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Math Symbols > Math > Flashcards

Math Flashcards

0
Q

Natural numbers

A

Counting numbers

1,2,3,4…

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1
Q

Counting numbers

A

Natural numbers

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2
Q

Whole numbers

A

Counting numbers plus 0

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3
Q

Integers

A

Whole numbers and their opposites

{…-3,-2,-1,0,1,2,3…}

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4
Q

Rational numbers

A

Can be written as fractions

Includes all whole, natural and integers

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5
Q

Irrational numbers

A

Can’t be written as a fraction

Ie. pie symbol

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6
Q

Real numbers

A

Include both rational and irrational numbers - anything goes

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7
Q

Coordinate

A

Location of a point on a number line

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8
Q

Plot a point

A

Place a dot at the location of the point on a number line

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10
Q

Absolute value

A

The distance of the number from 0 on the number line. |-4| is 4

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11
Q

If number have different signs…

A

Ignore the signs and subtract the smaller number from the larger. Attach the sign of the larger number.

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12
Q

Subtraction and division are not commutative

A

True, the order can’t be changed

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13
Q

Distributive property

A

Allows us to convert a product into an equivalent sum. 2(3+4) = 23 + 2*4

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14
Q

Additive inverse

A

The sum of a number and its opposite is 0

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15
Q

Multiplicative inverse

A

The product of a number and its reciprocal is 1

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16
Q

Order of operations

A 
Pemdas
Parentheses or other grouping symbols
Exponents, square roots, absolute values
Then, left to right
Multiply
Divide
Add
Subtract
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17
Q

Grouping symbols

A

Parentheses, brackets, fraction bar, absolute value, radical symbol

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18
Q

Set

A

Collection of distinct numbers, objects

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19
Q

Additive identity

A

The sum of a number and 0 is the number itself

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20
Q

Multiplicative identity

A

The product of a number and 1 is the number itself

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21
Q

Anything to the 0th power equals

A

1

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22
Q

Negative one to any even power equals

A

1

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23
Q

Absolute value

A

The distance of the number from 0 on the number line. |-4| is 4

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24
Q

Additive inverse

A

The sum of a number and its opposite is 0

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25
Q

Multiplicative inverse

A

The product of a number and its reciprocal is 1

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26
Q

Grouping symbols

A

Parentheses, brackets, fraction bar, absolute value, radical symbol

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27
Q

Pie equals?

A

3.14

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28
Q

slope intercept form

A

y = mx+b

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29
Q

y = mx+b

A

slope intercept form

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30
Q

y-y1 = m (x-x1)

A

point slope form (if given one point and slope)

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31
Q

point slope form (if given one point and slope)

A

y-y1 = m (x-x1)

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32
Q

x1+x2/2 y1+y2/2

A

midpoint formula

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33
Q

midpoint formula

A

x1 + x2/2 y1 + y2/2

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34
Q

Commutative Property of Addition

A

For all real numbers a and b,

a + b = b + a

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35
Q

Associative Property of Addition

A

For all real numbers a, b, and c,

a + (b + c) = (a + b) + c

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36
Q

Identity Property of Addition

A

There is a unique real number 0 such that for every real number a,
a + 0 = a and 0 + a = a

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37
Q

Additive Inverse Property (property of opposites)

A

For every real number a, there is a unique real number -a such that,
a + (-a) = 0 and (-a) + a = 0

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38
Q

Associative Property of Multiplication

A

For all real numbers a, b, and c,

ab)c = a(bc

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39
Q

Commutative Property of Multiplication

A

For all real numbers a and b,

ab = ba

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40
Q

Transitive Property of Equality

A

For all real numbers a, b, and c,

if a = b and b = c, then a = c.

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41
Q

Reflexive Property of Equality

A

For each real number a

a = a

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42
Q

Symmetric Property of Equality

A

For all real numbers a, b,

if a = b, then b = a

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43
Q

Closure Property

A

For all real numbers a and b,

a + b is a unique real number and ab is a unique real number

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44
Q

Distributive property with respect to addition

A

For all real numbers a, b, and c, a(b + c) = ab + ac

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45
Q

Distributive property with respect to subtraction

A

For all real numbers a, b, and c, a(b - c) = ab - ac

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46
Q

Identity Property of Multiplication

A

There is a unique number 1 such that for every real number a, 1(a) = a and (a)1 = a.

