CH 3 : Quadratic Functions

Factorization Theorem for a Polynomial With Real Coefficients

Every polynomial with real coefficients can be uniquely factored over the real #s as a product of linear factors and/or irreducible quadratic factors.

Finding the Domain of a Rational Function

Find domain of: f(x) = 3x²-12/x-1 Domain = all real #s except x ≠ 1 Interval notation (-∞, 1) ∪ (1, ∞) g(x) = x/x²-6x+8 D(x) = x²-6x+8 = 0 (x-4)(x-2) x = 4, 2 Domain = all real #s except x ≠ 2, 4 Interval notation (-∞,2) ∪ (2,4) ∪ (4, ∞)

Min & Max Values

k is the min value for any parabola that opens up. OR k is the max value for any parabola that opens down.

Descartes's Rule of Signs

Used to find the possible # of positive and negative zeros of a polynomial function. http://www.purplemath.com/modules/drofsign.htm

Volume

V = πr²h

Pick Up on 3.7 Polynomial and Rational Inequalities

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Rational Function

A function f that can be expressed in the form f(x) = N(x)/D(x) where the numerator N(x) and the denominator D(x) are polynomials and D(x) is not the zero polynomial, is called a rational function. The domain of f consists of all real number for which D(x) ≠ 0

Locating Horizontal Asymptotes of Rational Functions

A horizontal line is an asymptote only to the far left and the far right of the graph. "Far" left or "far" right is defined as anything past the vertical asymptotes or x-intercepts. Horizontal asymptotes are not asymptotic in the middle. It is okay to cross a horizontal asymptote in the middle. The location of the horizontal asymptote is determined by looking at the degrees of the numerator (n) and denominator (m). 1. If n<m, the x-axis, y=0 is the horizontal asymptote. 2. If n=m, then y=a_n_/b_m_ is the horizontal asymptote. That is, the ratio of the leading coefficients. 3. If n>m, there is no horizontal asymptote. However, if n=m+1, there is an oblique or slant asymptote. Examples: Find the horizontal asymptote of the graph of each rational function: a. f(x) = 5x+2/1-3x The numerator and denominator are both a degree of 1. By Rule 2, the line y = 5/-3 = -5/3 is the horizontal asymptote of the graph of f. b. h(x) = 2x/x²+1 The degree of the numerator is 1 and that of the denominator is 2. By Rule 1, the line y = 0 (the x-axis) is the horizontal asymptote.

Polynomial Factor

A polynomial D(x) is a factor of a polynomial F(x) if there is a polynomial Q(x) such that F(x) = D(x) × Q(x) Division Algorithm Example: x³-2 = (x-1)(x²+x+1) - 1 x³-2/x-1 = x²+x+1 - 1/x-1, x ≠ 1 Dividend = x³-2 Divisor = x-1 Quotient = x²+x+1 Remainder = -1

Real Zeros of a Polynomial Function

A polynomial function of degree n with real coefficients has, at most, n real zeros. Example: Find the number of distinct real zeros of the following polynomial functions: → f(x) = (x-1)(x+2)(x-3) = 0 x = -2, 1, 3 f(x) has 3 real zeros: 1, -2, and 3 → g(x) = (x+1)(x²+1) = 0 x = -1 Because x²+1 > 0 for all real x, the equation x²+1 = 0 has no real solution. So -1 is the only real zero of g(x)

Complex Polynomial & Complex Zero

A polynomial with complex coefficients. If P(z) = 0 for a complex number z, we say that z is a zero or a Complex Zero of P(x). In the complex # system, every nth-degree polynomial equation has exactly n roots AND, every nth-degree polynomial can be factored into exactly n linear factors.

Odd-Degree Polynomials with Real Zeros

Any polynomial P(x) of odd degree with real coefficients must have at least one real zero.

Number of Zeros Theorem

Any polynomial of degree n has exactly n zeros, provided a zero of multiplicity k is counted k times. Example (Constructing a Polynomial Whose Zeros are Given): Find a polynomial P(x) of degree 4 with a leading coefficient of 2 and zeros -1, 3, i, and -1. Write P(x) a. in completely factored form b. by expanding the product found in part a. a. P(x) = a(x-r₁) (x-r₂) (x-r₃) (x-r₄) P(x) = 2[x- (-1)] (x-3) (x-i) [x- (-i)] P(x) = 2(x+1) (x-3) (x-i) (x+i) b. P(x) = 2(x+1) (x-3) (x-i) (x+i) P(x) = 2(x+1) (x-3) (x²+1) P(x) = 2(x+1)(x³-3x²+x-3) P(x) = 2(x⁴-2x³-2x²-2x-3) P(x) = 2x⁴-4x³-4x²-4x-6

Axis Directions

End Behavior: x approaches infinity : x → ∞ This means x gets larger and larger without bound x approaches negative infinity x → -∞ This means that x can assume values less than -1, -10... without end. Summary: x → ∞ Left x → -∞ Right y → -∞ Down y → ∞ Up

Factorization Theorem for Polynomials

If P(x) is a complex polynomial of degree n ≥ 1 with leading coefficient a, it can be factored into n (not necessarily distinct) linear factors of the form P(x) = a(x - r₁)(x - r₂)...(x-r_n_) Where a, r₁, r₂,..., r_n_ are complex #s.

