Polynomial Functions Flashcards
Function VS Relation
(Every ___ is a ___ but not every ___ is a ___)
Function: (vertical line test)
- Relation with each independent variable has only 1 dependent variable
Relation:
- Values of independent variables are paired with values of the dependent variable
Every function is a relation but not every relation is a function
Zero exponent
A^0 = 1
-A^0 = -1
Product law
A^x • A^y = A^x+y
Quotient law
A^x/A^y = A^x-y
Negative exponent
A^-x = 1/A^x
(A/B)^-x = (B/A)^x
Power of a quotient
(A/B)^n = (A^n/B^n)
Power of a power
(A^x)^y = A^xy
Power of a product
(AB)^n = A^n B^n
Linear Functions
- Degree of 1
- Graph = Linear
- First diff = Constant
Linear function forms
Standard: Ax + By + c = 0
Slope (y-intercept): y = mx + b
Slope form y2 - y1/x2 - x1
F(0) = ?
3F(x) = ?
F(0) = x = 0
3F(x) = 3[…]
GCF
Greatest Common Factor
ST
Simple Trinomial
PSN (Play Station Network) (Product Sum Number)
DOS
Difference Of Square
CT
Complex Trinomial
Vertex form
Y = a(x - h)^2 + k
Factored form
Y = a(x - r)(x - s)
X intercept form
Factored form
Y = a(x - r)(x - s)
X intercept form
Standard to Vertex
Complete the Square
Mapping notation
(x,y) -> (x/k + d, ay + c)
Quadratic functions
- degree of 2
- graph is parabola
- second diff same
What are the transformations:
Y = af(k(x - d)) + c
A:
- vertical stretch/compress (0<a<1)
- (-a) = reflection on x axis (vertical)
K:
- horizontal stretch/compress
- (-k) = reflect on y axis (horizontal)
D:
- horizontal translate
C:
- vertical translate
Interval notation
(-2,6]
Set builder notation
{xeR , -2 < x =< 6}
What makes a “Polynomial function”/rules of polynomial function
- atleast 1 x
- no negative exponents
- no fraction exponents
- no variable exponents
- no sin/cos
Degree
Greatest power of x
Leading coefficient
Coefficient of greatest power of x
Smallest degree
Number of turning points + 1
End behaviour (Positive odd)
Q3 -> Q1
End behaviour (Negative odd)
Q2 -> Q4
End behaviour (Positive even)
Q2 -> Q1
End behaviour (Negative even)
Q3 -> Q4
“U”
Unison
Local minimum
Least y value on interval
Local maximum
Greatest y value on interval
Global minima/maxima
Absolute max/min points
Power function
Simplest type of polynomial function
Polynomial function
Name based on degree (ex: cubic)
Y = ax^n
- a is constant
- x is variable
Odd degree x intercepts ???
- atleast 1
- max n
- inflection through x-axis
Even degree x intercepts ???
- 0 to N
- bounce
Odd degree number of global max/min ????
Doesn’t exist
Even degree number of global max/min ????
Max = a<0
Min = a>0
Is an even degree function always an even function?
No (SAME WITH ODD)
Even functions info
- symmetry on y-axis
- f(-x) = f(x)
If you sub in (-x) into (x) then it’ll end the same as start
How to identify Odd functions
- rotationally symmetrical
- rotate 180 (vert + hori reflect)
- if you sub in (-x) for (x) the signs of each terms will switch
Transformations
Y = af (k (x-d)) + c
Y = af (k (x-d)) + c
A TRANSFORMATIONS
A < 0 = reflect in x-axis
A > 1 = vertical stretch by a factor of A
0 < A < 1 = vertical compress by a factor of A
Y = af (k (x-d)) + c
K TRANSFORMATIONS
- K > 1 = horizontal compress by a factor of 1/k
- 0 < k < 1 = horizontal stretch by a factor of 1/k
- k < 0 = reflect in y axis, horizontal reflection
Y = af (k (x-d)) + c
C TRANSFORMATIONS
- c > 0 = vertical translate up
- c < 0 = vertical translate down
Y = af (k (x-d)) + c
D TRANSFORMATIONS
- d > 0 = horizontal translate right
- d < 0 = horizontal translate left
Intervals of increase
Intervals where y increases as x increases
(Thing goes up even if below x axis)
Intervals of decrease
Intervals where y decreases as x increases
(Thing goes down even if above x axis)
Positive intervals
Interval where the function lies above x axis
Negative intervals
Intervals where the function lies below the x axis
Polynomial function of degree n, where n is a positive integer, the n-th differences:
- are equal (constant)
- have the same sign as leading coefficient
How do you find out he degree of the polynomial function
Number of finite differences
How do you find the sign of the leading coefficient?
The sign of the final finite differences
How do you find the value of the leading coefficient?
A3! = -6
The value of the leading coefficient is a
Order
If a polynomial function has a factor (x-a) that is repeated n times, then x = a is a zero of order n
INDIVIDUAL Exponent of the thing
Expand:
(a+b)^2
(a^2 +2ab + b^2)
(a-b)^2:
(a^2 +2ab - b^2)
Expand
a^2 - b^2
(a + b)(a - b)
