NC Pre Calculus - Mrs. Young's Math Classesmrsyoungsmathclasses.weebly.com/uploads/3/7/5/6/... ·...
Transcript of NC Pre Calculus - Mrs. Young's Math Classesmrsyoungsmathclasses.weebly.com/uploads/3/7/5/6/... ·...
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Pre Calculus 1 of 30
Linear Functions Slope-Intercept Point-Slope
bmxy += 11 )( yxxmy +−=
2
43
−= xy 3)1(23
++−= xy
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Pre Calculus 2 of 30
Slope
12
12
xxyy
xy
runrisem
−
−=
ΔΔ
==
Parallel Line 21 mm = Perpendicular Lines 2
11m
m −=
21
21=
221
−=
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Pre Calculus 3 of 30
Functions
Definition: A rule that assigns a unique output (y) for every input (x). Range: Set of Output (y-values) of the Function Domain: Set of Input (x-values) of the Function
To Find Domain: The domain is usually all real numbers (ℝ) or −∞,∞ except for rational, even roots and logarithmic functions.
Rational Functions (Fractions)
421)(−
=x
xf
All real numbers except values of x that make the denominator equal to zero.
042 ≠−x D: 2≠x or ),2()2,( ∞∪−∞
Even Root Functions xxf 312)( −=
All Positive Numbers & Zero 0123 ≥+x
D: 4≤x or ]4,(−∞
Logarithmic Functions )4(log)( 3 += xxf
Only Positive Numbers 04 >+x
D: 4−>x or ),4[ ∞−
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Pre Calculus 4 of 30
Even Function Odd Function Symmetric about the y-axis Symmetric about the Origin
)()( xfxf −= )()( xfxf −=−
12)1()1(12)1()1(
2)(
2
2
2
−=−−=−−=−=
−=
ff
xxf
2)1(2)1(2)1(2)1(
2)(
3
3
3
−=−=−==
=
ff
xxf
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Pre Calculus 5 of 30
Transformation of Functions
Cxf ±)(
)( Cxf ∓
)(xf−
)( xf −
1),( <axaf
1),( >axaf
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Pre Calculus 6 of 30
Inverse Operations
triginversetrictrigonomenonentialexarithmiclo
rootspowerdivisiontionmultiplica
nsubtractioaddition
pg
↔↔↔↔↔
xxaxxxx
a1
33
sinsinlog
−↔↔↔
÷↔×−↔+
Inverse Functions
abfbaf
==
− )()(
1 25)(52)(
1 +=
−=− xxf
xxf
3251)1(
15)3(2)3(1 =
+=
=−=−ff
Steps to find Inverse Function 1. Switch all x’s and y’s 2. Solve for y
Horizontal Line Test
g(x) has an inverse
f(x) does not have an inverse
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Pre Calculus 7 of 30
Absolute Value Function
⎩⎨⎧
<−
≥==
00,
)(xxxx
xxf
Domain: ),( ∞−∞ Range: ),0[ ∞ Even Function Note: Corner at the Origin
Greatest Integer Function xxf =)(
Domain: ),( ∞−∞ Range: set of integers Note: Jump Discontinuity at every integer
value of x
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Pre Calculus 8 of 30
Quadratic (Square) Function
2)( xxf =
Domain: ),( ∞−∞ Range: ),0[ ∞ Even Function
Square (Even) Root Function
xxf =)(
Domain: ),0[ ∞ Range: ),0[ ∞ Note: Square (Even) Root cannot have
Negative Numbers in the Radical
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Pre Calculus 9 of 30
Cubic Function
3)( xxf =
Domain: ),( ∞−∞ Range: ),( ∞−∞ Odd Function
Cubic (Odd) Root Function 3)( xxf =
Domain: ),( ∞−∞ Range: ),( ∞−∞ Odd Function Note: Cubic (Odd) Root have an Undefined
(Vertical) Slope at 0=x
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Pre Calculus 10 of 30
Exponential Function
xaxf =)(
Domain: ),( ∞−∞ Range: ),0( ∞ Note: The number e (a common base) is an
irrational number, approximately 2.718…
Logarithmic Function
xxf alog)( =
Domain: ),0( ∞ Range: ),( ∞−∞ Note: ln (natural log) is log with base e.