Conjugate Pairs Theorem

If P(x) is a polynomial function whose coefficieents are real numbers and if z = a+bi is a zero of P, then its conugate, z⁻ (line over z) = a-bi, is also a zero of P. Example: A polynomial P(x) of degree 9 with real coefficients has the following zeros: 2, of multiplicity 3; 4+5i, of multiplicity 2; and 3 - 7i. Write all nine zeros of P(x) Because complex zeros occur in conjugate pairs, the conjugate 4-5i of 4+5i is a zero of multiplicity of 2 and the conjugate 3+7i of 3-7i is a zero of P(x). The nine zeros of P(x), including the repetitions, are 2,2,2,4 + 5i, 4 + 5i, 4 - 5i, 4 - 5i, 3 +7i, and 3 - 7i

Locating Vertical Asymptotes of Rational Functions

If f(x) = N(x)/D(x) is a rational function, where N(x) and D(x) don't have a common factor and a is a real zero of D(x), then the line with equation x = a is a vertical asymptote of the graph of f. This means that the vertical asymptotes (if any) are found by locating the real zeros of the denominator Example: Find all vertical asymptotes of the graph of each rational function: a. f(x) = 1/x-1 There are no common factors in the numerator and denominator and the only zero of the denominator is 1. Therefore, x = 1 is a vertical asymptote of f(x). b. h(x) = 1/x²+1 Because the denominator x²+1 has no real zeros, the graph of h(x) has no vertical asymptotes.

Number of Turning Points

If f(x) is a polynomial of degree n, then the graph of f has, at most, (n-1) turning points. http://www.mathsisfun.com/algebra/polynomials-behave.html

Zeros of a Function

Let f be a function. An input (c) in the domain of (f) that produces output 0 is called a zero of the function Ex: f(x) = (x-3)², then 3 is a zero of (f) . In other words, a number (c) is a zero of (f) if f(c) = 0 - If f is a polynomial function and c is a real number, then the following statements are equivalent: 1. c is a zero of f 2. c is a solution of the equation f(x) = 0 3. c is an x-int of the graph of x. The point (c,0) is in the graph of f. Example: Find all real zeros of this polynomial function: f(x) = x³+2x²-x-2 (y-int = (0,-2)) f(x) = (x³+2x²) - (x+2) = x² (x+2) - 1 (x+2) = (x+2) (x²-1) = (x+2)(x+1)(x-1) (x+2)(x+1)(x-1) = 0 x = -2, -1, 1

Complex Zeros of a Polynomial Function

P(i) = i² +1 = (-1) +1 = 0 Consequently, i is a complex # for which P(i) = 0 That is, the complex zero of P(x) is i.

Graphing a Revenue Curve

The revenue curve for an economy of a country is given by R(x) = x(100-x)/x+10 Where x is the tax rate in % and R(x) is the tax revenue in billions of dollars.

Variation of Sign

When the terms of a polynomial are written in descending order, we say that a variation of sign occur when the signs of two consecutive terms differ. Example: In this polynomial 2x⁵-5x³-6x²+7x+3, the signs of the terms are + - - + +. Therefore there are two variation of signs as follows: [+ -] [- +] +

Polynomial Function of Degree n

f(x) = a_n_xⁿ + a_n-1_xⁿ⁻¹ + ... + a₂x² + a₁x + a₀ Where n is a nonnegative integer and the coefficients a_n_, a_n-1_, ..., a₂, a₁, a₀ are real #s with a_n_ ≠ 0 Leading term = a_n_xⁿ Leading coefficient = a_n_ (coef. of xⁿ) Constant term = a₀ A constant function f(x) = a(a≠0) may be written as f(x) = ax⁰ is a polynomial of degree 0. Polynomials of degree 3, 4, and 5 are also called cubic, quartic, and quintic polynomials respectively.

Quadratic Function

f(x) = ax²+bx+c Where a, b, and c are real #s with a ≠ 0

Standard Form of a Quadratic Function

f(x) = ax²+bx+c can be rewritten as f(x) = a(x-h)²+k a ≠ 0

Power Function

f(x) = axⁿ This is called a power function of degree n, where a is a nonzero real # and n is a positive integer. → Power functions of even degree = symmetry with respect to the y-axis (flipped across y-axis) → Power function of odd degree = symmetry with respect to the x-axis (flipped across x-axis)

The Leading-Term Test

http://hotmath.com/hotmath_help/topics/leading-coefficient-test.html

Asymptotes

http://www.coolmath.com/precalculus-review-calculus-intro/precalculus-algebra/17-rational-functions-finding-vertical-asymptotes-01.htm (the y-axis is a vertical asymptote and the x-axis is a horizontal asymptote in the graph on this website)

Oblique Asymptotes

http://www.isu.edu/~laquerht/classes/slant.pdf

Long Division & Synthetic Division

http://www.kkuniyuk.com/M1410203.pdf

Intermediate Value Theorem

http://www.mathsisfun.com/algebra/intermediate-value-theorem.html

Finding the Bounds on the Zeros

http://www.mathsisfun.com/algebra/polynomials-bounds-zeros.html

Remainder Theorem

http://www.mathsisfun.com/algebra/polynomials-remainder-factor.html

Multiplicity of a Zero

http://www.purplemath.com/modules/polyends2.htm

Solving a Polynomial Equation

http://www.purplemath.com/modules/solvpoly.htm

Rational Zeros Theorem

http://www.sparknotes.com/math/algebra2/polynomials/section4.rhtml