Ch 1 – Functions and Their Graphs• Different Equations for Lines• Domain/Range and how to find them• Increasing/Decreasing/Constant• Function/Not a Function• Transformations• Shifts• Stretches/Shrinks• Reflections

• Combinations of Functions• Inverse Functions

Ch 1 – Functions and Their Graphs1.1 Formulas for lines slope vertical

line

point- horizontal slope line

slope- parallel intercept slopes

general perpendicular form slopes

12

12

xx

yym

−−

=

( )1212 xxmyy −=−

bmxy +=

0=++ CByAx

ax =

by =

mm

1−=⊥

mm =||

1.2 Functions

domain (input)

range (output)

( )4,3 −

1.2 Functions domain (input)

range (output)

),1[ ∞−

),3[ ∞−

inclusive [

exclusivealway - exclusive ( ∞

1.2 Functions Increasing/decreasing/constant

on x-axis only (from left to right)

( )∞− ,1

never ][

always )(

1.2 and 1.3 Functions Functions Not functions

( ) ( )3,4,3,2 ( ) ( )4,2,3,2

4

3

2

4

2

3

1.2 and 1.3 Functions Function or Not a Function?

Domain? ( )∞∞− ,

Range? ),3[ ∞−

y-intercepts? ( )1,0 −

x-intercepts? ( ) ( )0,5 and 0,1−

increasing? ( )∞,2

decreasing? ( )2,∞−

1.2 and 1.3 Functions Finding domain from a given function.

Domain = except:

x in the denominator x in radical

( )∞∞− ,

( )4

32 −

=x

xxf

Can’t divide by zero

( ) 2or 2;, :domain −≠∞∞− x

( ) 62 −= xxf

Can’t root negative062 ≥−x

[ )∞,3 :domain

1.4 Shifts (rigid)

( )2010 −=− xy

horizontalshift

vertical shift

( )2hxaky −=−

( )2210 −=− xy

( )2012 −=− xy

1.4 Stretches and Shrinks (non-rigid)

( )23 xy =

stretch

vertical

( )2xcy =

( )23

1xy =

( )23xy =

horizontal

shrink stretch

2

3

1⎟⎠

⎞⎜⎝

⎛= xy

shrink

1.4 Reflections

xy −=

In the x-axis

( ) xxh =

In the y-axis

xy −=

( ) ( )xfxh −= ( ) ( )xfxh −=

If negative can be move to other side, flipped on x-axis.If can’t, flipped on y-axis.

1.5 Combination of Functions

( )( ) ( )( )xgfxgf =o

( ) ( ) ( )( )2 find ,3 and 2 Give −=+= gfxxgxxf o

( ) ( ) 6232 −=−=−g

( )( ) ( ) 462 −=−=− fgf

1.5 Combination of Functions

( )( ) ( )( )xgfxgf =o

( ) ( ) ( )( )2 find ,2 and 4 Give −=−= fgxxgxxf o

( ) ( ) 6422 −=−−=−f

( )( ) ( ) 1262 −=−=− gfg

1.6 Inverse Functions

( )other.each of inverses are

functions e verify thand 42 inverse theFind −= xxf

( ) . and switch and with Replace 1. yxyxf

42 −= yx.for Solve 2. y

yx 24 =+

yx

=+24

( )2

4+=x

xg

( )( ) xxgf =o that Show

( )( ) 42

42 −⎟

⎠

⎞⎜⎝

⎛ +=

xxgf o

( )( ) 44 −+= xxgf o

( )( ) xxgf =o

Ch 2 – Polynomials and Rational Functions• Quadratic in Standard Form• Completing the Square• AOS and Vertex• Leading Coefficient Test• Zeros, Solutions, Factors and x-intercepts• Given Zeros, give polynomial function• Given Function, find zeros• Intermediate Value Theorem, IVT• Remainder Theorem• Rational Zeros Test• Descartes’s Rule• Complex Numbers• Fundamental Theorem of Algebra• Finding Asymptotes

