION, AND ANY PHASE SHIFT OF EACH GRAPH

EX: S KETCH THE G RAPH OF ONE CYCLE 4.

EX: S KETCH THE G RAPH OF ONE CYCLE 5.

EX: S KETCH THE G RAPH OF ONE CYCLE 6.

EX: S KETCH THE G RAPH OF ONE CYCLE 7.