Polynomial Functions
Linear Graphs and Linear Functions
1.3
  Forms for equations of lines (linear functions)
  Ax + By = C
  y = mx +b
  (y – y1) = m(x – x1)
  x = a
  y = b
  Slope, ratio, rate (rate of change)
  Slope formula
y 2 − y1 y1 − y 2
m=
=
x 2 − x1 x1 − x 2
  CAUTION:
€
m≠
y 2 − y1
x1 − x 2
Standard Form
Slope-Intercept
Point-Slope
Vertical Line
Horizontal Line
Parallel and Perpendicular Lines
  Two lines are parallel if they have the same slope (m1=m2)
  Two lines are perpendicular if their slopes are negative
reciprocals (m1 = - 1/m2)
  Linear Extrapolation:
Prediction based on a linear model
An extrapolated point does not lie between the given data points
  Linear Interpolation:
Estimation based on a linear model
An interpolated point lies between the given data points
Polynomial Functions
Quadratic Functions
2.1
Forms for equations of parabolas (quadratic functions)
 ax2+bx+c, where a ≠ 0
Standard Form
 a(x-h)2+k, where a ≠ 0
Vertex Form
 Axis of symmetry: x=h a vertical line
 Vertex: the point (h, k)
  a minimum or a maximum of the function
Completing the Square
  Move the constant term to the right, ignore it for a bit.
  Make sure the leading term is “1”
  If a ≠ 1 then factor out a.
2
b
a x)
+c−
1
a( 2
2
(ax + bx) + c = a(x +
+c
  Add zero -- in a tricky way ;o)
2
a(x +
b
ax
+
1
(2
·
b 2
a) )
  Factor & Simplify (Notice: Now in vertex form)
a(x +
b 2
2a )
+ (c −
b2
4a )
·
b 2
a)
Polynomial Functions
Polynomials of Higher Degree & Division
2.2 & 2.3
Polynomial Function
Let a0, a1, a2, …, an-1, an be real numbers with an ≠ 0,
f(x) = anxn + an-1xn-1 + … + a2x2 + a1x + a0
is a polynomial function of x with degree n.
Polynomials are continuous with smooth rounded turns.
Leading Coefficient Test (LCT)
n even; an>0
n even; an<0
n odd; an>0
n odd; an<0
Real Zeros (Equivalent Statements)
1. x = a is a zero of the function f
2. x = a is a solution of the polynomial equation f(x) = 0
3. (x – a) is a factor of f(x)
4. (a, 0) is an x-intercept of the graph of f
Note: A polynomial function of degree of n, has at most n real
zeros and at most n-1 turning points.
Repeat Roots (Zeros)
  A factor (x – a)k, k > 1 yield a repeated zero x = a of
multiplicity of k.
  If k is odd, the graph crosses at x = a.
  If k is even, the graph touches the x-axis (but does not cross) at
x = a.
Intermediate Value Theorem (IVT)
Let a and b be real numbers such that a < b.
If f is a polynomial function such that f(a) ≠ f(b), then in the
interval [a, b] f takes every value between f(a) and f(b).
Factor Theorem:
A polynomial f(x) has a factor (x – k) iif f(k)=0.
Remainder Theorem:
If a polynomial f(x) is divided by x – k, the remainder is r = f(k).
Long Division always works!
Synthetic Division works only if the divisor is the form (x – k)
Polynomial Functions
Complex Numbers
2.4
Complex Numbers
i = −1
i2 = -1
€
i3 = -i
i4 = 1
a + bi, where a and b are real numbers, is a complex number
“a” is the real part and
“b” is the imaginary part
The (principal) square root of a negative number:
If a is a real number, where a > 0, then
€
−a = i a
Properties of Complex Numbers
1.
2.
3.
4.
5.
