Presentation on theme: "4.5 Graphs of Sine and Cosine Functions"— Presentation transcript:
14.5 Graphs of Sine and Cosine Functions JMerrill, 2010
2Amplitude: Sine Function xy1-1The maximum height of the sine function is 1. It goes one unit above and one unit below the x-axis, which is the center of it’s graph. This maximum height is called the amplitude.11
3Amplitude: Cosine Function xy11-1The maximum height of the cosine function is 1. It goes one unit above and one unit below the x-axis, which is the center of it’s graph. This maximumheight is called the amplitude.11
4Amplitude The amplitude of the normal sine or cosine function is 1. To change the amplitude of a sine or cosine function, you would need to vertically stretch or shrink the function.amplitude = |A|(Choose the line that is dead-center of the graph. The amplitude has the same height above the center line (axis of the wave) as the height below the center line.
5What is the amplitude? Examples Vert. stretch by 3Vert. shrink by ½Vert. shrink by π/4
6Period: Sine Function x y 1 -1 y1-1This one piece of the sine function repeats over and over, causing the sine function to be periodic. The length of this piece is called the period of the function.
7Period: Cosine Function xy11-1This one piece of the cosine function repeats over and over, causing the cosine function to be periodic. The length of this piece is called the period of the function.
8Period The period of a normal sine or cosine function is 2π. To change the period of a sine or cosine function, you would need to horizontally stretch or shrink the function.The period is found by: period =In the equation, b affects the frequency, which is related to the period.
9Period Examples of f(x) = sin bx The period of the sinx (parent) is 2πThe period of sin2x is π. p=If b > 1, the graph shrinks.This graph is happening twice as often as the original wave.
10Period Examples of f(x) = sin bx The period of y = sinx (parent) is 2πThe period of sin ½ x is π. p=If b < 1, the graph stretches.This graph is happening half as often as the original wave.
11What is the period? Examples Horiz. stretch by ½Horiz. shrink by 3Horiz. shrink by 2π/3Horiz. shrink by π/2
12Examples: y = A sin bx y = A cos bx Give the amplitude and period of each funtion:Y = 4 cos 2xA = 4,y= -4 sin 1/3 x
13Can You Write the Equation? Sine or cosine?Amplitude?Period?b?Equation?2
14Equation?Sine or Cosine?Amplitude?Period?b?Equation:28
15Translations of General Sine Waves What does a refer to?Referring to our previous equationsy = a sin bxy = a cos bxIf we translate the graphs h units horizontally, and k units vertically, then the resulting equations are:y = d + a sin b(x - c)y = d + a cos b(x - c)This is different than the book. My way is easier!!What does b refer to?
16ShiftsPhase ShiftWhen a graph is shifted c units horizontally, then x is replaced with (x-c)Remember that a phase shifts acts in the opposite direction—just like all other functions.1
17ShiftsVertical ShiftWhen a graph is shifted vertically, then y = d + blahblahblah is the equation.4
18ShiftsWhen the sine wave is shifted units to the left, what is the result?A cosine wave!So, sine and cosine curves are referred to as general sine waves.
19Amplitude revisitedIf the center of the wave is not at the x-axis, then amplitude can be found byAmplitude can still be measured by the vertical distance between the center of the wave to the peak (and/or valley)
20Axis of the Wave (Vertical Translation) If the x-axis is not the center of the wave, then you need to find the center. The center is the average of the peak and the valley pointsAxis of the wave: x =
21Example To find the amplitude To find the axis of the wave To verify the amplitude, what is the vertical distance from the axis of the wave to the peak or valley?33
22Write the Equation Axis of the wave? Amplitude? Period? 4 So, a = 2, b = ?
23Write the Equation Cosine Equation? To write the equation, look at the new x/y-axis (forget the old). Here, we changed the x-axis, but not the y-axis.Sine or cosine?CosineThe vertical shift is the amount we raised the x-axis.Equation?
24Write the Equation Axis of the wave? x = 2 Amplitude? Period? B? Sine or cosine?Can’t tell? Move the y-axis. Sine or cosine?Now, use the new set of axes and write the equation.