Polynomial Functions Flashcards

1
Q

Function VS Relation

(Every ___ is a ___ but not every ___ is a ___)

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

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2
Q

Zero exponent

A

A^0 = 1

-A^0 = -1

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3
Q

Product law

A

A^x • A^y = A^x+y

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4
Q

Quotient law

A

A^x/A^y = A^x-y

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5
Q

Negative exponent

A

A^-x = 1/A^x

(A/B)^-x = (B/A)^x

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6
Q

Power of a quotient

A

(A/B)^n = (A^n/B^n)

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7
Q

Power of a power

A

(A^x)^y = A^xy

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8
Q

Power of a product

A

(AB)^n = A^n B^n

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9
Q

Linear Functions

A
  • Degree of 1
  • Graph = Linear
  • First diff = Constant
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10
Q

Linear function forms

A

Standard: Ax + By + c = 0

Slope (y-intercept): y = mx + b

Slope form y2 - y1/x2 - x1

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11
Q

F(0) = ?

3F(x) = ?

A

F(0) = x = 0

3F(x) = 3[…]

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12
Q

GCF

A

Greatest Common Factor

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13
Q

ST

A

Simple Trinomial

PSN (Play Station Network) (Product Sum Number)

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14
Q

DOS

A

Difference Of Square

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15
Q

CT

A

Complex Trinomial

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16
Q

Vertex form

A

Y = a(x - h)^2 + k

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17
Q

Factored form

A

Y = a(x - r)(x - s)

X intercept form

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18
Q

Factored form

A

Y = a(x - r)(x - s)

X intercept form

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19
Q

Standard to Vertex

A

Complete the Square

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20
Q

Mapping notation

A

(x,y) -> (x/k + d, ay + c)

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21
Q

Quadratic functions

A
  • degree of 2
  • graph is parabola
  • second diff same
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22
Q

What are the transformations:

Y = af(k(x - d)) + c

A

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

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23
Q

Interval notation

A

(-2,6]

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24
Q

Set builder notation

A

{xeR , -2 < x =< 6}

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25
What makes a “Polynomial function”/rules of polynomial function
- atleast 1 x - no negative exponents - no fraction exponents - no variable exponents - no sin/cos
26
Degree
Greatest power of x
27
Leading coefficient
Coefficient of greatest power of x
28
Smallest degree
Number of turning points + 1
29
End behaviour (Positive odd)
Q3 -> Q1
30
End behaviour (Negative odd)
Q2 -> Q4
31
End behaviour (Positive even)
Q2 -> Q1
32
End behaviour (Negative even)
Q3 -> Q4
33
“U”
Unison
34
Local minimum
Least y value on interval
35
Local maximum
Greatest y value on interval
36
Global minima/maxima
Absolute max/min points
37
Power function
Simplest type of polynomial function
38
Polynomial function
Name based on degree (ex: cubic) Y = ax^n - a is constant - x is variable
39
Odd degree x intercepts ???
- atleast 1 - max n - inflection through x-axis
40
Even degree x intercepts ???
- 0 to N - bounce
41
Odd degree number of global max/min ????
Doesn’t exist
42
Even degree number of global max/min ????
Max = a<0 Min = a>0
43
Is an even degree function always an even function?
No (SAME WITH ODD)
44
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
45
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
46
Transformations
Y = af (k (x-d)) + c
47
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
48
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
49
Y = af (k (x-d)) + c C TRANSFORMATIONS
- c > 0 = vertical translate up - c < 0 = vertical translate down
50
Y = af (k (x-d)) + c D TRANSFORMATIONS
- d > 0 = horizontal translate right - d < 0 = horizontal translate left
51
Intervals of increase
Intervals where y increases as x increases (Thing goes up even if below x axis)
52
Intervals of decrease
Intervals where y decreases as x increases (Thing goes down even if above x axis)
53
Positive intervals
Interval where the function lies above x axis
54
Negative intervals
Intervals where the function lies below the x axis
55
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
56
How do you find out he degree of the polynomial function
Number of finite differences
57
How do you find the sign of the leading coefficient?
The sign of the final finite differences
58
How do you find the value of the leading coefficient?
A3! = -6 The value of the leading coefficient is a
59
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
60
Expand: (a+b)^2
(a^2 +2ab + b^2) (a-b)^2: (a^2 +2ab - b^2)
61
Expand a^2 - b^2
(a + b)(a - b)