6.
a + bi = c + di, iif a=c and b=d
Real Numbers are a subset of the Complex Numbers
Addition:
(a+bi) + (c+di) = (a+c) + (b+d)i
Subtraction:
(a+bi) – (c+di) = (a – c) + (b – d)i
Multiplication:
(a+bi)(c+di) = ac + adi + bci +bdi2
= (ac – bd) +(ad+bc)i
a+bi
Division: c+di
=
7.
=
a+bi
c+di
·
c−di
c−di
(ac+bd)+(bc−ad)i
c2 +d2
Complex conjugate of a + bi is a – bi
=
=
ac−adi+bci+bdi2
c2 −d2 i2
ac+bd
c2 +d2
+
(bc−ad)
c2 +d2 i
Recall: The Quadratic Formula
Let a, b, and c be real numbers with a �= 0
If ax2 + bx + √
c = 0,
−b± b2 −4ac
then x =
.
2a
Polynomial Functions
Zeros
2.5
Fundamental Theorem of Algebra
If f(x) is a polynomial of degree n, where n > 0, then f has at
least one zero in the Complex number.
This implies that:
f has precisely n linear factors
f(x) = an(x – c1)(x – c2)…(x – cn)
where c1, c2, …, cn are complex numbers
Note: Complex zeros come in conjugate pairs!!
If (a + bi) is a zero of f(x), then (a – bi) is also a zero.
Finding Zeros of a Polynomial…
  Rational Zeros Test (RZT)
If the polynomial f(x)=anxn+an-1xn-1+…+a1x+a0 has integer
coefficients, every rational zero of f has the form:
rational zero = p/q
where p and q have no common factors other than 1, and
p = a factor of the constant term a0
q = a factor of the leading coefficient an
Finding Zeros of a Polynomial…
  Descartes’s Rule of Signs
Let f(x)=anxn+an-1xn-1+…+a1x+a0 be a polynomial with real
coefficients and a0≠0.
The number of positive real zeros of f is either equal to the
number of variations in sign of f(x) or less than that number
by and even integer.
The number of negative real zeros of f is either equal to the
number of variations in sign of f(-x) or less than that number
by an even integer.
Finding Zeros of a Polynomial…
  Upper and Lower Bound Rules
Let f(x) be a polynomial with real coefficients and a positive
leading coefficient. Suppose f(x) is divided by (x – c), using
synthetic division.
If c >0 and each number in the last row is either positive or
zero, c is an upper bound for the real zeros of f.
If c < 0 and the numbers in the last row are alternately positive
and negative (zero entries count as positive or negative), c is
a lower bound for the real zeros of f.
Polynomial Functions
Rational Functions
2.6
Rational Functions
A rational function is a ratio (fraction) of polynomials;
f(x)=N(x)/D(x).
Domain:
x�R such that D(x) �= 0
Asymptotes
(V. A.) Vertical Asymptote – Vertical Line
x = a; where D(a) = 0
Find V.A. by looking for restrictions in the domain.
The function cannot touch nor cross a vertical asymptote.
Asymptotes
(H.A.) Horizontal Asymptotes – Horizontal Line
y = b; where lim f (x) = b
x→±∞
Find H.A. by looking at the end behavior of the function.
The function can touch or cross a horizontal asymptote.
If deg(N(x)) < deg(D(x)), then y=0 is the H.A.
If deg(N(x)) = deg(D(x)), then the H.A. is the ratio of leading
coefficients
If deg(N(x)) > deg(D(x)), then there is no H.A.
Asymptotes
(S. A.) Slant Asymptote
If y =
N (x)
D(x)
= ax + b + r(x)
And deg(N(x)) = deg(D(x)) + 1, then the rational function has
a slant asymptote; y = ax +b.
Find S.A. using long division.
A rational function NEVER has both a H.A. and S.A.
Sketching the Graph
  Identify domain.
  Simplify
  Find/Plot: vertical, horizontal, and slant asymptotes
  Find/Plot: x-intercept(s) and y-intercept
  Plot: At least one point on each side of the x-intercept(s) and
vertical asymptote(s)
  Fill in with smooth curve.