Transformations of the Graphs of Sine and Cosine Functions Credit to: JMerrill, 2010 EQ: How do I transform the graphs of trigonometric functions?

Amplitude: Sine Function (Sinusoidal) The maximum height of this sine function is 1. It goes one unit above and one unit below the x-axis, which is the center of its graph. This maximum height is called the amplitude. 1 1

Amplitude: Cosine Function (Cosinusoidal) The 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 maximum height is called the amplitude. 1 1

amplitude = ½|Max - Min| 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 compress the function. amplitude = ½|Max - Min| How to find: Choose the horizontal 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.

Examples: What is the amplitude? Vertical ______ by a factor of ___ Amp: ____ Vertical ______ by a factor of ___ Amp: ____ Vertical ______ by a factor of ___ Amp: ____

Period: Sine Function This 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. The normal sine function will repeat every ___ units.

Period: Cosine Function This 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. The normal cosine function will repeat every ___ units.

Period Therefore, 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 compress the function. The period is found by: period = In the equation, b affects the frequency, which is related to the period.

Period Examples of f(x) = sin bx The period of the sinx (parent) is 2π. The period of sin2x is π. p= If b > 1, the graph shrinks. Therefore, this graph is happening twice as often as the original wave. This means that two waves will fit in the same space as one wave for the normal sine function.

Period Examples of f(x) = sin bx The period of y = sinx (parent) is 2π. The 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. Therefore, only half of the graph could fit in the original period.

What is the period? Examples Horizontal ______ by a factor of ___ Per: ____ Horizontal ______ by a factor of ___ Per: ____ Horizontal ______ by a factor of ___ Per: ____ Horizontal ______ by a factor of ___ Per: ____

Examples: y = A sin bx y = A cos bx Give the amplitude and period of each funtion: Y = 4 cos 2x y= -4 sin 1/3 x

Can You Write the Equation? Sine or cosine? Amplitude? Period? b? Equation?

Equation? Sine or Cosine? Amplitude? Period? b? Equation:

Graphing: Find the Critical points To find the critical intervals (max/min, intercepts)

Can You Draw the Graph? y = 4 sin 2x A= _____ P=_____

Can You Draw the Graph? y = -6cos(½x) A= _____ P=_____

Translations of General Sine Waves What does a refer to? Referring to our previous equations y = a sin bx y = a cos bx If we translate the graphs h units horizontally, and k units vertically, then the resulting equations are: y = a sin b(x – c) + d y = a cos b(x – c) + d What does b refer to?

Shifts Phase Shift When 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

Shifts Vertical Shift When a graph is shifted vertically, then y = blahblahblah + d is the equation. This means every point gets shifted up/down d units. 4

Shifts When the sine wave is shifted units to the left, what is the result? So, sine and cosine curves are referred to as general sine waves.

Axis 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 points Axis of the wave: y =

Example To find the axis of the wave (center line): To find the amplitude To verify the amplitude, what is the vertical distance from the axis of the wave to the peak or valley?

Write the Equation Axis of the wave? Amplitude? Period? Equation?

Write the Equation Axis of the wave? Amplitude? Period? B? Sine or cosine? (If you can’t tell, move the y-axis the LEAST amount.) Now, use the new set of axes and write the equation.

Same Graph - 2nd Equation Axis of wave = 2 Move the y-axis to the left ½ unit. Now it’s a sine wave! Equation?

Equations of the Graph Since sine and cosine are both general sine waves, both equations are correct!

Graphing by Hand

Graphing by Hand

Graphing by Hand