Inside a log must always be positive.
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Pre Calculus 11 of 30
Sine Function
xxf sin)( =
Domain: ),( ∞−∞ Range: ]1,1[− Odd Function Period: 2π
Cosine Function
xxf cos)( =
Domain: ),( ∞−∞ Range: ]1,1[− Even Function Period: 2π
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Pre Calculus 12 of 30
Tangent Function
xxf tan)( =
Domain: All Real Numbers x ≠ 0.5π + nπ Vertical Asymptote: x ≠ 0.5π + nπ Range: ),( ∞−∞ Odd Function Period: π
Rational (Reciprocal) Function
xxf 1)( =
Domain: ),0()0,( ∞∪−∞ Range: ),0()0,( ∞∪−∞ Note: Denominator can never equal zero.
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Pre Calculus 13 of 30
Multiply Polynomials Distributing Method
Distribute all each term of the 1st polynomial to each term of the 2nd polynomial.
8414)122(7
+
+
xx
20262010126)42)(53(
2
2
−−
−+−
−+
xxxxx
xx
xxxxxxxxxx
xxxx
3132718183018353
)6)(353(
234
23423
22
−−−−
−−−−+
−−+
xxxxxxxxlog20log12106)log42)(53(
23
2
−−+
−+
Special Products
Sum & Difference
22
))((bababa
−
−+
Square of Binomial
22
2
2)(baba
ba+−
− 22
2
2)(baba
ba++
+
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Pre Calculus 14 of 30
Factoring Polynomials Common Factors
Find common factors and factor them out.
)6(148414+
+
xx
)52(42048
2
23
−−
−−
xxxxxx
)93(98127922
3223
yxyxxyxyyxyx
+−
+−
1=a Find factors of c that add up to b. Rewrite these factors as a product.
2082 −− xx 210 ⋅−
)2)(10( +− xx
962 +− xx 33 −⋅− )3)(3( −− xx
36162 ++ xx 123 ⋅
)12)(3( ++ xx
1≠a MUST find common factors first! Find factors of ac that add up to b. Then, divide those factors by a and reduce. Rewrite the reduced fractions as a product with the denominator multiplied by x and numerator added to x.
8166 2 +− xx )483(2 2 +− xx
322
32
36
1243
−⋅−
−⋅
−=⋅
)23)(2(2 −− xx
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Special Factoring Difference of 2 Squares
))((
22
bababa−+
−
Perfect Trinomial
2
22
)(2bababa
−
+−
2
22
)(2bababa
+
++
Standard to Vertex Form/Complete the Square
If 1=a , then:
423
25
12425
4255
125
2
2
2
squarehalf
+⎟⎠
⎞⎜⎝
⎛ +=
↓
+−++=
++=
xy
xxy
xxy
If 1≠a , then:
`
213
232
1129
4932
1162
2
2
bounce
2
squarehalf
+⎟⎠
⎞⎜⎝
⎛ +=
↓
+−⎟⎠
⎞⎜⎝
⎛ ++=
++=
xy
xxy
xxy
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Quadratic Formula If cbxax ++= 20 is not factorable use either complete the square or quadratic formula to solve:
aacbbx
242 −±−
=
Solve for x: 450 2 ++= xx
1,4235
295
)1(2)4)(1(4)5()5( 2
−−=
±−=
±−=
−±−=x
Complex (Imaginary) Numbers Numbers formed by multiplying a real number time i, where i, is the square root of negative 1.
1
11
4
3
2
=−=−=−=
iii
ii
4164424242424
==−⋅−
−=−−
=−
−=−
=
ii
Solve for x: 1250 2 ++= xx
i
i
x
522
5242204)1(2
)9)(1(4)4()4( 2
±−=
±−=
−±−=
−±−=
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max 1 31zeros 24
108)( 4
≥=≥=
+−=
n-n
xxxg
Polynomials: 012
21
1 ...)( axaxaxaxaxf nn
nn +++++= −
−
n, the largest exponent, is the degree (order) of the polynomial.
Leading Coefficient Test an is the coefficient of xn
Zeros (or x-intercept or roots) n is the maximum number of real zeros. There could be less than n. To find zeros, set the function equal to 0 and solve.
