9) P = π10) P = π/211) P = π/5; ± π/10 12) P = (2 π)/3; ± π/313) P = π/4; ± π/8 14) P = (3π 2 )/2; ±(3π 2 )/4 15) 16) 17) 18)
Translating Sine and Cosine Functions
The same translating rules apply to all functions Each constant in the equation does the same job y = a f (b (x – h)) + k a – vertical stretch b – horizontal stretch h – horizontal shift k – vertical shift
In periodic functions the horizontal shift is also called the “phase shift” The phase shift tells us how far around the unit circle we need to start to have the same results What is the value of h in each function? Describe the phase shift in terms of left or right. g (x) = f (x + 1) h = –1; left 1 m (x) = f (x – 3) h = 3; right 3 y = sin (x + π) h = –π; left π
A phase shift moves the graph sideways k moves the graph up or down
Translate the graph f (x) to be f (x – 1) 2 4
Parent Functions: y = a sin bx y = a cos bx Translated Functions y = a sin b (x – h) + k y = a cos b (x – h) + k Translating Rules |a| = amplitude = period (x is in radians and b > 0) h = phase shift, or horizontal shift k = vertical shift 2πb2πb
For the next class complete #3 – 30 every 3 rd, starting on page 746.
3) h = 1.6; right 1.66) h = 5π/7; right 5π/7 9) 12) 15) 18) Amp: 4, Per: π, Left 1, down 221) 24) 27) 30)
We can use the values for period, amplitude, phase shift, and vertical shift Begin with the parent function and place the values for a, b, h, and k in their appropriate places Write an equation for each translation: 1) y = sin (x), 4 units down y = sin (x) – 4 2) y = cos (x), π units left y = cos (x + π) 3) y = sin (x), period of 3, amp of 2, right π/2, down 1 y = 2 sin (x – π/2) – 1 2π32π3
MonthAverage High January42 February45 March52 April59 May68 June79 July84 August82 September74 October63 November50 December42 Plot a graph of the data (in degrees) and write a cosine function to model the information. Let a > 0. J J JFM MAASOND 50˚F J y = 21 cos (x – 180) + 63
For the next class complete #34 – 43 starting on page 746.