4. Graphing and Inverse Functions

4. Graphing and Inverse Functions 4.1 Basic Graphs 4.2 Amplitude, Reflection, and Period 4.3 Vertical Translation and Phase Shifts 4.4 The Other Trigonometric Functions 4.5 Finding an Equation From Its Graph 4.6 Graphing Combinations of Functions 4.7 Inverse Trigonometric Functions

1 4.1 Basic Graphs (long)

1) Graph of y = sin(x) and y = cos(x) 2) Period and zero 3) Amplitude 4) Domain and range of a function 5) Graph of the other four trig functions 6) Even and odd functions 7) Problems

2 4.1 Basic Graphs (long) 1) Graphs of y = sin(x) and y = cos(x). 9 y = sin x (using circle) 9 y = cos x (using circle)

See graphs of six trig functions on Table 6, sec 4.1 Technology demo.

3 4.1 Basic Graphs 2) Period and zero For any function y = f(x), the smallest positive number p for which f(x + p) = f(x) for all x is called the period of f(x).

For sine and cosine functions y = sin(x) and y = cos(x), the period is p = 2π.

4 4.1 Basic Graphs 2) Period and zero A zero of a function y = f(x) is any domain value x = c for for which f(c) = 0. If c is a real number, then x = c will be an X-intercept of the graph of y = f(x).

For sine functions y = sin(x) in cycle 0 ≤ x ≤ 2π , the zero are x = 0, π, 2π. For cosine functions y = cos(x) in cycle 0 ≤ x ≤ 2π , the zero are x = π/2, 3π/2

5 4.1 Basic Graphs 3) Amplitude If the greatest value of y is M and the least value is m, then, the amplitude of the graph is A = 1 |M – m| 2

For sine and cosine functions y = sin(x) and y = cos(x), The amplitude is A = 1 (A = 1 |1 – (–1)| = 1) 2

6 4.1 Basic Graphs 4) Domain and Range The domain of a function y = f(x) is the set of values that x can assume. For example, for sine and cosine functions y = sin(x) and y = sin(x), the domain is all real numbers.

The range of a function y = f(x) is the set of values that y can assume. For example, for sine and cosine functions y = sin(x) and y = sin(x), the range is {y | –1 ≤ y ≤ 1}

7 4.1 Basic Graphs (long) 5) Graphs of other four basic trig functions. 9 y = tan x (plotting points, identify zeros and undefined points) 9 y = csc x (using y = sin x) • y = sec x (using y = cos x) • y = cot x (similar to y = tan x )

Technology demo. See graphs of six trig functions on Table 6, sec 4.1 – Domain & range – Amplitude –Period –Zeros –Asymptotes 8 4.1 Basic Graphs 5) Graphs of other four basic trig functions Points for plotting y = tan(x). x tan(x) 0 0 π/4 1 π/3 3 ≈ 1.7 π/2 undefined 2π/3 − 3 ≈ –1.7 3π/4 –1 π 0

9 4.1 Basic Graphs 5) Graphs of other four basic trig functions Points for plotting y = csc(x). x sin(x) csc(x) 0 0 undefined π/4 1/ 2 2 ≈ 1.4 π/2 1 1 3π/4 1/ 2 2 ≈ 1.4 π 0 undefined 5π/4 –1/ 2 − 2 ≈ –1.4 3π/2 –1 –1 7π/4 –1/ 2 − 2 ≈ –1.4 π 0 undefined 10 4.1 Basic Graphs (long) 6) Even and odd functions

A function f is even function if f(–x) = f(x), for all x in the domain of the function. y = cos(x), y = sec(x) are even functions.

A function f is even function if f(–x) = –f(x), for all x in the domain of the function. y = sin(x), y = csc(x), y = tan(x) and y = cot(x) are odd functions.

