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Points & Polynomials. Lecture 3D Pre AP and GT Precalculus. Agenda: Hodgepodge Day. Homework Questions? Difference Quotient Continuity Zero Product Property How many points? Viete Relations Intermediate Value Theorem Challenge Problem. Difference Quotient.

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Points polynomials

Points & Polynomials

Lecture 3D

Pre AP and GT Precalculus


Agenda hodgepodge day
Agenda: Hodgepodge Day

  • Homework Questions?

  • Difference Quotient

  • Continuity

  • Zero Product Property

  • How many points?

  • Viete Relations

  • Intermediate Value Theorem

  • Challenge Problem


Difference quotient
Difference Quotient

  • The slope of a graph’s secant line

  • Difference Quotient of function y is symbolized with a prime after the function name:

  • Difference Quotient can be used to find parabola vertex


Difference quotient example
Difference Quotient Example

  • Find the difference quotient of h(t)=800t − 16t2

    (this is the equation for the height of an object with an initial velocity of 800 mps as it returns to earth)


Using the difference quotient
Using the difference quotient

  • Recall h(t)=800t − 16t2 and DQ = 800 −32t − 16Δt

  • What is the highest elevation this projectile reached?

  • When Δt=0 and h’(t)=0, a parabola its at its vertex

  • So… 800 −32t − 16(0)=0 implies 800 = 32t so max height reached when t=25

  • Max Height is 800(25)-16(252)=10,000


Example
Example

  • Find the difference quotient of y= x2+5x−2

    • y’ = 2x+5 +h

  • What is the vertex of y= x2+5x−2?

  • Let h=0

    y’ = 2x + 5

  • Set DQ to 0 and solve

    2x + 5 = 0 → x = −2.5

  • Use x to find y from original

    x = −2.5 → y = (−2.5)2 + 5(−2.5) − 2 = − 8.25

    Vertex is (−2.5, −8.25)


  • Continuity
    Continuity

    • Theorem: All polynomials are continuous

    • This is not a polynomial


    Zeroes zpp
    Zeroes & ZPP

    • Zero Product Property:

      If a*b*c=0 then a=0, b=0, or c=0

    • What does this mean for polynomials….

    • If p(x)=x(x+2)(x-5)=0 then x=0, x+2=0, or x−5=0

    • So 0, −2, and 5 are zeroes of the polynomial.


    Zeroes zpp1
    Zeroes & ZPP

    • Find a cubic polynomial which has zeroes 2, 3, -1

    • Reflection: Is this the ONLY cubic with those zeroes?

      • No there are many cubics with these zeroes


    How many points does it take to find the equation of an nth degree polynomial
    How many points does it take…To find the equation of an nth degree polynomial?

    • How many points to find a line?

      • 2 points – Point Slope Equation

  • How many points to find a quadratic?

    • 3 points – Simultaneous Equations

  • Can any 3 points be used to find a quadratic?

    • No, you can find a quadratic with any 3 non-collinear points

  • How many points in general to find an nth degree polynomial?

    • n+1 points


  • Example find quadratic
    Example: Find Quadratic

    • Find Quadratic through (-1,19) (0,12) (3,3)

    • General Form:


    Example find quadratic through 1 25 0 17 2 7
    Example: Find Quadratic through (-1,25) (0,17) (2,7)

    • Plug in Points:

    • Solve System


    Viete s formulae
    Viete’s Formulae

    • Polynomial Patterns

    • Given


    Example1
    Example

    • Find Equation of Cubic with zeroes of 4, 2, -3

    • General Form:


    Intermediate value theorem
    Intermediate Value Theorem

    • If p(x) is continuous and if p(a) is positive and p(b) is negative then p(x) has a zero on the interval (a,b)


    Imvt for zero existence
    IMVT for Zero Existence:

    • Establish function is continuous

    • Show that for point a that p(a) is positive

    • Show that for point b that p(b) is negative

    • Say by Intermediate Value Theorem, p(x) must have a zero on the interval (a,b)

    • Note: IMVT only establishes existence, not value


    Imvt example
    IMVT Example

    • Given: Show that the function p(x)=x2+5x−2 has a zero between 0 and 1.

      You Write:

    • p(x) is a polynomial and must be continuous

    • p(0)= −2 and p(1) = 5

    • By the IMVT, p(x) must have a zero on the interval (0,1)

    • Extension: use Quadratic Equation to find that exact zero!


    Challenge problem
    Challenge Problem

    • The quartic function

      has four total roots (2 double roots).

      What is p+q?


    Homework
    Homework

    • Pg 150 #86,88

    • Pg 152 #102

    • Supplement on Web


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