interval notation interval notation to/from inequalities number line plots open & closed...
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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