11 4.1 Basic Graphs 7) Problems (1) graph y = cot x (x-scale: π/4, 0 < x < 2π)[2]

(2) Stretch the graph of y = cot x to –4π < x < 4π [8]

a) and b) will be done for six trig functions

(3) Use graph to find all values of x, 0 ≤ x ≤ 2π, such that (a) sin x = 0 [14] Ans. 0, π, 2π (b) sec x = 1 [20] Ans. 0, 2π

(4) Use the unit circle and the fact that cosine is an even function to find the value of [26, 28]

4π (a) cos(–120°) Ans. –0.5 (b) cos(–3 ) Ans. –0.5 (5) Use the unit circle and the fact that sine is an odd function to find the value of [30, 32]

7π 2 (a) sin(–90°) Ans. –1 (b) sin(–4 ) 2 (6) Prove the identity (a) cos(–θ ) tan(θ ) = sinθ [40]

13 4.2 Amplitude, Reflection and Period In this section, we consider sine and cosine functions only

1) Amplitude 2) Reflecting About the x-Axis 3) Period 4) Summary 5) Problems

More specifically, we discuss functions of the form: y = Asin(Bx) and y = Acos(Bx)

14 4.2 Amplitude, Reflection and Period 1). Amplitude e.g.1 Sketch the graph of y = 2sin x, 0 ≤ x ≤ 2π. 1 e.g.2 Sketch the graph of y = 2 cos x, 0 ≤ x ≤ 2π. (using calculator) Explanation. Scale the y-coordinate!

Conclusion. If A > 0, then functions y = Asin x and y = Acos x Have amplitude A and range [–A, A]

15 4.2 Amplitude, Reflection and Period 2) Reflecting About the x-Axis e.g.1’ Sketch the graph of y = –2sin x, 0 ≤ x ≤ 2π. 1 e.g.2’ Sketch the graph of y = –2 cos x, 0 ≤ x ≤ 2π. (using calculator) Explanation. Reflecting about the x-axis!

The graph of y = –2sin x is the reflection of the graph of y = 2sin x with respect to x-axis. 1 The graph of y = –2 cos x is the reflection of the graph of 1 y = 2 cos x with respect to x-axis. 16 4.2 Amplitude, Reflection and Period 3) Period e.g.4 Sketch the graph of y = sin (2x), 0 ≤ x ≤ 2π. The period is π

2π e.g.5 Sketch the graph of y = sin (3x), 0 ≤ x ≤ 2π. The period is 3

(using calculator) Explanation. 1 e.g.6 Sketch the graph of y = cos ( 2 x) for one cycle. The period is 4 π

17 4.2 Amplitude, Reflection and Period 4) Summary

If B is a positive number, the graph of y = Asin(Bx) and y = Acos(Bx) will have: 2π amplitude = |A|, period = B

What if B is negative? Use the property that sine is an odd function, and cosine is an even function.

18 4.2 Amplitude, Reflection and Period 5) Problems

(1) For the given function, graph one complete cycle; label the axes accurately; identify the period: π (a) y = cos(3x) [14] (b) y = cos( 2 x) [18] 2π is Period is 3 Period is 4

(2) Give the amplitude and the period of each of the graph:

5 (a) [20] 4 3 2 1 0 –4π –2π -1 2π 4π -2 -3 -4 Amplitude: 4; Period 4π -5

19 4.2 Amplitude, Reflection and Period 5) Problems (3) Give the amplitude and the period of each of the graph:

(b) [22] 2.5 2 1.5 1 0.5 0 –2π –π -0.5 2π 4π 3π 4π -1 Amplitude: 2; Period 2π -1.5 -2 -2.5 (4) Graph one complete cycle; label the axes so that the amplitude and the period are easy to read: y = 1 sin(3x) 2

(5) Graph the function over the given interval. Label the axes so that the amplitude and the period are easy to read: [34]

y = 3cos(πx), –2 ≤ x ≤ 4 20 4.3 Vertical Translation and Phase Shift In this section, we consider sine and cosine functions only

1) Vertical translations 2) Phase shift 3) Summary 4) Problems

More specifically, we discuss functions of the form: y = k + Asin(B(x – h)) and y = k + Acos(B(x – h))

21 4.3 Vertical Translation and Phase Shift 2) Vertical translation

Summary The graphs of y = k + sin(x) and y = k + cos(x) will be translated vertically k units upward if k > 0, or k units downward if k < 0.

