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Families of Functions Families of Lines The family y = mx + b , with - - PowerPoint PPT Presentation
Families of Functions Families of Lines The family y = mx + b , with - - PowerPoint PPT Presentation
Families of Functions Families of Lines The family y = mx + b , with m fixed and b varying. m = 1, b = 2, 1, 1, 2, 3 m = 0.2, b = 2, 1, 1, 2, 3 Families of Lines The family y = mx + b , with b fixed and m varying m = 2,
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m = –2, – 1, 1, 2, 3 b = – 2 m = –2, – 1, 1, 2, 3 b = 1 The family y = mx + b, with b fixed and m varying
Families of Lines
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n = 1, 3, 5 n = 2, 4, 6
The Family y = xn
The family y = xn, n = 1, 2, 3, …
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n = 100 n = 101
The Family y = xn
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The Family y = xn
The family y = xn, n = –1, –2, –3, … n = – 1, – 3, – 5 n = –2, –4, –6
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n = – 100 n = – 101
The Family y = xn
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The family , n = 1, 2, 3, …
n x
n = 1, 3, 5 n = 2, 4, 6
The Family y = n x
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n = 101
The Family y = n x
n = 100
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Polynomials
2 4 5 1 4 x x x + − + 2 3 4 3 x x x + − 3 4 3 2 x x x + − +
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2 1 2 3 2 x x x + +
Rational Functions
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3 2 3 4 7 x x x − +
Rational Functions
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8 5 3 x +
Rational Functions
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Algebraic Functions
2 2
(2 ) 1 x x + −
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Algebraic Functions
2/3
2 3
(2 ) (1 ) x x x + − +
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a b c θ csc( ) c a θ = 1 sin( ) θ = sec( ) c b θ = 1 cos( ) θ = cot( ) b a θ = cos( ) sin( ) θ θ = tan( ) a b θ = sin( ) cos( ) θ θ =
A Quick Review of Trigonometry
sin( ) a c θ = cos( ) b c θ =
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a b c θ x y z ha hb hc θ These triangles are similar (all angles equal), so the sides are proportional, that is: a b c h x y z = = ≡ Thus the trig .functions are all equal. θ
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θ
θ
ht len x y ht y len x =
y x
ht len
= If a six ft pole gives a 5 ft shadow, and a building gives a shadow
- f 250 ft, then the height of the building is
6 5
250 300
=
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2 2 2 c a b = + a b c θ Divide both sides by
2
cos( ) θ to obtain the identity:
2 2
tan( ) 1 sec( ) θ θ + =
Trigonometric Identities
2 2 2 2 sin( ) cos( ) 1 2 a b c θ θ + + = = sin( ) sin( ) x x − =− cos( ) cos( ) x x − =
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1 2 3
3 π 6 π
1 sin cos 6 3 2 π π = = 3 sin cos 3 6 2 π π = = Two important triangles. 1 1 2
4 π 4 π
1 sin cos 4 4 2 π π = =
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(1,0) (– 1,0) P(cos(θ), sin(θ)) θ cos(θ) and sin(θ) are the projections onto the x and y axes respectively, as the point P goes around the unit circle.
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sin(x) cos(x) sin(x) is equal to cos(x) shifted to the right by 2 π cos( ) sin sin 2 2 x x x π π = − =− − That is:
Graphs of the Trigonometric Functions
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a b c θ
2 π θ −
tan(θ) = cot 2 π θ
− sec(θ) = csc 2 π θ
− cot(θ) = tan 2 π θ
− csc(θ) = sec 2 π θ
−
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tan(x) cot(x)
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sec(x) csx(x)
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