Ch 2 – Polynomials and Rational Functions

2.1 Finding the vertex of a Quadratic Function

( ) 782 2 ++= xxxf

a

bx

2−=

1. By writing in standard form (completing the square)

2. By using the AOS formula€

f x( ) = 2 x 2 + 4x + 4( ) + 7 − 8

€

f x( ) = 2 x + 2( )2

−1

( ) 222

8−=−=x

( )1,2 −−

€

f x( ) = 2 −2( )2

+ 8 −2( ) + 7 17168 −=+−=

2.1 Writing Equation of Parabola in Standard Form

( )( ) form. standardin equation its write,3,-6

throughpasses and 2,1at ex with vertparabola aGiven

.for solve and form standard into and ,, Substitute akhyx

€

y = a x − h( )2

+ k

( ) 246 +=− a

a48 =−

2−=a €

y = −2 x −1( )2

+ 2( ) 2136 2 +−=− a

2.2 Leading Coefficient Test

Leading Coefficient aPositive Negative

Leading exponent nOdd

Even

( ) ...+= naxxf

2.2 Zeros, solutions, factors, x-intercepts

There are 3 zero (or roots),solutions, factors,and x-intercepts.

function theofsolution a is 2=x

function theof zero a is 2=x

( ) function theoffactor a is 2−x

( ) function theofintercept -an is 0,2 x

2.2 Zeros, solutions, factors, x-intercepts

Find the polynomial functions with the followingzeros (roots).

€

x = −1

2, 3, 3

If the above are zeros, then the factors are:

( ) ( )( )332

1−−⎟

⎠

⎞⎜⎝

⎛+= xxxxf

Can be rewritten as( ) ( )( )( )3312 −−+= xxxxf 912112 23 ++−= xxx

2.2 Zeros, solutions, factors, x-intercepts

Find the polynomial functions with the followingzeros (roots). 112,112,3 −+=x

Writing the zeros as factors:

€

f x( ) = x − 3( ) x − 2 + 11( )[ ] x − 2 − 11( )[ ]

Simplifying.

€

f x( ) = x − 3( ) x − 2( ) − 11[ ] x − 2( ) + 11[ ]

€

f x( ) = x − 3( ) x − 2( )2

−11[ ]

€

f x( ) = x − 3( ) x 2 − 4x + 4 −11[ ]

€

= x − 3( ) x 2 − 4x − 7[ ]

( ) 2157 23 ++−= xxxxf

2.2 Intermediate Value Theorem (IVT)

IVT states that when y goes from positive to negative,There must be an x-intercept.

2.3 Using Division to find factors

Long Division

Synthetic Division

2.3 Remainder Theorem

( ) ( ) ( ) remainder. theis then by divided is When kfkxxf −

( ) ( ) ?7583 offactor a 2 Is 23 −++=+ xxxxfx

9- 1 2 3

2- 4- 6-

7- 5 8 3 2- :dividion synthetic Using

( )( ) graph. on thepoint a bemust 9- 2,- Also,

factor. anot is 2 Therefore,

remainder theis 9

+

−

x

2.3 Rational Zeros Test

tcoefficien leading of factors

ermconstant t of factors ZerosRational ==

qp

( ) pqxxf n ++= ...

( ) .3832 of zeros rational possible theFind 23 +−+= xxxf

2

3,

2

1,3,1

2,1

3,1

2 of Factors

3 of Factors±±±±=

±±±±

=

Possible

2.3 Descartes’s Rule

( ) 4653 23 −+−= xxxxf

3201

3003

==

Count number of sign changes of f(–x) for number ofpositive zeros

+ – + – 1 2 3 = 3 or 1 positive zeros

( ) ( ) ( ) ( ) 4653 23 −−+−−−=− xxxxf

Count number of sign changes of f(–x) for number of negative zeros.

– – – – 0 negative zeros (+) (–) (i)

2.3 Complex Numbers

conjugate. itsby 53Multiply i−

Complex number = Real number + imaginary number

Treat as difference of squares.

( )( )ii 5353 +−

( ) ( ) ( ) 3425953 22 =−−=− i

2.3 Complex Numbers

. 24

53 form standardin Write

i

i

−+

i

i

i

i

24

24

24

53

++

⋅−+

( )416

1020612

−−

−++=

iii

i

10

13

10

1

20

262+=

+=

2.5 Fundamental Theorem of Algebra

A polynomial of nth degree has exactly n zeros.

( ) 345 xxxf += has exactly 4 zeros.