Ex. Graph one complete cycle; label the axes accurately, and identify the vertical translation. [6] y = 6 – sin(x)

22 4.3 Vertical Translation and Phase Shift 2) Vertical translation

Ex. Graph one complete cycle; label the axes accurately, and identify the amplitude, period, and vertical translation. [10] y = –2 + 2sin(4x)

2π is Period2, is Amplitude is Period2, is 3

23 4.3 Vertical Translation and Phase Shift 3) Phase shift

Phase shift of basic sine and cosine functions.

Summary The graphs of y = sin(x – h) and y = cos(x – h) will be translated horizontally h units to the right if h > 0, or h units to the left if h < 0. The value of h is called the phase shift.

24 4.3 Vertical Translation and Phase Shift 3) Phase shift

Ex. Graph one complete cycle; label the axes accurately, and identify the phase shift. [14]

π π = xy + 6 )sin( Phase isshift − 6

1.5 = xy )sin(

0.5

0 − π π π 3π 2π 6 -0.5 2 2 -1

-1.5

25 4.3 Vertical Translation and Phase Shift Now consider both period and phase shift of basic sine and cosine functions. Ex. Given the equation, y = sin(2x + π), identify the amplitude, period, and phase shift. Label the axes accordingly and sketch one completely cycle of the curve. [24]

Explanation (how do get phase shift using argument).

π shift Phase; Period1; Amplitude = Period1; = π Phase; shift = − 2

Graph it.

26 4.3 Vertical Translation and Phase Shift

Summary. Combine period with phase shift of basic sine and cosine functions. Let B > 0, C any real number. Then the graph of y = sin(Bx + C) and y = cos(Bx + C) will have 2π C Period = B , Phase shift = − B

A recommendation.

27 4.3 Vertical Translation and Phase Shift The general form.

Graphing the sine and cosine functions. The graphs of y = k + Asin(B(x – h)) and y = k + Acos(B(x – h)), where B > 0, will have the following characteristics 2π amplitude = |A|, Period = B C phase shift = − B , vertical translation = k

28 4.3 Vertical Translation and Phase Shift

Ex. Use problem 24 for reference, graph one cycle of the function: y = –1 + sin(2x + π) [36] (adding vertical translation)

Ex. Graph one complete cycle; label the axes accurately; identify the amplitude, period, vertical translation, and phase shift: [42] 1 π y = 3 + 2sin( x − 22 )

Amplitude: 2; period: 4π; v-translation: upward 3; PS: π

29 4.4 The Other Trigonometric Functions

1) Tangent & cotangent 2) Secant & cosecant

30 4.4 The Other Trigonometric Functions 1) Tangent & cotangent

Ex. Graph one complete cycle; label axes accurately; and draw asymptotes. [2] y = 3cot(x) (mean of 3: growth factor)

Ex. Graph one complete cycle; label axes accurately; state the period of the graph; draw asymptotes. [24] 1 1 y = 3 cot( 2 x) (mean of ½ and 1/3) Amplitude: none; period: 2π

31 4.4 The Other Trigonometric Functions 1) Tangent & cotangent

Ex. Graph one complete cycle; label axes accurately; draw asymptotes; state the period, vertical translation, and phase shift of the graph. [54] 3 1 3ππ y = 2 –2 cot( 2 x − 2 )

3 3 shift Phase, upward :nTranslatio-V ;2 Period = ;2 :nTranslatio-V upward 2 Phase, shift = 3

Reflected

32 4.4 The Other Trigonometric Functions 1) Tangent & cotangent The general form.