Maximums & Minimums
n–1 determines the possible number of both hills (maximums) and valleys (minimums) the function may have.
max 1min 2 31zeros 34
8)( 24
+≥=≥=−=
n-n
xxxf
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Rational Functions Horizontal Asymptote
1. Multiply (if necessary). 2. Take leading coefficient from numerator and
denominator. 3. Simplify.
a. If x remains on bottom, y = 0. b. If x’s cancel, y = coefficients. c. If x remains on top, no horizontal
asymptote.
Vertical Asymptote 1. Factor (if possible). 2. Simplify. 3. Set denominator equal to zero.
Graphing 1. Graph asymptotes as dashed lines. 2. Test points for direction. Note: Any section
in the center must be tested with at least two points.
16)4(3)( 2
2
−
+=xxxf
Horizontal Asymptote
3
331648243
2
2
2
2
=
==−
++=
yxx
xxxy
Vertical Asymptote
440
4)4(3
)4)(4()4(3 2
=−=
−
+=
−+
+=
xx
xx
xxxy
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Pre Calculus 19 of 30
Properties of Exponents & LogarithmsProduct Rule
Add the two exponents. baba xxx +=
Quotient Rule Subtract the smallest exponent from the largest.
abba
b
a
xx
xx
−−=
1or
Power Rule Multiply the exponents.
( ) abba xx = Negative Rule
If on bottom, goes to the top. If on top, goes to bottom.
aa
xx 1
=− aa x
x=
−
1
Radicals (Root) ↔ Rational Exponent Radical or root becomes the denominator in the exponent.
ba
b a
bb
xx
xx
=
=1
Logs Product Rule
abba logloglog =+ Quotient
baba logloglog =−
Power Rule abab loglog =
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Pre Calculus 20 of 30
Exponential Growth & Decay Growth
trAy )1( +=
4.124)20.01(50 5 =+=y
9.1916)20.01(50 20 =+=y
y = current amount at t A = initial amount r = rate t = time
Decay trAy )1( −=
4.16)20.01(50 5 =−=y
6.0)20.01(50 20 =−=y
Compound Interest
nt
nrPy ⎟⎠
⎞⎜⎝
⎛ += 1
40.080,22$408.01000,10
104
=⎟⎠
⎞⎜⎝
⎛ +=⋅
y
46.253,22$36508.01000,10
10365
=⎟⎠
⎞⎜⎝
⎛ +=⋅
y
y = current amount at t P = Principle (initial
deposit) r = rate t = time n = number of times
compounded
Compound Continuously rtPey =
41.255,22$000,10 1008.0 == ⋅ey
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Pre Calculus 21 of 30
Trigonometric Values
..tan
..cos
.
.sin
adjopp
hypadj
hypopp
=
=
=
θ
θ
θ
..cot
..sec
.
.csc
oppadj
adjhyp
opphyp
=
=
=
θ
θ
θ
θ
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Pre Calculus 22 of 30
Common Trigonometric Values
0 𝜋2
𝜋 𝜋2
2𝜋
𝜋3
2𝜋3
4𝜋3
5𝜋3
𝜋4
3𝜋4
5𝜋4
7𝜋4
𝜋6
5𝜋6
7𝜋6
11𝜋6
I II III IV
Examples
sin 𝜃 =32
𝜃 =𝜋3,2𝜋3
tan 𝜃 = −33
𝜃 =5𝜋6,11𝜋6
Degrees 0º 30º 45º 60º 90º
Radians 0 𝜋6
𝜋4
𝜋3
𝜋2
θsin 02= 0
12=12
22
32
42= 1
θcos 42= 1
32
22
12=12
02= 0
θtan 0 13=
33
22= 1 3
1= 3 undefined
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Trigonometric Identities
Reciprocal Identities
xx
xx
sin1csc
csc1sin
=
=
xx
xx
cos1sec
sec1cos
=
=
xx
xx
tan1cot
cot1tan
=
=
Quotient Identities
xxx
cossintan =
xxx
sincoscot =
Pythagorean Identities
1cossin 22 =+ xx
xx 22 sectan1 =+ xx 22 csccot1 =+
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Pre Calculus 24 of 30
Law of Sines
Cc
Bb
Aa
sinsinsin==
Area of an Oblique Triangle
BacCabAbcArea sin21sin
21sin
21
===
Law of Cosines
Abccba cos2222 −+= Baccab cos2222 −+= Cabbac cos2222 −+=
Heron’s Area Formula
))()(( csbsassArea −−−= ,
where 2cbas ++
=
Alternative Form
bcacbA
2cos
222 −+=
acbcaB
2cos
222 −+=
abcbaC
2cos
222 −+=
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Pre Calculus 25 of 30
Vectors
Components
𝒗 = 𝑞! − 𝑝!, 𝑞! − 𝑝! = 𝑣!, 𝑣!
