Graphs of Cosine Functions (part 2) I. Graphing the cosine function f(x) = cos θ. A) Use the unit circle to find cos θ (width values). 1) cosine is the width, x, when θ is the angle. 2) θ = 0 θ = π/2 θ = π θ = 3π/2 θ = 2π 0° 90° 180° 270° 360° (1, 0) (0 , 1) (-1, 0) (0 , -1) (1, 0) x = 1 x = 0 x = -1 x = 0 x = 1 3) To graph, we plot these 5 points (min / max / x-int) and connect them with a curved line. 1 2π π/2 3π/2 π -1

Graphs of Cosine Functions (part 2) I. Graphing the cosine function f(x) = cos θ. B) The cos graph repeats after every 2π (one cycle) 1) The graph from 0 to 2π is called the Period. C) The height of the graph is called the Amplitude. 1) This is the range (the y values of the graph). D) Sideways shifts are Horizontal Translations. 1) These horizontal shifts are phase shifts E) Standard form of a sine function. 1) f(x) = A cos (B(x + C)) + D a) Notice we have to factor out the B term.

Graphs of Cosine Functions (part 2) II. Transformations of the Cosine Function. A) We need to get it into standard form. 1) f(x) = A cos (B(x + C)) + D or y = D + A cos (B(x + C)) B) Amplitude (the A term). This is the height. Amp = | A |. 1) Go the A value above and below the origin. C) Period (the B term). Remember to factor out the B value. 1) The period = 2π/B. a) Big B values are horiz shrinks, small B’s stretch it. D) Phase Shift (the C term). This moves the graph sideways. 1) Set the (x + C) part = 0 to find the shift (changes sign). E) Vertical Shift (the D term). Moves the graph up or down. 1) Moves all the min / max / x-int points up or down by D.

Graphs of Cosine Functions (part 2) III. Graphs of the cos function y = D + A cos (B(x + C)).

Graphs of Cosine Functions (part 2) III. Graphs of the cos function y = D + A cos (B(x + C)).