Interval Notation Interval Notation to/from Inequalities Number Line Plots open & closed endpoint conventions Unions and Intersections Bounded vs. unbounded include endpoints

Distributive Property: Multiplying Polynomials & FOIL Vertical Form Multiplication: Horizontal Form: compute & add partial products

Factoring: Guess & Check Though 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: How do you check?

Factoring: Templates & Heuristics Binomial Squared: Difference of Squares: FWIW: Sum of Squares: Difference of Cubes: Sum of Cubes: Heuristics Factors of a Factors of c products of inside and outside terms must sum to b

Slope = Rate of Change A straight line is determined by its slope m (a constant rate of change) and a point (x 0,y 0 ) Slope : Slope m > 0 – line is increasing Slope m < 0 - line is decreasing Horizontal lines have zero slope (m = 0); Vertical lines have undefined (∞?) slopes Parallel lines have equal slopes Perpendicular lines have slopes which are negative reciprocals

Equations for Lines Point (x 0,y 0 ) – Slope (m) Form: Slope (m) - Intercept (0,b) Form: Find the equation of a line 1.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) 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 (x 0,y 0 ). Use the point slope form then convert to slope intercept form.

II. Solving Quadratics – Factoring This works best if the quadratic is easy to factor Example: Solve Factor Zero 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 format Solve Take the square root of both sides and solve i.e. difference of two squares

IV a. Solving Quadratics – Completing the Square To 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: …

IV b. Vertex Form of Quadratic Where if parabola opens up if parabola opens down And 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!

Finding Vertex Form Expand And match coefficients. For example Solve the system Sketch the graph – vertex? opens up or down? Check with your grapher Find the roots (if any) Try

V. Solving Quadratics – Quadratic Formula Given use the quadratic formula So … Deriving the Quadratic Formula from the Vertex Form Match coefficients and solve... 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). Moreover … vertical shift horizontal shift vertical stretch if a < 0 then x-axis reflection 3 Step Process Expand Equate Solve the System

Finding the Vertex Form: Example: Completing the Square (when a = 1): add & subtract Example: perfect square =

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 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

Higher Polynomial Inequalities Answer:

Functions & Their Representations A functions from a set A to a set B, denoted is a rule or mapping that assigns to every element a unique element Uniqueness “means” Or no one x gets mapped to two different y’s This definition will be a question on the 2 nd test

Functions Representing Functions Equations, Tables, Graphs, Sets of Ordered Pairs Detecting Functions Uniqueness Criterion Vertical Line Test Determining Domains No division by 0 No square roots of negative numbers Relevant 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 ?

Composition of Functions Given functions and the composite functions and are defined as follows Examples: if and Evaluate and Do the same for and

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

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. 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.

One-to-One Functions If 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)

Finding the Inverse 1.Given y as a function of x, swap x’s and y’s 2.Solve for y Example: Check :

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

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?

Finite Limits Definition 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 :

Function Rules for Limits Constant Rule: Identity Rule: Algebraic Rules for Limits “The limit of the is the of the limits” Constant Multiple Rule: sum difference product quotient power sum difference product quotient* power *provided the denominator is not 0 “transporter rule” constants don’t change this is trivial – why?

The Limits of Quotients The limit of a quotient is the quotient of the limits provided the denominator is not zero. : the interesting case! * *

Continuity A function is continuous if it has no holes or breaks A 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?