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# Chapter 3 - PowerPoint PPT Presentation

Chapter 3. Complex Numbers Quadratic Functions and Equations Inequalities Rational Equations Radical Equations Absolute Value Equations. Willa Cather –U.S. novelist.

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Presentation Transcript

• Complex Numbers

• Inequalities

• Rational Equations

• Absolute Value Equations

• “Art, it seems to me, should simplify. That indeed, is very nearly the whole of the higher artistic process; finding what conventions of form and what detail one can do without and yet preserve the spirit of the whole – so that all one has suppressed and cut away is there to the reader’s consciousness as much as if it were in type on the page.

• Complex Numbers

• R = real numbers

• I = imaginary numbers

• C = Complex numbers

• “Positive anything is better than negative nothing.”

• a + bi

• Where a and b are real numbers

• 0 + bi = bi is a pure imaginary number

• a+bi = c + di

• iff

• a = c and b = d

• Add or subtract the real and imaginary parts of the numbers separately.

• “All who have accomplished great things have had a great aim, have fixed their gaze on a goal which was high, one which sometimes seemed impossible.”

• Multiply as if two polynomials and combine like terms as in the FOIL

• Note i squared = -1

• a – bi is the conjugate of a + bi

• The product is a rational number

• Multiply numerator and denominator by complex conjugate of denominator.

• Write answer in standard form

• “A pessimist is one who makes difficulties of his opportunities and an optimist is one who makes opportunities of his difficulties.”

• Use Mode – Complex

• Use i second function of decimal point

• Use [Math][Frac] and place in standard form a + bi

• Can add, subtract, multiply, and divide complex numbers with calculator.

• Algebraically

• This section contains much information

• General Form

• a,b,c,are real numbers and a not equal 0

• Two distinct solutions

• One Solution – double root

• Two complex solutions

• Solve for exact and decimal approximations

• Factoring

• Use zero Factor Theorem

• Set = to 0 and factor

• Set each factor equal to zero

• Solve

• Check

• Graphing

• Solve for y

• Graph and look for x intercepts

• Can not give exact answers

• Can not do complex roots.

Solving Quadratic Equations #3Square Root Property

• For any real number c

• 1. Use LCD and remove fractions

• 2. Isolate the squared term

• 3. Use the square root property

• 4. Determine two roots

• 5. Simplify if needed

• “Act as if it were impossible to fail.”

• Make one side of the equation a perfect square and the other side a constant.

• Then solve by methods previously used.

• 1. If necessary, divide so leading coefficient of squared variable is 1.

• 2. Write equation in form

• 3. Complete the square by adding the square of half of the linear coefficient to both sides.

• 4. Use square root property

• 5. Simplify

• Solve quadratic equations using the technique of completing the square.

• “Aerodynamically, the bumble bee shouldn’t be able to fly, but the bumble bee doesn’t know it so it goes flying anyway.”

College AlgebraVery Important Concept!!!

• The

• Formula

• Derive

• the

• For all a,b, and c that are real numbers and a is not equal to zero

• “All things are possible until they are proved impossible and even the impossible may only be so, as of now.”

• 1. Factoring

• 2. Square Root Principle

• 3. Completing the Square

• Negative – complex conjugates

• Zero – one rational solution (double root)

• Positive

• Perfect square – 2 rational solutions

• Not perfect square – 2 irrational solutions

• “It is one of man’s curious idiosyncrasies to create difficulties for the pleasure of resolving them.”

CalculatorPrograms

• ALG2

• “Positive thinking is the key to success in business, education, pro football, anything that you can mention. I go out there thinking that I’m going to complete every pass.”

• Solve by Extracting Square Roots

If a,b,c are real numbers and not equal to 0

• Two distinct solutions

• One Solution – double root

• Two complex solutions

• Solve for exact and decimal approximations

• Graphically

• Numerically

• Programs

• ALGEBRAA

• ALG2

• others

D’Alembert – French Mathematician applied, real-life problems.

• “The difficulties you meet will resolve themselves as you advance. Proceed, and light will dawn, and shine with increasing clearness on your path.”

Vertex applied, real-life problems.

• The point on a parabola that represents the absolute minimum or absolute maximum – otherwise known as the turning point.

• y coordinate determines the range.

• (x,y)

Axis of symmetry applied, real-life problems.

• The vertical line that goes through the vertex of the parabola.

• Equation is x = constant

Objective applied, real-life problems.

• Graph, determine domain, range, y intercept, x intercept

Parabola with vertex (h,k) applied, real-life problems.

• Standard Form

Standard Form of a Quadratic Function applied, real-life problems.

• Graph is a parabola

• Axis is the vertical line x = h

• Vertex is (h,k)

• a>0 graph opens upward

• a<0 graph opens downward

Find Vertex applied, real-life problems.

• x coordinate is

• y coordinate is

Vertex of quadratic function applied, real-life problems.

Objective: Find minimum and maximum values of functions in real life applications.

• 1. Graphically

• 2. Algebraically

• Standard form

• Use vertex

3. Numerically

Roger Maris, New York Yankees Outfielder real life applications.

• “You hit home runs not by chance but by preparation.”

Objective: real life applications.

• Solve Rational Equations

• Check for extraneous roots

• Graphically and algebraically

Objective real life applications.

Check for extraneous roots

• Graphically and algebraically

Problem: radical equation real life applications.

Problem: radical equation real life applications.

Problem: radical equation real life applications.

Objective: real life applications.

• Solve Equations

Objective real life applications.

• Solve equations

• involving

• Absolute Value

Procedure:Absolute Value equations real life applications.

• 1.Isolate the absolute value

• 2. Set up two equations joined by “or”and so note

• 3. Solve both equations

• 4.Check solutions

Elbert Hubbard real life applications.

• “Positive anything is better than negative nothing.”

Elbert Hubbard real life applications.

• “Positive anything is better than negative nothing.”

Addition Property of Inequality real life applications.

• If a < b then a + c < b + c

Multiplication property of inequality real life applications.

• If a < b and c > 0, then ac > bc

• If a < b and c < 0, then ac > bc

Objective: real life applications.

• Solve Inequalities Involving Absolute Value.

• Remember < uses “AND”

• Remember > uses “OR”

• and/or need to be noted

Objective: Estimate solutions of inequalities graphically. real life applications.

• Two Ways

• Change inequality to = and set = to 0

• Graph in 2-space

• Or Use Test and Y= with appropriate window

Objective: real life applications.

• Solve Polynomial Inequalities

• Graphically

• Algebraically

• (graphical is better the larger the degree)

Objectives: real life applications.

• Solve Rational Inequalities

• Graphically

• algebraically

• Solve models with inequalities

Zig Ziglar real life applications.

• “Positive thinking won’t let you do anything but it will let you do everything better than negative thinking will.”

Zig Ziglar real life applications.

• “Positive thinking won’t let you do anything but it will let you do everything better than negative thinking will.”

Mathematics 116 Regression real life applications.Continued

• Explore data: Quadratic Models and Scatter Plots

Objectives real life applications.

• Construct Scatter Plots

• By hand

• With Calculator

• Interpret correlation

• Positive

• Negative

• No discernible correlation

Objectives: real life applications.

• Use the calculator to determine quadratic models for data.

• Graph quadratic model and scatter plot

• Make predictions based on model

Napoleon Hill real life applications.

• “There are no limitations to the mind except those we acknowledge.”