Interval Notation

• Interval Notation to/from Inequalities

• Number Line Plots open & closed endpoint conventions• Unions and Intersections

• Bounded vs. unbounded

,a b is equivalent to a x b ,a b is equivalent to a x b

, ,a b is equivalent to x a or b x

, ,a b is equivalent to x a an b xd

include endpoints

Distributive Property: Multiplying Polynomials & FOIL

Vertical Form Multiplication:

Horizontal Form:

a b c d a b c a b ac abc bd dd

2 3 2x x 22 3x x 23 9 6x x

3 23 2x x x 4 3 22 6 4x x x 4 3 22 5 2 11 6x x x x

compute & addpartial products

32 222 4 3 22 33 6 3 9 43 222 6x xx x x xx xx xx x

Factoring: Guess & CheckThough there are some general templates for factoring

polynomials, it all comes down to guessing an answer then checking by multiplying the trial factors.

Using guess & check factor the following:2 3 2x x 22 3x x 23 3 36x x 26 7 2x x

How do you check?

Factoring: Templates & Heuristics

Binomial Squared: Difference of Squares:FWIW: Sum of Squares:Difference of Cubes:Sum of Cubes:

Heuristics

2 2u v u v u v 22 22u uv v u v

2 2u v u vi u vi

3 3 2 2u v u v u uv v 3 3 2 2u v u v u uv v

2 _ __( ) _( )x bx x xa c Factors of a

Factors of c

products of insideand outside terms must sum to b

Slope = Rate of ChangeA straight line is determined by its slope m (a constant rate of change) and a point (x0,y0)

Slope :

Slope m > 0 – line is increasingSlope m < 0 - line is decreasing

Horizontal lines have zero slope (m = 0);Vertical lines have undefined (∞?) slopes

Parallel lines have equal slopesPerpendicular lines have slopes which are negative

reciprocals

2 1

2 1

y rise y y

x run xm

x

Equations for LinesPoint (x0,y0) – Slope (m) Form:

Slope (m) - Intercept (0,b) Form:

Find the equation of a line1. Given two points P = (-2, 3) and Q = (3, -6)2. Given a slope m = - 3 and a point (4, 1)3. Given an slope m = 2 and the y-intercept (0, 1)

00 0

0

y yy x x y

x xm m

0

y by x

xm bm

check with your grapher

Review/ Recall - Linear Functions

Slope-Intercept form: Point Slope Form:

Slope of a Line:

To find the equation of any line you need the slope m and a point (x0,y0) . Use the point slope form then convert to slope intercept form.

y mx b 0 0y m x x y

1 0

1 0

y y y rise

x x x run

II. Solving Quadratics – FactoringThis works best if the quadratic is easy to factor Example: Solve

FactorZero Factor Property: A product equals 0 if and only if one of its factors is zero so it follows that either or so solving the zeros are

III. Solving Quadratics – Extracting Square Roots This only works if the quadratic is in the proper formatSolve Take the square root of both sides and solve

2 2 0x x 2 1 0x x

2 0x 1 0x 2 1x and x

2 22 1 9 0 2 1 9x or x

2 1 9 2 1 3 2 1x x x or

i.e. difference of two squares

IV a. Solving Quadratics – Completing the SquareTo solve (note the particular form of the quadratic and the missing for the term) add to both sides of the equation and factor the left side

Solve this by extracting the square root - see previous slide

Example: …

2x bx c a 2x 2/ 2b

2 22

2 2

b bx bx c

2 2

2 2

b bx c

22 3 1 0x x 2 3 1

2 2x x

IV b. Vertex Form of Quadratic

Where if parabola opens up if parabola opens downAnd are coordinates of vertex - observed that when , depending on whether the parabola opens up or down, this is the minimum or maximum point on the curve.

Therefore the vertex form is easy to sketch by hand!

2 kay x h

0a 0a

,h k,x kh y

Finding Vertex FormExpand And match coefficients. For example

Solve the system

Sketch the graph – vertex? opens up or down? Check with your grapherFind the roots (if any) Try

2 2 22x xa k kah ha ahx

2 2 22 3 5 2x x xa ah kx ah

2

2

2 3

5

h

h

a

a

a k

22 12 8y x x

V. Solving Quadratics – Quadratic Formula

Given use the quadratic formula

So …

Deriving the Quadratic Formula from the Vertex Form

Match coefficients and solve . . .

22 3 5 0x x 2 4

2

b b acx

a

22 2 22kx bx c x xa a a ha ax kh h

2

2b

c

a

ah k

h

Can you complete the derivation?

Quadratic Functions

general form vertex form

Example: Convert to vertex form

In the vertex form (h, k) is the vertex of the quadratic, the maximum or minimum value depending on whether the quadratic opens down (a < 0) or opens up (a > 0). Moreover …

22ax x a x hcb k 2 22ah ah kax x

2y a x h k vertical shift

horizontal shift

vertical stretch

if a < 0 then x-axis reflection

2

2b ah

c ah k

22 7 3x x

3 Step ProcessExpandEquate

Solve the System

Finding the Vertex Form:

Example:

Completing the Square (when a = 1): add & subtract

Example:

2 2 22y a x h k ax ahx ah k

222 3 5y x x a x h k

2 22 2 5 55 8 5 8

2 2y x x x x

2

2b

perfect square = 2

52

x

Useful Facts About Quadratics

It’s easy to compute the zeros of the quadratic from the vertex form – why?

