Points & Polynomials

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

Example: Find Quadratic through (-1,25) (0,17) (2,7) • Plug in Points: • Solve System

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

Points & Polynomials