precalculus functions and graphs, mark dugopolski, 4th edition
DESCRIPTION
Some Basic Functions and Their PropertiesSome Inverse FunctionsPolynomial FunctionsSome Basic Rational FunctionsSome Basic Functions of Algebra with TransformationsThe Sine and Cosine FunctionsTrigonometric FunctionsInverse Trigonometric FunctionsFunctions in Polar CoordinatesTRANSCRIPT
xii Function Gallery
FUNCTION
gallery... Some Basic Functions and Their Properties
Constant Function Identity Function Linear Function
Square Function Square-Root Function
Cube Function Cube-Root Function Greatest Integer Function
y
x1 2 3–1–2–3
1
3
2
5 f(x) = 4
Domain 1-�, �2Range 546Constant on 1-�, �2Symmetric about y-axis
Absolute Value Function
y
x1 2 3–1–2–3
1
3
2
–2
–1
–3
f(x) = x
Domain 1-�, �2Range 1-�, �2Increasing on 1-�, �2Symmetric about origin
y
x1 2 3–1–2–3
3
4
5
2
–1
f(x) = 3x + 2
Domain 1-�, �2Range 1-�, �2Increasing on 1-�, �2
y
x1 2 3–1–2–3
1
2
3
f(x) = ⏐x⏐
Domain 1-�, �2Range 30, �2Increasing on 30, �2Decreasing on 1-�, 04Symmetric about y-axis
y
x1 2 3–1–2–3
1
2
3
4
5
f(x) = x2
Domain 1-�, �2Range 3-0, �2Increasing on 30, �2Decreasing on 1-�, 04Symmetric about y-axis
y
x1 2 3 4 5
1
2
3 √f(x) = x
Domain 30, �2Range 30, �2Increasing on 30, �2
y
x1 2–2
2
6
8
4
–4
–2
–6
–8
f(x) = x3
Domain 1-�, �2Range 1-�, �2Increasing on 1-�, �2Symmetric about origin
y
x1 2 3–1–2–3
1
2
–2
–1
√f(x) = x3
Domain 1-�, �2Range 1-�, �2Increasing on 1-�, �2Symmetric about origin
y
x–1–3
–2
–3
1 2 3–2
2
1f (x) = [[ x]]
Domain 1-�, �2Range 5n ƒ n is an integer6Constant on 3n, n + 12 for every integer n
Function Gallery xiii
FUNCTION
gallery... Some Inverse Functions
Linear
Powers and Roots
y
x–1–2–3–4–1
–2
–3
–4
1 2 3
2
1
3
f –1(x) = x – 3
f (x) = x + 3
f (x) = 5x
f –1(x) = x5
y
x–1–4–1
–4
1 2 3
2
1
3
y
x–1–2–3–4–1
–2
–3
1 2 3
2
1
3
f –1(x) = x+32
f (x) = 2x – 3
x
y
–1
–1
1 2 3 4
2
1
3
4
for x ≥ 0 f (x) = x2
f –1(x) = x
f (x) = x3
–1–2–3–4 1 2 3 4
y
–1
–2
–3
–4
2
1
3
4
f –1(x) = x 3
x
for x ≥ 0
f –1(x) = x
f (x) = x4
4
x
y
–1
–1
1 2 3 4
2
1
3
4
xiv Function Gallery
FUNCTION
gallery... Polynomial Functions
Linear: f1x2 = mx + b
Quadratic: f1x2 = ax 2 + bx + c or f1x2 = a1x - h22 + k
y
x1–1
1
3
2
–1
–2
–3
f(x) = x3y
x1 2–2
1
3
2
–1
–2
–3
f(x) = x3 – xy
x1–1
1
3
2
–2
–3
f(x) = –x3 + 4x
Quartic or Fourth-Degree: f1x2 = ax 4 + bx 3 + cx 2 + dx + e