Graphing the tangent and cotangent functions. The graphs of y = k + Atan(B(x – h)) and y = k + Acot(B(x – h)), where B > 0, will have the following characteristics:

π Period = B , phase shift = h, vertical translation = k

33 4.4 The Other Trigonometric Functions 2) Secant & cosecant

Ex. Graph one complete cycle; label axes accurately; draw asymptotes [4]. 1 y = 2 sec(x) (mean of 1/2: shrink factor)

Ex. Graph one complete cycle; label axes accurately; state the period of the graph; draw asymptotes. [20] 1 y = 3csc( 2 x) (mean of 3 and ½) Amplitude: none; period: 4π

34 4.4 The Other Trigonometric Functions 2) Secant & cosecant

Ex. Graph one complete cycle; label axes accurately; draw asymptotes; state the period, vertical translation, and phase shift of the graph. [58] π y = –3 – 2sec(πx + 3 )

1 shift Phase3, downward :nTranslatio-V ;2 Period = ;2 :nTranslatio-V downward Phase3, shift = − 3

Reflected

35 4.4 The Other Trigonometric Functions 2) Secant & cosecant The general form.

Graphing the tangent and cotangent functions. The graphs of y = k + Asec(B(x – h)) and y = k + Acsc(B(x – h)), where B > 0, will have the following characteristics:

2π Period = B , phase shift = h, vertical translation = k

36 4.5 Finding an Equation from Its Graph

Each graph shows at least one complete cycle of the graph of and equation containing a trig function. In each case, find an equation to match the graph. If you are using a graphing calculator, use it to verify.

37 4.5 Finding an Equation from Its Graph y Ex. [2] 6 5

2 1 Ans. = − 2 xy +1 1 x -1-2-3-4-5-6 1 2 3 4 56 -1

-2

-3

-4

-5

-6 38 4.5 Finding an Equation from Its Graph

Finding an equation from its graphs of a trig function. 1) How to find amplitude? (max – min)/2 2) How to find vertical translation? (max + min)/2 3) How to find period? end – start 4) How to find phase shift? start

39 4.5 Finding an Equation from Its Graph

Ex. [10] Ans. y = –2sin(x) 2.5 2 1.5 1 0.5 0 -0.5 π π 3π 2π -1 2 2 -1.5 -2 -2.5 Ex. [12] Ans. y = cos(2x)

1.5

0.5

0 π π 3π π -0.5 4 2 4

-1

-1.5

40 4.5 Finding an Equation from Its Graph

Ex. [22] Ans. y = 3 – 2cos(πx) 6

0 00.511.522.533.5

Ex. [30] Ans. y = 3 – 3sin(2x – π/2) 7

0 π -1 π 3π π 5π 4 2 4 4

41 4.6 Graphing Combinations of Functions

Graph functions of form y = y1 + y2, y1 and y2 are algebraic and/or trigonometric functions. 1) Plotting points

2) Using zero(s) of y1 and y2 3) Check using the calculator

42 4.6 Graphing Combinations of Functions Sketch each graph for 0 ≤ x ≤ 4π 1 (1) y = x2 – sin(x) [8] (relatively easy)

(2) y = 3cos(x) + sin(2x) [14] (need time)

x (3) y = cos(x) + cos( )2 [18] (need time)

• Plot point (for the combined function) first. See shape. • Use the calculator to verify.

43 4.6 Graphing Combinations of Functions

• Plot point (for the combined function) first. See shape. • Use the calculator to verify. πx)2sin( (4) y = sin(πx) + 2 0 ≤ x < 4 [36.a]

period of y = sin(πx) 2 period of y = sin(2πx) 1

44 4.7 Inverse Trigonometric Functions

• Inverse function • Graph of sine function • Graph of cosine function • Graph of tangent function • Inverse of sine, cosine and tangent • problems

45 4.7 Inverse Trigonometric Functions Function – see section A.1 (appendix A.1).

