4. Graphing and Inverse Functions
4.1
4.2
4.3
4.4
4.5
4.6
4.7
Basic Graphs
Amplitude, Reflection, and Period
Vertical Translation and Phase Shifts
The Other Trigonometric Functions
Finding an Equation From Its Graph
Graphing Combinations of Functions
Inverse Trigonometric Functions
1
4.1 Basic Graphs (long)
1)
2)
3)
4)
5)
6)
7)
Graph of y = sin(x) and y = cos(x)
Period and zero
Amplitude
Domain and range of a function
Graph of the other four trig functions
Even and odd functions
Problems
2
4.1 Basic Graphs (long)
1)
Graphs of y = sin(x) and y = cos(x).
9
9
y = sin x (using circle)
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
1
A = 2 |M – m|
For sine and cosine functions y = sin(x) and y = cos(x),
1
The amplitude is A = 1 (A = 2 |1 – (–1)| = 1)
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
9
•
•
y = tan x
y = csc x
y = sec x
y = cot x
(plotting points, identify zeros and undefined points)
(using y = sin x)
(using y = cos 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
0
π/4
π/2
3π/4
π
5π/4
3π/2
7π/4
π
sin(x)
0
1/ 2
1
1/ 2
0
–1/ 2
–1
–1/ 2
0
csc(x)
undefined
2 ≈ 1.4
1
2 ≈ 1.4
undefined
− 2 ≈ –1.4
–1
− 2 ≈ –1.4
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
Ans. 0, π, 2π
(a) sin x = 0 [14]
(b) sec x = 1 [20] Ans. 0, 2π
(c) sec x is undefined
[24] Ans. π/2, 3π/2
12
4.1 Basic Graphs
7) Problems
(4) Use the unit circle and the fact that cosine is an even
function to find the value of
[26, 28]
(a) cos(–120°)
Ans. –0.5
(b) cos(–
4π
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]
(a) sin(–90°)
Ans. –1
(b) sin(– 74π )
2
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)
2)
3)
4)
5)
Amplitude
Reflecting About the x-Axis
Period
Summary
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.
e.g.2 Sketch the graph of y = 12 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.
e.g.2’ Sketch the graph of y = – 12 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.
The graph of y = – 12 cos x is the reflection of the graph of
y = 12 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
e.g.5 Sketch the graph of y = sin (3x), 0 ≤ x ≤ 2π. The period is
2π
3
(using calculator)
Explanation.
e.g.6 Sketch the graph of y = cos ( 12 x) for one cycle. The period is 4π
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:
(a) y = cos(3x) [14]
(b) y = cos( π2 x)
[18]
Period is 23π
Period is 4
(2) Give the amplitude and the period of each of the graph:
(a)
[20]
5
4
3
2
1
0
–4π
–2π
-1
2π
4π
-2
-3
Amplitude: 4; Period 4π
-4
-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
Amplitude: 2; Period 2π
–π
-0.5
2π
4π
3π
4π
-1
-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 = 12 sin(3x)
(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)
2)
3)
4)
Vertical translations
Phase shift
Summary
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)
Amplitude is 2, Period is 23π
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]
y = sin( x + π6 )
Phase shift is − π6
y = sin(x )
1.5
1
0.5
0
− π6 -0.5
π
2
π
3π
2
2π
-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).
Amplitude = 1; Period = π ; Phase shift = −
π
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
Period = 2Bπ , Phase shift = − CB
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
B
amplitude = |A|, Period =
phase shift = − CB , 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]
y = 3 + 2sin( 12 x − π2 )
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]
y = 13 cot(12 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]
y = 23 – 12 cot(π2 x − 32π )
Period = 2; V - Translatio n : upward 23 , 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].
y = 12 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]
y = 3csc( 12 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]
y = –3 – 2sec(πx + π3 )
Period = 2; V - Translatio n : downward 3, Phase shift = − 13
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
B
Period = , 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
6
Ex. [2]
5
4
3
Ans. y = − 12 x + 1
2
1
-6
-5
-4
-3
-2
-1
1
2
3
4
5
x
6
-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
-1
2
π
3π
2
2π
-1.5
-2
-2.5
Ex. [12]
Ans. y = cos(2x)
1.5
1
0.5
0
-0.5
π
π
4
2
3π
4
π
-1
-1.5
40
4.5 Finding an Equation from Its Graph
Ex.