Cubic: f1x2 = ax 3 + bx 2 + cx + d
y
x1 2 3–1–2–3
1
3
2
–2
–1
–3
f(x) = x
Slope 1, y-intercept 10, 02
y
x1 2 3–1–2–3
1
3
2
–2
–1
f(x) = 3x – 2
Slope 3, y-intercept 10, -22
y
x1 2 3 4–1–2
1
3
4
2
–1
f(x) = –2x + 4
Slope -2, y-intercept 10, 42
y
x1 2 3–1–2–3
1
3
4
2
–1
f(x) = x2
Vertex 10, 02Range 30, �2
y
x1 2 4–2
1
3
2
–3
–4
f(x) = (x – 1)2 – 4
Vertex 11, -42Range 3-4, �2
y
x2–1–2–4
1
3
4
2
–1
–2
f(x) = –x2 – 2x + 3
Vertex 1-1, 42Range 1-�, 44
y
x1 2–1–2
1
3
4
5
2
f(x) = x4 y
x1–1
2
1
f(x) = x4 – x2 y
x1 2–1–2
1
3
–1
–2
f(x) = –x4 + x2 + 2
Function Gallery xv
FUNCTION
gallery... Some Basic Rational Functions
Horizontal Asymptote x-axis and Vertical Asymptote y-axis
Various Asymptotes
y
x1 2 3–1–3
1
2
3
–1
f(x) = x–1
y
x1–1–2–3
1
–2
–3
–1
f(x) = – x–1
y
x21 3–3 –2 –1
1
2
f(x) =x2––1
y
x1 2 3–1–2–3
1
3
–2
–1
f(x) = x——–2x – 1
y = 2
y
x2 3–3
1
2
3
–2
–3
–1
f(x) =x2 – 1——–x
x = –1
x = 1
y
x21 3–3 –2 –1
1
–2
–3
–1
f(x) = x2 – 1——–x
y = x
xvi Function Gallery
FUNCTION
gallery... Exponential and Logarithmic Functions
Exponential: f1x2 = a x, domain (-H , H ), range 10, H2
Logarithmic: f -11x2 = loga1x2, domain 10, H2, range 1-H , H2
y
x1 2 3–1–2
1
3
4
5
6
2
g(x) = 2x
y
x1 2–1–2
1
3
4
5
6
2
h(x) =x1–
2⎞⎠
⎞⎠
y
x1 2–1–2
1
3
4
5
6
2
j(x) = ex
y
x1 2 3 4 5 6 7
1
–2
–1
2
g–1(x) = log2(x) y
x1 2 3 4 5 6 7
1
–2
–1
2h–1(x) = log1/2(x)
y
x1 2 3 4 5 6 7
1
–2
–1
2 j –1(x) = ln(x)
Increasing on 10, �2 x-intercept 11, 02Decreasing on 10, �2
x-intercept 11, 02Increasing on 10, �2 x-intercept 11, 02
Increasing on 1-�, �2y-intercept 10, 12 Decreasing on 1-�, �2
y-intercept 10, 12 Increasing on 1-�, �2y-intercept 10, 12
Function Gallery xvii
y
x–1–1
–2
–3
1 2 3–2
2
1
3
Quadratic
4
y = (x – 3)2
y = –x2
y = x2
y
x–1
–2
–3
1 2 3
2
1
3
Cubic
4 5
y = –x3
y = x3
y = (x – 4)3
y
x–1–3–1
–2
–3
1 2 3
2
1
3
Absolute value
y = ⏐x⏐
y = ⏐x⏐ – 3
y = ⏐x – 2⏐
Exponential
y
x
–11
1
2
2
–2
y = 2x
y = 2 – x
y = –2x
y
x–1–1
–2
1 2
2
1
3
Logarithmic
–2
4
y = log2(x)
y = log2(x) + 3
y = log2(x + 4)
y
x–1–3–1
–2
–3
1 2 3–2
1
3
Square root
y = x + 4
y = x + 3 – 4
y = x
√
√
√
Reciprocal
y
x–1–1
1
1
y = – 1–x
y = 1–x
Rational
y
x–1–1
1
1
2
–2y = – 1—
x2
y = 1—x2
y
x–1
–2
–3
1 2 3
2
1
3
Fourth degree
4 5
y = x4
y = (x – 4)4
y = – x4 – 1
y
x–1–1
1
1
Semicircle
y = – 4 – x2
y = 4 – x2√
√
y
x–1
–2
–3
1 2 3
2
1
3
Greatest integer
4 5
y = [[x]]
y = [[x – 3]]
FUNCTION