2.5 Finding all zeros

1. Start with Descartes’s Rule

( ) 8122 235 +−++= xxxxxf

2. Rational Zeros Test (p/q)

+ – i

2 1 2

0 1 4

8,4,2,11

8,4,2,1PRZ ±±±±=

±±±±±

=

3. Test a PRZ (or look at graph on calculator).( )

084211

84211

812210111

−

−

−−x

( )

08421

8421

11−x

( )

0401

802

842122

−−

−+x

42 +x

iix 2,2 −=

2.6 Finding Asymptotes

Vertical Asymptotes

Horizontal Asymptotes

( )...

...

+

+==

m

n

bx

ax

D

Nxf

mn <

Where f is undefined. Set denominator = 0

Degree larger in D, y = 0. BOBOO

Degree larger in N, no h asymptotes. BOTNN

Degrees same in N and D, take ratio of coefficients.

mn >

mn=

b

ay =

Ch 3 – Exponential and Log Functions

• Exponential Functions• Logarithmic Functions• Graphs (transformations)• Compound Interest (by period/continuous)• Log Notation• Change of Base• Expanding/Condensing Log Expressions• Solving Log Equations• Extraneous Solutions

Ch 3 – Exponential and Log Functions

3.1 Exponential Functions

( ) ( )hxfkxh −=−Same transformation as

If negative can be move to other side, flipped on x-axis.If can’t, flipped on y-axis.

( ) 23 1 −= −xxf Shifted 1 to right, 2 down.

( ) xxf 3−= Flipped on x-axis.

( ) xxf −= 3 Flipped on y-axis.

3.1 Compounded Interest

nt

n

rPA ⎟

⎠

⎞⎜⎝

⎛ += 1

Compound by Period

rtPeA=

Compound Continuously

3.1 Compounded Interest

nt

n

rPA ⎟

⎠

⎞⎜⎝

⎛ += 1

A total of $12,000 is invested at an annual interest rate of 3%. Find the balance after 5 years if the interest is compounded (a) quarterly and (b) continuously.

000,12=P

yearper 03.0=ryearper times4=n

years 5=t

€

A = 12,000( ) 1+0.03( )

4( )

⎛

⎝ ⎜

⎞

⎠ ⎟

4( ) 5( )

21.934,13≈

rtPeA=

€

A = 12,000( )e 0.03( ) 5( ) 01.942,13≈

3.2 Logarithms

yax =

Used to solve exponential problems (when x is an exponent).

xy alog=

3.2 Logarithms

yax =

Used to solve exponential problems (when x is an exponent).

xy alog=

a

ba b log

loglog =

Change of base

3.3 Logarithms

yy

x444

3

4 loglog35log5

log −+=

Expanding Log Expressions

( ) ( )x

xxx

22lnln2ln2

+=−+

Condensing Log Expressions

3.4 Solving Logarithmic Equations

( ) 4223 =x

Solve the Log Equation

142 =x

x in the exponent, use logs

14ln2ln =x

14ln2ln =x

2ln

14ln

2ln

2ln=x 807.3≈x

3.4 Solving Logarithmic Equations

0232 =+− xx ee

Solve the Log Equation

( )( ) 012 =−− xx ee

( ) 02 =−xe ( ) 01 =−xe

2=xe

2lnln =xe

2ln=x 1ln=x

1lnln =xe

1=xe

0=

3.4 Solving Logarithmic Equations

43log2 5 =xSolve the Log Equation

23log5 =x

23log 55 5 =x

253 =x

3

25=x

3.4 Solving Logarithmic Equations

( ) ( ) xxx ln232ln2ln =−+−Solve the Log Equation

( )( ) 2ln322ln xxx =−−

( )( ) 2322 xxx =−−

22 672 xxx =+−

0672 =+− xx

( )( ) 016 =−− xx

1,6=x

( )( ) invalid=− 21ln

6=x

3.4 Solving Logarithmic Equations

( ) ( ) xxx ln232ln2ln =−+−Solve the Log Equation

( )( ) 2ln322ln xxx =−−

( )( ) 2322 xxx =−−

22 672 xxx =+−

0672 =+− xx

( )( ) 016 =−− xx

1,6=x

( )( ) invalid=− 21ln

6=x