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47
Q

Multiplicative Inverses (reciprocals)

A

Two numbers whose product is 1

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48
Q

Property of Reciprocals

A

For every NONZERO number a, there is a unique number 1/a such that a • 1/a = 1 and 1/a • a = 1

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49
Q

Property of the Reciprocal of the Opposite of a Number

A

For every nonzero number a, -1/a = 1(-a)
This is read “The reciprocal of the opposite of a is 1 over the opposite of a. Note: this is what allows us to say 3/(-4) = - 3/4

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50
Q

Slope

A

steepness of a line , rise over run

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51
Q

Y-Intercept

A

Where a line crosses the y-axis

52
Q

Zero Slope

A

A horizontal line

53
Q

No Slope/Undefined

A

A vertical line

54
Q

Slope (from a table)

A

Change in y over change in x

55
Q

m=(what does m stand for)

A

Slope

56
Q

b=(what does b stand for)

A

y-Intercept

57
Q

What is the y-intercept

A

Where the line crosses the y axis

58
Q

y=3x+2 What is the slope

A

3

59
Q

y=6x+5 What is the y-intercept

A

5

60
Q

On a table how do you find b?

A

When x is zero

61
Q

Is a point a solution of a line?

A

If the line goes through that point when graphed

62
Q

Rate of change definition

A

change in y/change in x

63
Q

Looking at a graph, how can you tell if it is a function?

A

Vertical line should not intersect graph in more than one place. If it does, then it is not a function.

64
Q

The output variable and the input variable:

Which one is dependent, and which is independent?

A

The output variable is dependent
(it depends upon the input)

The input variable is independent
(may have its value freely chosen regardless of any other variable values)

The output is a function of (depends upon) the input

65
Q

The DOMAIN (or INPUT) is on the _____ axis.

A

The domain is x (x-axis)

66
Q

The RANGE (or OUTPUT) is on the _____ axis.

A

The range is f(x) or y (y-axis)

67
Q

Pythagorean Theorem for a Right Triangle

A

L2 + H2 = D2

68
Q

Area of a Circle

A

(pi) r2

69
Q

Area of a Triangle

A

2

70
Q

Vertical Shift of Function (up/down):

A

Add or subtract from the function

EG: f(x) → f(x) + 5 will move up 5 units

71
Q

Horizontal Shift of Function (left/right):

A

Add or subtract the reverse from x

EG: f(x) → f(x-5) will move right 5 units

72
Q

Reflect Function Across x-axis:

A

Multiply function by -1

EG: f(x) → -f(x) will mirror across x-axis

73
Q

Reflect Function Across y-axis:

A

Multiply x by -1

EG: f(x) → f(-x) will mirror across y-axis

74
Q

Vertically Stretch Graph of a Function:

A

Multiply function by a number greater than 1

EG: f(x) → 3f(x) will vertically stretch the graph

75
Q

Vertically Shrink Graph of a Function:

A

Multiply function by a number between 0 and 1

EG: f(x) → 0.5f(x) will vertically shrink the graph

76
Q

Horizontally Stretch Graph of a Function:

A

Multiply x by a number between 0 and 1

EG: f(x) → f(0.5x) will horizontally stretch the graph

77
Q

Horizontally Shrink Graph of a Function:

A

Multiply x by a number greater than 1

EG: f(x) → f(3x) will horizontally shrink the graph

78
Q

Odd and Even Functions

A

f(x) = f(-x) is EVEN
(symmetry about the y-axis)

f(x) = -f(x) not possible except for 0
(symmetry about the x-axis)

-f(x) = f(-x) and f(-x) = -f(x) are ODD
(symmetry about the origin)

79
Q

How to find the inverse of a function:

A
  1. Replace f(x) with y
  2. Solve for x in terms of y (x on one side, alone)
  3. Interchange x and y, then replace y with f-1(x)
80
Q

Is this a function?

X: 3, 2, 4, 6, 8, 12

Y: 3, 3, 7, 12, 4, 8

A

Yes - passes Vertical Line Test

All Domain values are unique

81
Q

Is this a function?

