1. Analytic Trigonometry
Barnett Ziegler Bylean

2. Graphs of trig functions
Chapter 3

3. Basic graphs
Ch 1 - section 1

4. Why study graphs?

5. Assignment
Be able to sketch the 6 basic trig functions WITHOUT referencing notes or using a graphing calculator.
Be able to answer questions concerning:
domain/range
x-int/y=int
increasing/decreasing
symmetry
asmptote
without notes or calculator.

6. Hints for hand graphs X-axis - count by π/2 with domain [-2π, 3π]
Y-axis – count by 1’s with a range of [-5,5]

7. Defining trig functions in terms of (x,y)
Input x
(cos(ө),sin(ө) ) ө
Output cos(𝑥)= 𝑥 𝜃 =𝑦,
sin(x)= 𝑦 𝜃 =y,
tan(x) ,sec(x), csc(x), cot(x)

8. y=sin(x) Domain/range X-intercept Y-intercept Other points
Input x
Domain/range
X-intercept
Y-intercept
Other points
Periodic/period
Increase
Decrease
Symmetry (odd)
(cos(ө),sin(ө) ) ө
Output
y = sin(x) – using π/2 for the x-scale
Output cos(𝑥)= 𝑥 𝜃 ,sin(x)= 𝑦 𝜃 , tan(x) ,sec(x), csc(x), cot(x)

9. y = cos(x) Input x (cos(ө),sin(ө) ) Output cos(𝑥)=𝑦 ө Domain/range
X-intercept
Y-intercept
Other points
Periodic/period
Increase
Decrease
Symmetry (odd)

10. y = tan(x) and y = cot(x) Input x y = tan(x) y = cot(x)
restricted/asymptotes:
Range ?
y-intercept
x-intercept
(cos(ө),sin(ө) ) ө

11. y = sec(x) and y = csc(x) Input x sec(x) csc(x) restricted/asymptotes?
range?
(cos(ө),sin(ө) ) ө

12. Transformations of sin and cos
Chapter 3 – section 2

13. Review transformations
Given f(x)
What do you know about the following
f(x-3) f(x + 5)
f(3x) f(x/7)
f(x) f(x) – 4
3f(x) f(x)/3

14. Trigonometric Transformations - dilations
Y = Acos(Bx) y = Asin(Bx)
Multiplication causes a scale change in the graph
The graph appears to stretch or compress

15. Vertical dilation : y = Af(x)
If the multiplication is external (A) it multiplies the y-co-ordinate (stretches vertically) – the x intercepts are stable (y=0), the y intercept is not stable for cosine
The height of a wave graph is referred to as the amplitude (direct correlation to physics wave theory) - It is how much impact the x has on the y value - louder sound, harder heartbeat etc.
Amplitude is measured from axis to max. and from axis to min.

16. Examples of some graphs
y=3(sin(x)
y=sin(x)
𝜋 2
y=-2sin(x)

17. y = 3cos(x) 1
Scale π/2

18. Horizontal Dilations
If the multiplication is inside the function it compresses horizontally against the y-axis – the x-intercepts are compressed – the y- intercept is stable – this affects the period of the function
Period – the length of the domain interval that covers a full rotation – The period for sine and cosine is 2π – multiplying the x – coordinates speeds up the rotation thereby compressing the period -
New period is 2π /multiplier
Frequency – the reciprocal of the period-

19. Examples of some graphs
y=cos(2x)
y= cos(x/2)
y=cos(x)

20. Sketch a graph (without a calculator)
y = 3cos(2x)
y = - sin(πx)
𝑦= cos⁡ 𝑥 3 2

21. Transformations - Vertical shifts
Adding “outside” the function shifts the graph up or down – think of it like moving the x-axis
f(x) = sin(x) g(x) = cos(x) - 4

22. Pertinent information affected by shift
the amplitude and period are not affected by a vertical shift
The x and y intercepts are affected by shift –
The maximum and minimum values are affected by vertical shift

23. Finding max/min values
Max/min value for both sin(x) and cos(x) are 1 and -1 respectively
Amplitude changes these by multiplying
Shift change changes them by adding
Ex: k(x)= 4cos(3x -5) – 2
the max value is now 4(1)- 2= 2
the min value is now 4(-1) – 2 =-6

24. Example
Graph k(x) = 4 + 2cos(𝛑x)

25. Summary

26. Writing equations Identify amplitude Identify period
Identify axis shift

27. Horizontal shifts
Chapter 3 – section 3

28. Simple Harmonics
f(x) = Asin(Bx + C) or g(x) = Acos(Bx + C) are referred to as Simple Harmonics.
These include horizontal shifts referred to as phase shifts
The shift is -C units horizontally followed by a compression of 1/B - thus the phase shift is
-C/B units
The amplitude and period are not affected by the phase shift

29. Horizontal shift
f(x) = cos(x + 𝜋 4 )
g(x) = cos(2x – 𝜋 4 )

30. find amplitude, max, min, period and phase shift
f(x) = 3cos(2x – π/3)
y = 2 – 4sin(πx + π/5)

31. Tangent/cotangent/secant/cosecant revisited
Chapter 3 – section 6

32. Basic graphs asymptotes Period Increasing/decreasing tan(x) cot(x)
sec(x) csc(x)

33. k + A tan(Bx+C) or k + A cot(Bx+C)
No max or min - effect of A is minimal
Period is π/B instead of 2π/B
Phase shift is still -C/B and affects the x intercepts and asymptotes
k moves the x and y intercepts

34. Examples
y = tan(3x)
y = cot( 𝑥 2 − 𝜋 3 )

35. k+ Asec(Bx+ C) or k + Acsc(Bx + C)
local maxima and minima affected by k and A
Directly based on sin and cos so Period is 2π/B
Shift is still -C/B

36. Examples
y = 3 + 2sec(3πx)
y = 1 – csc (2x + π/3)