4.1 Quadratic Functions and Models Hwk: Std (1, 6, 9, 14, 17, …), +62, -61
Basics:
the rule of the function f(x) = ax^2+bx+c can be rewritten in the form f(x) = a(x+h)^2 + k
by completing the square
The graph of f is a parabola with vertex (h,k).
where h = -b/(2a).
Opens upward if a>0 and downward if a<0
Vertex by compl sq or by -b/2a
Ident graph using transformations
Build func using successive xfm (#35)
Find "rule of the func" given vertex and one point
Graph and determine vertex and x-intercpets exactly
App: #38 How many sold to maximize profit? What is the max profit?
freefall eqn: find max h, and at what time it occurs, and when does it hit ground #43
econ func example #50
Excursion
4.2A Synthetic Division Hwk:
Odds #1-15
Show technique --to find quotient and remainder
Ex with fraction
Show that 1st polynomial is a factor of the second (so you are factoring the polynom)
4.2 Polynomial Functions and Roots Hwk: Std (1, 6, 9, 14, 17, …)
Also =
4.3 Graphs of Polynomial Functions Hwk: Std (1, 6, 9, 14, 17, …), -37a, 49
4.4 Polynomial Models OMIT
4.5 Rational Functions Hwk: Worksheet (Lab 3)
4.6 Complex Numbers Hwk: Std (1, 6, 9, 14, 17, …), -6, 29, 34, +8. For #74, see notation in #73
+, -
X
powers of i. Divide by 4, take only the remainder. Find i271
/ and recip: Mult by conjugate of denom.
Solve 1 equ for 2 vars e.g. 5x-3yi = 3-7i
Solve x^4=4 and write sols in form a+bi
Find (1+i)2 (it equals 2i); divide by 2 and take sqrt to show sqrt(i) = 1/sqrt(2)*(1+i)
Find a formula for z*zbar
4.7 Fundamental Thm of Algebra Hwk: Std (1, 6, 9, 14, 17, …),
X1-6 use remainder thm [i.e. rmdr on syndiv by c or long div by x-c is f(c)]
Ex: X3 find the rmdr when f(x)=3x4-6x3+2x-1 is div by g(x)=x+1 w/o syn div or long div
The point: the root c corresp to the factor x-c
Ex: X like 9: Find roots and multiplicity: 5x12(x+p)3[x-(1-sqrt(3))]
FTA: Every nonconstant polynom has a root in the complex nr system
Consequence: Every nonconstant polynom has exactly n roots in the complex nr system
[Linear] factorization over the complex nrs: f(x) = d(x-c1)(x-c2)…(x-cn)
Ex X20: Find all the roots in the complex nr syst and write f(x) as a product of linear factors
f(x) = x^4-x^2-6
Complex conjugate roots thm: Let f(x) be a polynom with real coeffs. If the complex nr z = a+bi is a root of f(x), then its conjugate z-bar = a-bi is also a root of f(x).
Ex: Find the polynom with real coeffs that satisfies the conditions (don't mult out):
Roots include 5 and 2-i, degree 3, f(1)=3 [two possibilities, w pos and neg leading coeff]
Consequence: Every nonconst polynom with real coeffs can be factored into a product of linear and irreducible quadratic factors (irreducible in the sense that the quadratic factors have no real roots)
Why? The product (x-z)(x-zbar) has real coeffs: (x-a-bi)(x-a+bi) = (x-a)^2 - b^2 i is gone.
Ex: Find the irreducible quadratic factor that comes from the root 2-i and its conjugate
Ex: X55: 2-i is given as a root of x4-4x3+6x2-4x+5; find all the roots
Return to: Merced College; Don Power Updated 08/16/05 by Don Power