• A function is one-to-one if it passes vertical line test. • If a function is one-to-one, then it has an inverse function • e.g. find the inverse of function y = x2 –4

46 4.7 Inverse Trigonometric Functions Definition – Inverse of a Function If y = f(x) is a one-to-one function, then the inverse of f is also a function and can be denoted by y = f -1(x).

47 4.7 Inverse Trigonometric Functions Graph of sine function.

1.5

0.5

0 −2ππ−π π π 2π - 2 -0.5 2

-1

-1.5

π • If we take the middle part (bounded by x = − 2 and

π x = 2 ), then the sine function is one-to-one. − π ≤ x ≤ π • The domain in the middle part is 2 2 • The corresponding range is –1 ≤ y ≤ 1

48 4.7 Inverse Trigonometric Functions Graph of cosine function.

1.5

0.5

0 −2π −π π π π 2π - 2 -0.5 2

-1

-1.5 • If we take the middle branch (bounded by x = 0 and x = π), then the cosine function is one-to-one. • The domain in the middle branch is 0 ≤ x ≤π • The corresponding range is –1 ≤ y ≤ 1

49 4.7 Inverse Trigonometric Functions Graph of tangent function.

0 3π −π π π π 3π - 2 - 2 -5 2 2 -10

-15

-20 • If we take the middle branch, then the tangent function is one-to-one.

π π • The domain in the middle branch is − 2 < x < 2 • The corresponding range is –∞ < y < ∞

50 4.7 Inverse Trigonometric Functions Summary on page 242 = sin −1 xy = cos−1 xy = tan −1 xy arcsin = arcsin x = arccos x = arctan x

sine inverse cosine inverse tangent inverse π π π 2 2

-1.5 -1 -0.5 0 0.5 1 1.5 -4 -3 -2 -1 0 1 2 3 4 π π − -1.5 -1 -0.5 0 0.5 1 1.5 2 − 2

Domain: –1 ≤ x ≤ 1 Domain: –1 ≤ x ≤ 1 Domain: all real numbers

:Range π y ≤≤− π π π 2 2 :Range ≤ y0 ≤ π :Range − 2 ≤ y ≤ 2

x in radians!!! 51 4.7 Inverse Trigonometric Functions

When solving inverse trig functions, 1) Be careful with domain. 2) Notations – Inverse of sine: sin–1 , or arcsin – Inverse of cosine: cos–1 , or arccos – Inverse of tan: tan–1 , or arctan 3) When using calculator, know the mode

52 4.7 Inverse Trigonometric Functions Evaluate without using calculator. Answer in radians. −1 1 (1) cos ()2 [6] Ans. π/3

(2) arctan(1 ) [16] 3 Ans. π/6

−1 3 (3) sin (− 2 ) [18] Ans. –π/3

1 (4) arccos()− 2 [I made] Ans. 2π/3

53 4.7 Inverse Trigonometric Functions Evaluate using calculator. Answer in radians. (5) sin–1(–0.1702) [26] Ans. –9.8°

(6) arctan(–0.3799) [30] Ans. –20.8°

(7) cos–1(–0.7660) [38] Ans. –140°

–1 x (8) Simplify 5|sec(θ)|, if θ = tan 5 for some real number x [44]. Ans. 5sec(θ)|

54 4.7 Inverse Trigonometric Functions Evaluate without using calculator. −1 3 (9) (coscos 5 ) [46] Ans. 3/5

−1 1 (10) e.g.4 (a) (sinsin 2 ) Ans. 1/2

−1 (11) e.g.4 (b) ( 135sinsin o ) Ans. 45°

−1 (12) ( 60tantan o ) [60] Ans. 60°

(13) −1 2π [62] π ()tantan 3 − 3

−1 7π 5π (14) ()coscos 6 [56] 6 55 4.7 Inverse Trigonometric Functions Evaluate without using calculator. (right triangle method) −1 3 (15) (costan 5 ) [66] Ans. 4/3

−1 1 3 (16) (sincos 2 ) [70] 2

Write an equivalent expression that involves x only. (17) cos(tan–1 x) [80] 1 x2 +1