[22]
Ans. y = 3 – 2cos(πx)
6
5
4
3
2
1
0
0
Ex.
0.5
1
1.5
2
[30]
2.5
3
3.5
Ans. y = 3 – 3sin(2x – π/2)
7
6
5
4
3
2
1
0
-1
π
4
π
2
3π
4
π
5π
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) y = x12 – sin(x)
[8] (relatively easy)
(2) y = 3cos(x) + sin(2x)
[14] (need time)
(3) y = cos(x) + cos( )2x
[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.
sin( 2πx )
(4) y = sin(πx) +
0≤x<4
[36.a]
2
period of y = sin(πx)
period of y = sin(2πx)
2
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
1
0.5
−2π
−π
π
-2
0
π
-0.5
2
π
2π
-1
-1.5
•
•
•
If we take the middle part (bounded by x = − π2 and
x = π2 ), then the sine function is one-to-one.
π
π
The domain in the middle part is − 2 ≤ x ≤ 2
The corresponding range is –1 ≤ y ≤ 1
48
4.7 Inverse Trigonometric Functions
Graph of cosine function.
1.5
1
0.5
−2π
−π
π
-2
0
π
-0.5
2
π
2π
-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.
20
15
10
5
-
3π
2
−π
- π2
0
π
-5
2
π
3π
2
-10
-15
-20
•
•
•
If we take the middle branch, then the tangent function
is one-to-one.
The domain in the middle branch is − π2 < x < π2
The corresponding range is –∞ < y < ∞
50
4.7 Inverse Trigonometric Functions
Summary on page 242
y = sin −1 x
= arcsin x
y = cos −1 x
= arccos x
cosine inverse
sine inverse
π
-1
-0.5
tangent inverse
π
π
2
-1.5
y = tan −1 x
= arctan x
0
0.5
1
2
1.5
− π2
-4
-1.5
-1
-0.5
0
0.5
1
1.5
-3
-2
-1
0
1
2
3
4
− π2
Domain: –1 ≤ x ≤ 1
Domain: –1 ≤ x ≤ 1
Domain: all real numbers
Range : − π2 ≤ y ≤
Range : 0 ≤ y ≤ π
Range : −
π
2
x in radians!!!
π
2
≤ y≤
π
2
51
4.7 Inverse Trigonometric Functions
When solving inverse trig functions,
1) Be careful with domain.
2) Notations
–
–
–
Inverse of sine:
Inverse of cosine:
Inverse of tan:
sin–1 , or arcsin
cos–1 , or arccos
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
( )
[16]
Ans. π/6
( )
[18]
Ans. –π/3
1
3
(2)
arctan
(3)
sin −1 −
(4)
arccos (− 12 ) [I made]
3
2
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°
(8) Simplify 5|sec(θ)|, if θ = tan–1 5x for some real number x
[44].
Ans. 5sec(θ)|
54
4.7 Inverse Trigonometric Functions
Evaluate without using calculator.
−1 3
(9) cos cos 5
[46]
Ans. 3/5
(
)
(10) e.g.4 (a)
(11) e.g.4 (b)
(12)
sin (sin −1 12 )
sin −1 (sin 135o )
tan −1 (tan 60o )
Ans. 1/2
Ans. 45°
[60]
Ans. 60°
(13)
tan
(tan 23π )
[62]
− π3
(14)
cos (cos 76π )
[56]
5π
6
−1
−1
55
4.7 Inverse Trigonometric Functions
Evaluate without using calculator. (right triangle method)
−1 3
(15) tan cos 5
[66]
Ans. 4/3
(
)
(
(16) cos sin
−1 1
2
)
[70]
3
2
Write an equivalent expression that involves x only.
1
(17) cos(tan–1 x)
[80]
2
x +1
56