X: 3, 2, 4, 3, 8, 12

Y: 3, 3, 7, 12, 4, 8

A

No - does not pass Vertical Line Test

Domain contains duplicates (3 corresponds to two values in the range- 3 and 12)

82
Q

Standard Form of a Linear Function

A

y or f(x) = mx + b

m is the slope

b is the y-intercept

83
Q

How to calculate slope from coordinates of 2 points on the line:

A

y2 - y1
M = ———–
x2 - x1

84
Q

Point-Slope Form

A

y or f(x) = m(x-x1) + y1

(x-x1) ends up being x

y1 ends up being b or y-intercept

85
Q

How to find the root of a linear function:

A

Calculate y = mx + b as

0 = mx + b

86
Q

Parallel lines have slopes that are ______

A

Parallel lines have slopes that are EQUAL

EG: m1 = m2

87
Q

Perpendicular lines have slopes that are ______

A

Perpendicular lines have slopes that are

NEGATIVELY RECIPROCAL

88
Q

How to find the point of intersection of 2 lines:

A

For two lines y1 = m1x1+b1 and y2 = m2x2+b2

                                           b2-b1

Point of intersection (x0) is ————

                                           m1-m2

(then can use this as x to find y)

89
Q

In regression analysis,

r is ________

and r2 is ________

A

r is the CORRELATION COEFFICIENT

(a number between -1 and 1 that measures how well the best fitting line fits the data points)

r2 is the COEFFICIENT OF DETERMINATION

(a number that determines if the best fitting line can be used as a data model. Closer to 1, the better the fit)

90
Q

Standard form of a Quadratic Function

A

y or f(x) = ax2+bx+c

a≠0, if a=0 then it is a horizontal line

91
Q

Standard Form vs. Vertex Form of a Quadratic Function

A

Standard Form

y or f(x) = ax2+bx+c

Vertex Form

y or f(x) = a(x-h)2+k

-h,k are the x,y of the vertex

92
Q

Finding the Vertex of a Quadratic Function

A

-b

x = ——

 2a

Plug this into the equation to find y

93
Q

Finding the roots of a Quadratic Function:

A

The root(s) are at

0 = ax2+bx+c

Use the Quadratic Formula:

-b ± √b2 - 4ac

x= ——————–

          2a
94
Q

What is the Quadratic Formula?

What is it used for?

A

Quadratic Formula

-b ± √b2 - 4ac

x= ——————–

           2a

Quadratic Formula is used to find the roots of a quadratic function

95
Q

What is the Discriminant and what can it tell you?

A

The Discriminant is the b2 - 4ac part of the Quadratic Function

If the Discriminant is positive, there are two roots

If the Discriminant is zero, there is one root, the graph is sitting on the x-axis

If the Discriminant is negative, the graph does not intersect the x-axis (there is no root)

96
Q

What kind of function is this:

f(x) = mx + b

A

Linear Function

97
Q

Linear Regression Analysis

What is the correlation coefficient and how is it represented?

A

correlation coefficient = r

Measures how well the best fitting line fits the data points. Ranges from -1 to 1.

98
Q

Linear Regression Analysis

What is the coefficient of determination and how is it represented?

A

Coefficient of Determination = r2 (the square of the correlation coefficient). Determines if the best fitting line can be used as a model (is it good enough?)

The closer r2 is to 1, the better the fit.

99
Q

What kind of function is this:

f(x) = ax2 + bx + c

A

Quadratic Function

(a ≠ 0)

The simplest form of a quadratic function is

f(x) = x2

100
Q

What kind of function is this:

ax4 + ax3 + ax2 + ax + a

A

Polynomial Function

of degree 4 - quartic polynomial

101
Q

Standard form of a Polynomial Function

A

ax4 + ax3 + ax2 + ax + a

(the exponent cannot be negative,

the exponent cannot be a fraction,

x cannot be in the denominator)

102
Q

If the first (largest) term in a polynomial function is

ax4 the function is ____________

ax3 the function is ____________

ax2 the function is ____________

ax the function is ____________

ax0 ________________

A

If the first (largest) term in a polynomial function is

ax4 the function is quartic (parabola)

ax3 the function is cubic (snakelike)

ax2 the function is quadratic (parabola)

ax the function is linear (line)

ax0 is a horizontal line at y=a

103
Q

Polynomial Function

bx4 + ax3 + ax2 + ax + g

What is b?

What is 4?

What is g?

What is bx4?

