4Graph y = sin sin 0° 0.707 45° 90° 1 135° 0.707 180° 225° 20°0.70745°190°1-270º-90º90º180º270º360º135°0.707-1180°-2225°-0.707-1270°-0.707315°360°
5y = sin x 2 1 -90º 90º -270º 270º -1 -2 Maximum Maximum intercept MinimumMinimum-2
6y = sin xPeriod: the least amount of space (degrees or radians) the function takes to complete one cycle.21-90º90º-270º270º-1-2Period: 360°
7In other words, how high does it go from its axis? y = sin xAmplitude: half the distance between the maximum and minimum2Amplitude = 11-90º90º-270º270º-1-2In other words, how high does it go from its axis?
9Graph y = cos cos 1 0.707 -0.707 -1 -0.707 0.707 1 2 1 -1 -2 120.7071-0.7072-1-1-0.707-20.70721
10y = cos x -2 - 2 2 1 -1 -2 Maximum Maximum intercept intercept MinimumMinimum-2
11y = cos x -2 - 2 Period: 2 Period: the least amount of space (degrees or radians) the function takes to complete one cycle.21-2-2-1-2Period: 2
12How high does it go from its axis? y = cos xHow high does it go from its axis?2Amplitude = 11-2-2-1-2
13Try it on your calculator! y = sin xy = cos xTry it on your calculator!21-1-2
14y= sin and y = cos are the mother functions. Changing the equations changes the appearance of the graphsWe are going to talk about the AMPLITUDE, TRANSLATIONS, and PERIOD of relative equations
15Mother Functionrelative functionchange?y1 = sin xreflection over x-axisy2 = - sin xy1 = sin xy2 = 4 sin xamplitude = 4amplitude =y2 = sin xy1 = sin xgeneralization?y = a sin xamplitude = a
16is the horizontal translation Mother Functionrelative functionchange?y1 = sin xy2 = sin (x - 45)horizontal translation, 45 degrees to the right.horizontal translation, 60 degrees to the left.y1 = sin xy2 = sin (x + 60)horizontal translation, 30 degrees to the left.y2 = sin (2x + 60)y1 = sin xhorizontal translation, 90 degrees to the right.y1 = sin xy2 = sin (3x - 270)generalization?y = sin (bx - c)to the rightis the horizontal translationy = sin (bx – (- c))to the left
17‘d’ is the vertical translation Mother Functionrelative functionchange?y1 = cos xy2 = 2 + cos xvertical translation, units up.vertical translation, units down.y1 = cos xy2 = -3 + cos xgeneralization?y = d + cos x‘d’ is the vertical translationwhen d is positive, the graph moves up.when d is negative, the graph moves down.
18Mother Functionrelative functionchange?y1 = sin xy2 = sin 2xPeriod = 180orPeriod = 720ory1 = sin xy2 = sin xgeneralization?Period =y = sin bxor
19Summary: y = d + a sin (bx - c) y = d + a cos (bx - c) a is the amplitudeperiod =oris the horizontal translationd is the vertical translation
20Analyze the graph of amplitude = period = horizontal translation: vertical translation:none
21Analyze the graph of 3 (to the left) amplitude = period = horizontal translation:vertical translation:none
22Analyze the graph of 3 amplitude = period = horizontal translation: nonevertical translation:Up 2
23Graph and Analyze y = -2 + 3 cos (2x - 90°) 3 x y high 1 amplitude = 45°3amplitude =90°-2midperiod == 180°135°low-5horizontal translation:180°-2mid(to the right)225°1highvertical translation:down 21) horiz. tells you where to start3) divide period by 4 to find increments= 452) add the period to find out where to finishtable goes in increments of 454) plot points and graph= 225
25You must know how to analyze the equation before you can graph it. The most important thing to remember about graphing is determining the starting point and the stopping point on the t-table.
26Ex #6c Graph y = 1 + 3 sin (2 + ) = 3 up 1 x y mid 1 amplitude = high= 4period =1midhorizontal translation:-2lowup 1vertical translation:mid11) horiz. tells you where to startOn Calculator, go to table setup and change independent to ask.3) divide period by 4 to find incrementstable goes in increments of2) add the period to find out where to finish4) plot points and graph