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## PowerPoint Slideshow about ' Points & Polynomials' - kelly-hill

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Agenda: Hodgepodge Day

- Homework Questions?
- Difference Quotient
- Continuity
- Zero Product Property
- How many points?
- Viete Relations
- Intermediate Value Theorem
- Challenge Problem

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

- 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

- 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 What is the vertex of y= x2+5x−2? Let h=0 Set DQ to 0 and solve Use x to find y from original

- Find the difference quotient of y= x2+5x−2
- y’ = 2x+5 +h

y’ = 2x + 5

2x + 5 = 0 → x = −2.5

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

Vertex is (−2.5, −8.25)

Continuity

- Theorem: All polynomials are continuous
- This is not a polynomial

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 & 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 to find a quadratic? Can any 3 points be used to find a quadratic? How many points in general to find an nth degree polynomial?

- How many points to find a line?
- 2 points – Point Slope Equation

- 3 points – Simultaneous Equations

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

- n+1 points

Example: Find Quadratic

- Find Quadratic through (-1,19) (0,12) (3,3)
- General Form:

Viete’s Formulae

- Polynomial Patterns
- Given

Example

- Find Equation of Cubic with zeroes of 4, 2, -3
- General Form:

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:

- 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

- 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

- The quartic function
has four total roots (2 double roots).

What is p+q?

Homework

- Pg 150 #86,88
- Pg 152 #102
- Supplement on Web

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