MAT129 Fall 2024 Notes

Monday  October  7	 Topics:	5.1, 5.2, 5.3, 5.4, 5.5, 5.7			
Notes:	10.07a, 10.07b, 10.07c, 10.07d
5.1 - Quadratic Functions, pg. 344 (see day 2),  Learning Objectives
	* Recognize characteristics of parabolas.
	* Understand how the graph of a parabola is related to its quadratic function.
	* Determine a quadratic function’s minimum or maximum value.
	* Solve problems involving a quadratic function’s minimum or maximum value.
5.2  - Power Functions and Polynomial Functions, pg. 360,  Learning Objectives 
	* Identify power functions.
	* Identify end behavior of power functions.
	* Identify polynomial functions.
	* Identify the degree and leading coefficient of polynomial functions.
5.3  - pg 375,  Learning Objectives
	* Recognize characteristics of graphs of
	* Use factoring to find zeros of polynom
	* Identify zeros and their multiplicities.
	* Determine end behavior.
	* Understand the relationship between d
	* Graph polynomial functions.
	* Use the Intermediate Value Theorem.
5.4  - pg 393,  Learning Objectives
	* Use long division to divide polynomials.
	* Use synthetic division to divide polynomials.
5.5  - pg 402,  Learning Objectives
	* Evaluate a polynomial using the Remainder Theorem.
	* Use the Factor Theorem to solve a polynomial equation.
	* Use the Rational Zero Theorem to find rational zeros.
	* Find zeros of a polynomial function.
	* Use the Linear Factorization Theorem to find polynomials wi
	* Use Decartes' Rule of Signs.
	* Solve real-world applications of polynomial equations.
5.7 - pg 435
	* Find the inverse of an invertible polynomial function.
	* Restrict the domain to find the inverse of a polynomial function.

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Quadratic Functions

Notes

EVERYTHING YOU EVER WANTED TO KNOW ABOUT A QUADRATIC BUT WERE AFRAID TO ASK

      Quadratics In General

      Ways to Solve Quadratics

      Quadratic Equation, Formula, & The Discriminant

      Completing the Square

A Line times a Line Is A Parabola

Solve a Quadratic by Completeing the Square Instructions gif

Squaring and Square Root Functions

quadratic equation

Uses of Quadratics:

      To Create Other Functions

      To Express Projectile Motion

 

Images

How the general equation y=ax² + bx + c and the standard equation y=a(x-h)² +k relate

 

Games or Manipulatives

Oscar Reutersvard's Penrose Square -- Solve the quadratic. Connect the dots.

          easy print pdf

 

Sketch Pads -- Dynamic Analytic Geometry Parabols, Quadratic Functions parabola   -- Multi page: 2 parabolas: 1 in general form, 1 in standard forms,     * both generated by equations defined by parameters.     * WITH SHOW/HIDE ACTION BUTTON     * graphs of 2 parabolas: 1 in general form, 1 in standard forms,     * 1 parabola, set a, then generat parabola w/2 points.     * 1 parabola controlled by 3 points, used for determinant analysis     * 1 vertical axis of symmetry, 1 horizontal "axis of symmetry" TracePoly   -- drag point f(x) to state and show new ( x, f(x))

Spread Sheets -- Dynamic Computation polynomial spreadsheet --     * graph polynomial function written in factored form     * complete synthetic division computation quadratic --     * enter a, b, c, get x1, x2, h, k     * enter a, h, k, get x1, x2, a, b, c, and the discriminant

Forms Web Pages -- Dynamic Computation Computes Quadratic Equation, Formula, & Discriminant

Polynomial Definition & Notation Simple definition (for algebra I and geometry):   polynomial   Precalc definition: an algebraic expression or function, named by its highest degree, produced by the MULTIPLICATION of constants, monomials, binomials, ..., with the use of one variable x,   anxn + an-1xn-1 + an-2xn-2 + ... +a2x2 + a1x1 + a0x0,   in which the constants an, an-1, an-2, ... , a2, a1, a0 are real, and serve as coefficients or the constant term, and in which all exponent of x are counting numbers.

Notes

Polynomial Functions: Notes

      Definition & Notation

      Polynomial Computation

      Possible Roots

      Quadratic, Cubic, Quartic Formulas

      Complex Roots

Polynomial Functions

      How to sketch a polynomial function

      Examples of sketching polynomial functions

      Problems to sketch and their solution

Intercepts/Roots/Factors Vocabulary & Images

Games and Manipulatives

Who Has? Polynomial F(x) Game CHANGE LINK

 

Spread Sheets -- Dynamic Computation polynomial spreadsheet --     graph a polynomial function written in factored form     complete synthetic division synthetic division spreadsheet -- just complete synthetic division of a polynomial or of a quadratic by a linear binomial rational function spread sheet     Graph and table of (Axm)/(Bxn)     Watch the power of the top or bottom equal or dominate the other.     Examine hoirzontal asymptotes or infinite increase or decrease.     Think endbehavior and explore.     Graph and table of rational function, polynomial A(x) divided by B(x).     Examine asymptotes.     Examine the role of factors A(x) and 1/B(x).     Graph and table of rational functions written as A(x) / B(x) + C(x)     Really play with C(x), the asymptote     Complete synthetic division and rewrite of quotient and remainder.

            Polynomial functions notes

            parabola.gsp

            zeros definition

            TracePoly gps   -- input coefficients, drag point f(x) to state and show new ( x, f(x))

            polynomial spreadsheet -- graph a polynomial function written in factored form, complete synthetic division

            poly.xls

• Graph a polynomial defined by degrees and coefficients.
  y= a_nxⁿ + a_{n -1}x^{n -1} + a_{n - 2}x^{n -2} + a_{n - 3}x^{n -3} + a_{n - 4}x^{n - 4} + a_{n - 5}x^{n - 5} + a_n-6x^{n - 6} + a_{n -7}x^{n -7}
• Graph a polynomial displayed in factored form.
  y = A(x - b)^B(x - c)^C(x - d)^D
• Synthetically divide and then view the quotient polynomial and remainder.
  Divide (a_nxⁿ + a_{n -1}x^{n -1} + a_{n - 2}x^{n -2} + a_{n - 3}x^{n -3} + a_{n - 4}x^{n - 4} + a_{n - 5}x^{n - 5} + a_n-6x^{n - 6} + a_{n -7}x^{n -7}) ÷ (x-c)
• Solves linear or quadratic equations. Multiplies 2 binomials or 3 binomials.

            parabola.gsp  -- Multi page: 2 parabolas: 1 in general form, 1 in standard forms,

• both generated by equations defined by parameters.
• WITH SHOW/HIDE ACTION BUTTON
• graphs of 2 parabolas: 1 in general form, 1 in standard forms,
• 1 parabola, set a, then generat parabola w/2 points.
• 1 parabola controlled by 3 points, used for determinant analysis
• 1 vertical axis of symmetry, 1 horizontal "axis of symmetry

     

5.5 Zeros of Polynomial Functions, pg. 402

      Possible Roots, Factor Theorem, Rational Zero Theorem

     

5.4 Dividing Polynomials, pg. 393
      synthetic division spreadsheet -- synthetic division of a polynomial or of a quadratic by a linear binomial

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