𝒗 = 𝑣!𝒊 + 𝑣!𝒋
𝒗 = 𝒗 cos 𝜃 𝒊 + 𝒗 sin 𝜃 𝒋
Magnitude & Direction
𝒗 = 𝑣!! + 𝑣!!
𝜃 = tan!!𝑣!𝑣!
𝒖 = −1 − 2, 3 − (−5) = (−3, 8)
𝒖 = −3𝒊 + 8𝒋
𝒖 = (−3)! + 8! = 73 = 8.54
𝜃 = tan!!8−3
= 110°
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Pre Calculus 26 of 30
Conics General Equation 𝐴𝑥! + 𝐵𝑥𝑦 + 𝐶𝑦! + 𝐷𝑥 + 𝐸𝑦 + 𝐹 = 0 A, B, C, D, E, F are constants
Circle 𝑥! + 𝑦! with equal coefficients Ex: 3𝑥! + 3𝑦! + 2𝑥 − 3𝑦 = 0
Ellipse 𝑥! + 𝑦! with unequal coefficients Ex: 2𝑥! + 4𝑦! + 2𝑥 + 64 = 0
Parabola Either 𝑥! or 𝑦! Ex: 3𝑥! − 4𝑥 + 𝑦 − 5 = 0
Hyperbola Either 𝑥! − 𝑦! or 𝑦! − 𝑥! Ex: 3𝑥! − 𝑦! + 6𝑥 − 3𝑦 + 81 = 0
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Pre Calculus 27 of 30
Circle
Fact: Each point on a circle is equidistance from the center by r units.
(𝑥 − ℎ)! + (𝑦 − 𝑘)! = 𝑟!
Use complete the square to rewrite any conic section into its standard form.
𝑥! + 𝑦! − 4𝑥 + 6𝑦 − 23 = 0𝑥! − 4𝑥 + 4 + 𝑦! + 6𝑦 + 9 = 23 + 4 + 9
(𝑥 − 2)! + (𝑦 + 3)! = 36
9𝑥! − 𝑦! − 36𝑥 + 6𝑦 − 12 = 09(𝑥! − 4𝑥 + 4) − (𝑦! − 6𝑦 + 9) = 12 + 4 + 9
9(𝑥 − 2)! − (𝑦 − 3)! = 25
(𝑥 − 2)!
259
−(𝑦 − 3)!
25= 1
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Pre Calculus 28 of 30
Ellipse Fact: The sum of the distances from the foci to any point on the ellipse is 2𝑎.
(𝑥 − ℎ)!
𝑏!+(𝑦 − 𝑘)!
𝑎!= 1
(𝑥 − ℎ)!
𝑎!+(𝑦 − 𝑘)!
𝑏!= 1
𝑐 = 𝑎! − 𝑏! 𝑒 = !!< 1
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Pre Calculus 29 of 30
Parabola Fact: Each point on the graph (blue line) is equidistance from focus and directrix. Parabolas have a reflective property that focuses energy to one point, the focus. Thus, they are used as satellite dishes to focus low energy radio waves.
𝑦 − 𝑘 ! = 4𝑝 𝑥 − ℎ
(𝑥 − ℎ)! = 4𝑝(𝑦 − 𝑘)
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Pre Calculus 30 of 30
Hyperbola Fact: The difference of the distances from the foci to any point on the curve is 2𝑎.
(𝑥 − ℎ)!
𝑎! −(𝑦 − 𝑘)!
𝑏! = 1
(𝑦 − ℎ)!
𝑎! −(𝑥 − 𝑘)!
𝑏! = 1
𝑐 = 𝑎! + 𝑏! 𝑒 = !!> 1