The vertex of a quadratic (h, k) is the unique maximum/minimum value depending on whether the quadratic opens down or up

The x-coordinate of the vertex of a quadratic (i.e. h) is always the midpoint of its two roots

Find (h,k) for y = 3(x + 1)(x - 5)Any 3 non-linear points in the plane uniquely

determine a quadratic (see next panel)

Quadratic Inequalities

To solve find the zeros!

Zeros at and and the parabola opens up so the answer is

However if the inequality were the answer would be

Example: Solve

2 2 0x x 2 2 2 1x x x x

2x 1x 1, 2

2 2 15 0x x

2 2 0x x , 1 2,

Why up?The vertex is the midpoint between the zeros

Higher Polynomial Inequalities

Given any polynomial inequality

First factor the polynomial

For each factor create a signed number line and compute the signed product of the signed number lines

3 22 2 0x x x

3 2 2

2

2 2 2 2

1 2 1 1 2 0

x x x x x x

x x x x x

Higher Polynomial Inequalities

Answer:

1x

1x

2x 1

2

1

2

1

1

1 1 2x x x

2, 1 1,

Functions & Their Representations

A functions from a set A to a set B, denoted is a rule or mapping that assigns toevery element a unique element

Uniqueness “means”

Or no one x gets mapped to two different y’s

f

:f A Bx A y B

1 2 1 2&f x y f x y y y

This definition will be a question on the 2nd test

FunctionsRepresenting Functions

Equations, Tables, Graphs, Sets of Ordered PairsDetecting Functions

Uniqueness Criterion Vertical Line Test

Determining Domains No division by 0No square roots of negative numbersRelevant domains

Determining Range (image of x)

Functional Notation

dependent variable independent variable argument

means take the argument, square it and multiply by 3, subtract the argument and add 1

So what is ?

y f x

23 1f x x x

f x h

Composition of FunctionsGiven functions and the composite functions and are defined as follows

Examples: if andEvaluate and

Do the same for and

f x g x

f g x g f x

f g x xgf

g f x xfg

23f x x x 2

2

1

xg x

x

f g x g f x

inner functionouter function

1

3

xf x

x

1 3

1

xg x

x

One-to-One Functions

A function is one-to-one if and only if for each there is a unique such that

That is: implies

Example: Use above to show is 1:1

:f A B y Range f x A

f x y

1 2f x y f x 1 2x x

1

3

xy

x

One-to-One FunctionsA function is one-to-one if and only if for each there is a unique such that

That is: implies .

Detecting 1:1 Functions!

1. Horizontal Line Test - which of the 12 Basic Functions are 1:1?

2. By definition: Show and are 1:1.

:f A B y Range f x A

f x y

1 2f x y f x 1 2x x

1

3

xy

x

5 3y x

One-to-One FunctionsIf is a one to one function, then the inverse function, denoted is the function with domain Ran(f) and range A (i.e. : Ran(f) → A) defined by

Important: Every 1:1 function has an inverse!

Important! - the inverse of f(x) is not the same as the reciprocal of f(x)

:f A B1f

1f b a iff f a b

1f

1f x

1f x

Finding the Inverse

1. Given y as a function of x, swap x’s and y’s2. Solve for yExample: Check :

5 3

5 3

5 3

1 3

5 5

y x

x y

y x

y x

?

1f f x x

1 15 3f f x f x

1 35 35 5

3 3

x

x x

Useful Properties of Inverse Functions

Inverse Reflection Property: and are symmetric with respect to the 45 degree line y = x.

Inverse Composition Rule: One-to-one functions f(x) and g(x) are inverses of each other if and only if

f a b 1f b a

f g x x

g f x x

Example

1.Verify that is one-to-one

2. Find its inverse . Start with

3. Verify and

4. Find the vertical and horizontal asymptotes for both functions. What do you observe about them?

2 1

1

xf x

x

1f x

1f f x x 1f f x x

2 1

1

xy

x

Finite LimitsDefinition of Limit: A function f(x) has a limit L as x approaches c written

if and only if f(x) gets closer and closer to L as x gets closer and closer to c but never equals c.

There is a difference between f(c) the value of a function at c and the limit of f(x) as x approaches c, i.e. the behavior of f(x) near c.

Examples :

limx cf x L

3

lim 2 1 ___x

x

2

2lim ___

1x

xx

2

1

1lim ___

1x

xx

Function Rules for Limits

Constant Rule:

Identity Rule:

Algebraic Rules for Limits

“The limit of the is the of the limits”

Constant Multiple Rule:

limx ck k

limx cx c

sumdifferenceproductquotientpower

sumdifferenceproduct

quotient*power

*provided the denominator is not 0

lim limx c x c

f x xk fk

“transporter rule”

constants don’t change

this is trivial – why?

The Limits of QuotientsThe limit of a quotient is the quotient of the limits

provided the denominator is not zero.

1

11

lim 33lim

1 lim 1x

xx

xx

x x

1

3lim

1x

x

x

2

1

1lim

1x

x

x

2

1

1lim

1x

x

x

: the interesting case!0

0form

* *

ContinuityA function is continuous if it has no holes or breaksA function is continuous at a point c iff exists, exists and the two the same (thus no breaks or holes)

A function is continuous on an interval I if and only if it is continuous at each point c on I

Example is continuous at 2? At 1? At -1?

limx cf x

f c

2 2

2x x

f xx