gallery... Some Basic Functions of Algebra with Transformations
y
x–1–3–1
–2
–3
1 2 3–2
2
1
3
Linear
y = x
y = x + 3
y = – x
xviii Function Gallery
FUNCTION
gallery... The Sine and Cosine Functions
y
x
–1
1
y = sin x
π–2
π 3π—2
20 π
Amplitude 1
πPeriod 2Phase shift 0
y
D – A
D
D + A
x0 C
2π—B
2π—B
C +
Amplitude A
PeriodPhaseshift C
y = A sin[B(x – C)] + D
A, B, C, D > 0
y
x
–1
1
y = cos x
π–2
π 3π—2
20 π
Amplitude 1
πPeriod 2Phase shift 0
y
D – A
D
D + A
x0 C
2π—B
2π—B
C +
Amplitude A
PeriodPhaseshift C
y = A cos[B(x – C)] + D
A, B, C, D > 0
FUNCTION
gallery... Periods of Sine, Cosine, and Tangent 1 B 7 1 2 y
x
1
–1
Period2�B
y = sin x
y = sin (Bx)
2�� 2�B
�B
Fundamental cycles
y
x
1
–1
Period2�B
y = cos x
y = cos (Bx)
2�
� 2�
B�B
y
x
1
–1
y = tan (Bx)Period
�B
y = tan x
�2
�2
�2B
�2B
––
Function Gallery xix
FUNCTION
gallery... Trigonometric Functions
Domain(k any integer)RangePeriod
Fundamental cycle
1- q , q2 3-1, 14
2p
30, 2p4
1- q , q2 3-1, 14 2p
30, 2p4
x �p
2+ kp
1- q , q2 p
c-p2
, p
2d
Domain (k any integer)Range Period Fundamental cycle
x � kp
1- q , -14 � 31, q22p30, 2p4
x �p
2+ kp
1- q , -14 � 31, q2 2p 30, 2p4
x � kp 1- q , q2 p
30, p4
y
x
–4
4321
–1–2–3
π 2π
y = csc(x)
y
x
–1
1
π 2π
y = sin(x)
y
x
–4
4321
–1–2–3
π 2π
y = sec(x)
y
x
–1
1
π 2π
y = cos(x)
y
x
–4
4321
–1–2–3
ππ–2
y = cot(x)
y
x
–4
4321
–1–2–3
π–2
π–2
–y = tan(x)
xx Function Gallery
FUNCTION
gallery... Inverse Trigonometric Functions
1–1
y
x
Domain [–1, 1]
Range –2– ⎡
⎣, –2⎡⎣
� �
y = sin–1 x
�2
�2
–
Domain [–1, 1]
Range [0, p]
y
x–1 1
y = cos–1 x�
�
2 1–1
y
x
Domain
Range –2– , –2
� �
y = tan–1 x�2
�2
–
(–�, �)⎞⎠⎝
⎛
–
Domain Range
y
x
–⎡⎣
⎡⎣, 0 0,)) �
–1 1
–2�–2
�
y = csc–1 x�2
�2
(–�, –1] � [1, �)
1–1
y
x
Domain (– , –1] [1, )
Range ⎡⎣
⎡⎣0, –2
, –2
�
�) )��
���
�
�
2
y = sec–1 x
1–1
y
x
Domain (– , )
Range (0, )� �
�
y = cot–1 x�2
�
Function Gallery xxi
FUNCTION
gallery... Functions in Polar Coordinates
–10 –8 –6 –4 –2 2 4 6 8–2
2
4
6
8
–4
–6
–8
–10
r = θθ ≥ 0
Spiral
10
10
a–2
y
xa–2
a
a–2
–
r = a cos θa > 0
Circle
a–2
y
xa–2
a–2
–
a
r = asin θa > 0
Circle
a
y
xa–a
–a
r = a cos 3θa > 0
Three-Leaf Rose
a
y
x–a
–a
–2a
Cardioid
a > 0r = a(1 – cos θ)
a
y
xa–a
–a
r = a cos 2θa > 0
Four-Leaf Rose
a
y
x–a
–a
–(a + b)
Limaçon
b > a > 0r = a – b cos θ
y
x√⎯a– √⎯a
r2 = a cos 2θa > 0
Lemniscate
a
y
x–a
–a
–(a + b) a – b
Limaçon
a > b > 0r = a – b cos θ