Chapter 2: Polynomial and Rational Fncs.
2.1 Quadratic Fncs.
A. OBJECTIVE: to analyze and sketch the graph of a quadratic; to write the
equation for a quadratic fnc.; to find the max/min of a quadratic.
B. FACTS/FORMULAS:
1. Quad. Fnc. has degree 2 (X2)
2. A quadratic function is a function that can be written in the
standard form y = ax2+bx+c where a ≠ 0. The graph of a quadratic
equation is a parabola and is “U” shaped.
3. The lowest or highest point on a parabola is the vertex.
4. The axis of symmetry divides the parabola into mirror images and
passes through t he vertex.
5.
For y = ax2+bx+c, the vertex’s y-coordinate is the minimum
valueof the function if a > 0 and the maximum value if a < 0.
6. The vertex form of a quadratic equation is given by
y = a(x - h)2 +k.
7. The intercept form of a quadratic equation is given by
y = a(x - p)(x - q).
C.
Ex:
2.2 Higher Degree Polynomials
A. OBJ: to sketch and analyze graphs of polynomial equations; to
find the intercepts; to evaluate end behaviors and to graph
transformations
B. FACTS:
1.
Graph of a polynomial equation is a smooth continuous
curve.
2. F(x) = xn is either even or odd
a) If it is even, the end behavior on both sides are either
approaching ±∞
b) If it is odd, the end behavior is +∞ on one side and -∞
on the other side.
3.
Zeros of a fnc = x-intercepts, root/solution, and a factor
(x-a)
a) Double root – repeat zero, graph just touches the xaxis.
C.
Ex:
2.3 Polynomials and Division
A. OBJ: to divide polynomials using long and synthetic division and
to interpret the quotient.
B. FACTS/FORMULAS:
1. Long divison:
a) F(x) = d(x) * q(x) + r(x)
1) Divisor * quotient + remainder
b) F(x)/d(x) = q(x) + r(x)/d(x)
2.
Synthetic division:
a) Works only for (x-k) divisors
3. Missing powers must be replaced with zero as a placeholder.
4. Remainder Thm = direct substitution
a) If f(x) ÷ (x-k), then r = f(k)
b) If r = 0, then k is a zero of the fnc and (x-k) is a factor of
f(x).
C. Ex:
2.4 Complex #’s
A. OBJ: to identify imaginary unit “i”; to +/-/*/÷ complex #’s; to find
complex solutions to quadratic equations.
B. FACTS/FORMULAS:
Imaginary unit “i” = √−1 and i2 = -1
Complex #’s = a ± bi = real unit ± imaginary unit
a) +/- using like terms
b) * all terms
c) ÷ using conjugates
1.
2.
3.
C.
Ex:
1) √𝑎 ∗ √𝑏 ≠ √𝑎𝑏 if a<0 and b<0
Solve quadratics using factoring or quad. form.;
graphing will not give imaginary solutions.
2.5 Zeros of Polynomial Fncs..
A. OBJ: to find rational zeros of a fnc.; to determine the number of
zeros; to write an equation given the zeros.
B. FACTS:
1. Degree of the polynomial determines the number of zeros
2. Imaginary solutions come in pairs
C.
𝑝
=
𝑎0
3.
Rational Zero Test:
4.
Descarte’s Rule of Signs depends on the variations of signs
of f(x) and f(-x)
𝑞
𝑎𝑛
2.6 Rational Fncs
A. OBJ: to identify the domain of a rational fnc.; to graph a rational
fncs. and identify the vertical and horizontal asymptotes.
B. FACTS/FORMULAS:
Rational fnc =
2.
Domain is all real numbers except the values that make the
denominator = zero.
Asymptotes
a) Vertical when x = a, f(x) → ±∞
b) Horizontal when y = f(x) = b, x → ±∞
3.
C.
𝑁(𝑥)
1.
Ex:
𝐷(𝑥)
2.7 Nonlinear Inequalities.
A. OBJ: to solve polynomial and rational inequalities
B. Facts/Formulas:
1. For polynomial inequalities, find the zeros of the fnc. and
test the intervals to the left, right and between the zeros.
2. Solutions are written as intervals vs. compound inequalities
3. Exceptions to the rule: one, none or many solutions.
4. For rational inequalities, set one side = to zero, find the LCD
and solve for key values (when numerator and denominator
= zero).
C. Ex: