Unit 2 Formulas (Pre-calc) Semester 1

Vertex Form from quadratic formula

x=-b/2a
y=f(-b/2a)

Axis of symmetry

x=h

Opens Upwards

a>0

Opens Downwards

a<0

Quadratic function

ax^2+bx+c

Perfect square formula

(x-m)^2=x^2-2mx+m^2

End Behavior

+-x^even or odd
Positive= down
Negative= up
Even= down
Odd= up

Multiplicity

How many times it shows up as a zero,
Odd crosses
Even touches

Long division

Divide coefficient by first term then multiply top by exterior
(If degree of devisor is too high, it finishes)

Synthetic Division

(x-c) and c must is constant

Remainder Theorem

Synthetically divide, if no remainder then x is a zero

Rational Zero Theorem

p/q
p=factors of a0 (Constant)
q= factors of a1 (Leading coefficient)

i=

i

i^2=

-1

i^3=

-i

i^4=

1

If f(x) is a polynomial of a degree N > or equal to 1 with complex coefficients than it has at least one complex zero

Theorem of Algebra

n > or equal to 1 and an does not equal zero then an(x-cn)... are complex

Linear factorization theorem

If a polynomial with real coefficients and if a + bi is a zero then is conjugate is also a zero

Conjugate zeros theorem

If polynomial is a degree of n > or equal to 1 with complex coefficients then f(x) has exactly n complex zeros counted by multiplicity

Number of zeros of a polynomial

Polynomial with real coefficients and a nonzero constant

Descartes Rule of signs

Same number of sign changes in f(x) or less than the number of sign changes in f(x) by a positive even integer

Number of positive real zeros

Same number of sign changes in f(-x) or less than the number of sign changes in f(-x) by a positive even integer

Number of negative real zeros

How to find bounds

Synthetically divide

All positive

Upper bound

Alternate signs

Lower bound

y= p(x)/q(x)

Rational function

x -> c^+

x approaches c from the right

x -> c^-

x approaches c from the left

x -> infinity

x increases with no bound

x -> -infinity

x decreases with no bound

Factoring denominator

Find vertical asymptotes by

There is a hole

If numerator and denominator cancel out

n = degree of numerator

m = degree of denominator

n>m

There is no horizontal asymptote

n<m

Horizontal asymptote is y=0

n=m

line y is horizontal asymptote

y=an/

bm

Determine if a horizontal asymptote crosses by

plugging in y and solving , if its a real number it crosses

Degree of numerator is 1 greater than degree of denominator

Slant asymptote

Divide numerator by denominator, _____ will be y= quotient

Equation of slant asymptote

F is symmetric to y-axis if

f(-x)=f(x)

F is symmetric to origin if

f(-x)=-f(x)

Quadratic formula

-b+-(sqrtb^2-4ac)/2a