WARM UP State the sign (positive or negative) of the function in each quadrant. 1. cos x 2. tan x Give the radian measure of the angle ° ° Find a transformation that will transform the graph of y 1 to the graph of y 2 7. y 1 = x 2 and y 2 = 3x 2 8. y 1 = x 3 and y 2 = x 3 – 2

WARM UP State the sign (positive or negative) of the function in each quadrant. 1. cos x: +, -, -, + 2. tan x: +, -, +, - Give the radian measure of the angle °: 3π/ °: 5π/2 Find a transformation that will transform the graph of y 1 to the graph of y 2 7. y 1 = x 2 and y 2 = 3x 2 :vertically stretch by 3 8. y 1 = x 3 and y 2 = x 3 – 2:translate down 2 units

Graphs from the Unit Circle Let’s finish up our projects, then, answer the questions at the bottom of your exercise paper If you have already finished, work with another group as a “consultant”

Graphs from the Unit Circle What component from the unit circle do the x- values on the function graph represent? The vertical legs of the triangles in the unit circle (which are the y-values on the function graph), represent what function of the related angle measure? After how many radians does the graph start to repeat? How do you know it repeats after this point? Where does the curve cross the x-axis? Does the curve have any minimum and maximum values? What are the x- values at the maxima and minima and what are the y- values at the maxima and minima?

Graphs from the Unit Circle For any point P on the Unit Circle, how can we express the x- and y-coordinates of P in terms of trig ratios? What curve did you graph? If you had graphed the horizontal leg of the right angle for each point on the Unit Circle, what curve would you have graphed? Let’s look here: – tml tml

What you’ll learn about The Basic Waves Revisited Sinusoids and Transformations Modeling Periodic Behavior with Sinusoids … and why Sine and cosine gain added significance when used to model waves and periodic behavior.

Sinusoid

Amplitude of a Sinusoid

Period of a Sinusoid

Sinusoid Amplitude Period

Periodic Functions More simply, a periodic function is a function that repeats its values in regular intervals or periods

Periodic Functions Looking at the average temperature for Philadelphia over two years, when does the average temperature repeats its values in regular intervals or periods?

Periodic Functions Looking at the Unit Circle, let’s choose an angle Θ at random. The values of sin Θ and sin (Θ + 2π) are always ___(what ?)____.

Example Horizontal Stretch or Shrink and Period

Example Vertical Stretch or Shrink and Amplitude Find the amplitude of each function and use the language of transformations to describe how the graphs are related. a)y 1 = cos x b)y 2 = ½ cos x c)y 3 = –3 cos x

Example Vertical Stretch or Shrink and Amplitude Solving algebraically, the amplitudes are: a)1 ( for y 1 = cos x) b)½ (for y 2 = ½ cos x) c)|–3| = 3 (for y 3 = –3 cos x) The graph of y 2 is a vertical shrink of the graph of y 1 by a factor of ½ The graph of y 3 is a vertical stretch of the graph of y 1 by a factor of 3, and a reflection across the x-axis, performed in either order. (We do not call this a vertical stretch by a factor of –3, nor do we say that the amplitude is –3.)

Example Vertical Stretch or Shrink and Amplitude a)y 1 = cos x (blue) b)y 2 = ½ cos x (red) c)y 3 = –3 cos x (black)

Class Work Find the period and amplitude of each function and use the language of transformations to describe how the graphs are related. a)y 1 = sin x b)y 2 = – 2 sin (x/3) c)y 3 = 3 sin (2x)

Class Work a)y 1 = sin x  The period is 2π and the amplitude is 1 b)y 2 = – 2 sin (x/3)  The period is 2π /(1/3) = 6π and the amplitude is 2  The graph of y 2 is a horizontal stretch of the graph of y 1 by a factor of 3, a vertical stretch by a factor of 2, and a reflection across the x-axis c)y 3 = 3 sin (2x)  The period is 2π /2 = π and the amplitude is 3  The graph of y 3 is a horizontal shrink of the graph of y 1 by a factor of ½ and a vertical stretch by a factor of 3

Frequency of a Sinusoid The frequency is simply the reciprocal of the period.

HOMEWORK P 392 # EXIT TICKET Write the definition of the period and amplitude of a sinusoid in your own words.