A

b is the leading coefficient

4 is the degree/order

g is the constant term

bx4 is the leading term

104
Q

f(x) = axn

A

is a monomial function

is a power function

(n > 0

b ≠ 0)

105
Q

f(x) = axn

if n=0, graph is ________________

if n=1, graph is ________________

if n=2, graph is _________________

if n=3, graph is _________________

A

f(x) = axn

if n=0, graph is a horizontal line at y=a

if n=1, graph is linear with slope of a (odd function)

if n=2, graph is parabola, branches facing up when a is a is positive, down when a is negative (even function)

if n=3, graph is snakelike, increasing when a is positive, decreasing when a is negative (odd function)

106
Q

Even-exponent Power Functions

xn → n could equal _____

the shape is _______

graph gets ______ the _______ the exponent

When x>1 or xx>1, _______ are ________

A

xn → n could equal 2, 4, etc.

the shape is a parabola

graph gets flatter (on the bottom) the higher the exponent

When x>1 or xx>1, branches are flatter

107
Q

Odd-exponent Power Functions

xn → n could equal _____

the shape is _______

graph gets ______ the _______ the exponent

When x>1 or xx>1, _______ are ________

A

xn → n could equal 1, 3, 5, etc.

the shape is snakelike

graph gets flatter (on the bottom) the higher the exponent

When x>1 or xx>1, traces are flatter

108
Q

Intermediate Value Theorem

polynomial functions

A

If the result of f(a) and f(b) are opposite signs (+/-), then there must be at least one root between them

(as long as a≠b)

109
Q

Factor Theorem

polynomial functions

A

f(c) will equal zero ONLY IF (x-c) is a factor of the polynomial.

In other words, the factors (x-c) are the only places where the function will equal zero.

110
Q

(x-c)3 has a _________ of _____

if x=4, the factor of the polynomial is ______

if x = -3, the factor of the polynomial is ______

A

(x-c)3 has a multiplicity of 3

if x=4, the factor of the polynomial is (x-4)

if x = -3, the factor of the polynomial is (x+3)

111
Q

(x-c)3 will _____ the x-axis at the x=c

(x-c)2 will _____ the x-axis at the x=c

A

(x-c)3 will cross the x-axis at the x=c

(x-c)2 will touch the x-axis at the x=c

112
Q

How do you represent a polynomial factor that does not cross or touch the x-axis anywhere?

A

The constant factor k

f(x) = k(x-c1)(x-c2)(x-c3)

Adding or subtracting from the constant factor k shifts the graph up or down the y-axis

113
Q

A polynomial of degree/order “n” can have a maximum of ___ roots

A

A polynomial of degree/order “n” can have a maximum of n roots

114
Q

A polynomial of degree/order “n” can have a maximum of ___ turning points

A

A polynomial of degree/order “n” can have a maximum of n-1 turning points

115
Q

A quadratic function can have ____ turning points

A cubic function can have ____ turning points

A quartic function can have ____ turning points

A

A quadratic function can have 1 turning point

A cubic function can have 2 turning points

A quartic function can have 3 turning points

116
Q

Polynomial Functions

When the absolute value of x is large, end/long-run behavior of the graph will tend to ______

A

When the absolute value of x is large, end/long-run behavior of the graph will tend to follow the leading term

117
Q

Every polynomial of a degree of ≥1 with complex coefficients has at least one zero in the complex number system.

This is called _______________

A

The Fundamental Theorem of Algebra

Every polynomial of a degree of ≥1 with complex coefficients has at least one zero in the complex number system.

118
Q

What kind of function is this:

    p(x)

f(x)= ———–

    q(x)
A

Rational Function

119
Q

What kind of function is this:

    ax3+bx2+cx+d

f(x)= ————————-

    ax4+bx3+cx2+dx+e
A

Rational Function

120
Q

What is the domain of a rational function?

A

The domain of a rational function is the set of all real numbers that are NOT roots of the denominator

(the denominator≠0)

121
Q

x2-8x+12

  1. Y-intercept:
  2. Horizontal Asymptote(s):
  3. End behavior of graph:
  4. Degree of numerator/denominator:
A

x2-8x+12

  1. Y-intercept: x=0 is not a root of the denominator, so evaluate function at x=0. y=-6
  2. Horizontal Asymptote(s): oblique asymptote, divide the equation to find it. x+18
  3. End behavior of graph: x3/x2 which would be a line increasing as x increasing that crosses the graph at x=2 and x=6
  4. Degree of numerator/denominator: 3/2
122
Q

Asymptote of

bn

A

Horizontal asymptote at

y=a/b

123
Q

Asymptote of

bN

A

Horizontal asymptote at

y=0

124
Q

Asymptote of

bn

A

Oblique asymptote at

divide the equation to find it

125
Q

Asymptote of

bn

A

No line asymptote

Math Symbols (1 decks)

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