Math Flashcards
Natural numbers
Counting numbers
1,2,3,4…
Counting numbers
Natural numbers
Whole numbers
Counting numbers plus 0
Integers
Whole numbers and their opposites
{…-3,-2,-1,0,1,2,3…}
Rational numbers
Can be written as fractions
Includes all whole, natural and integers
Irrational numbers
Can’t be written as a fraction
Ie. pie symbol
Real numbers
Include both rational and irrational numbers - anything goes
Coordinate
Location of a point on a number line
Plot a point
Place a dot at the location of the point on a number line
Absolute value
The distance of the number from 0 on the number line. |-4| is 4
If number have different signs…
Ignore the signs and subtract the smaller number from the larger. Attach the sign of the larger number.
Subtraction and division are not commutative
True, the order can’t be changed
Distributive property
Allows us to convert a product into an equivalent sum. 2(3+4) = 23 + 2*4
Additive inverse
The sum of a number and its opposite is 0
Multiplicative inverse
The product of a number and its reciprocal is 1
Order of operations
Pemdas Parentheses or other grouping symbols Exponents, square roots, absolute values Then, left to right Multiply Divide Add Subtract
Grouping symbols
Parentheses, brackets, fraction bar, absolute value, radical symbol
Set
Collection of distinct numbers, objects
Additive identity
The sum of a number and 0 is the number itself
Multiplicative identity
The product of a number and 1 is the number itself
Anything to the 0th power equals
1
Negative one to any even power equals
1
Absolute value
The distance of the number from 0 on the number line. |-4| is 4
Additive inverse
The sum of a number and its opposite is 0
Multiplicative inverse
The product of a number and its reciprocal is 1
Grouping symbols
Parentheses, brackets, fraction bar, absolute value, radical symbol
Pie equals?
3.14
slope intercept form
y = mx+b
y = mx+b
slope intercept form
y-y1 = m (x-x1)
point slope form (if given one point and slope)
point slope form (if given one point and slope)
y-y1 = m (x-x1)
x1+x2/2 y1+y2/2
midpoint formula
midpoint formula
x1 + x2/2 y1 + y2/2
Commutative Property of Addition
For all real numbers a and b,
a + b = b + a
Associative Property of Addition
For all real numbers a, b, and c,
a + (b + c) = (a + b) + c
Identity Property of Addition
There is a unique real number 0 such that for every real number a,
a + 0 = a and 0 + a = a
Additive Inverse Property (property of opposites)
For every real number a, there is a unique real number -a such that,
a + (-a) = 0 and (-a) + a = 0
Associative Property of Multiplication
For all real numbers a, b, and c,
ab)c = a(bc
Commutative Property of Multiplication
For all real numbers a and b,
ab = ba
Transitive Property of Equality
For all real numbers a, b, and c,
if a = b and b = c, then a = c.
Reflexive Property of Equality
For each real number a
a = a
Symmetric Property of Equality
For all real numbers a, b,
if a = b, then b = a
Closure Property
For all real numbers a and b,
a + b is a unique real number and ab is a unique real number
Distributive property with respect to addition
For all real numbers a, b, and c, a(b + c) = ab + ac
Distributive property with respect to subtraction
For all real numbers a, b, and c, a(b - c) = ab - ac
Identity Property of Multiplication
There is a unique number 1 such that for every real number a, 1(a) = a and (a)1 = a.
Multiplicative Inverses (reciprocals)
Two numbers whose product is 1
Property of Reciprocals
For every NONZERO number a, there is a unique number 1/a such that a • 1/a = 1 and 1/a • a = 1
Property of the Reciprocal of the Opposite of a Number
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
Slope
steepness of a line , rise over run
Y-Intercept
Where a line crosses the y-axis
Zero Slope
A horizontal line
No Slope/Undefined
A vertical line
Slope (from a table)
Change in y over change in x
m=(what does m stand for)
Slope
b=(what does b stand for)
y-Intercept
What is the y-intercept
Where the line crosses the y axis
y=3x+2 What is the slope
3
y=6x+5 What is the y-intercept
5
On a table how do you find b?
When x is zero
Is a point a solution of a line?
If the line goes through that point when graphed
Rate of change definition
change in y/change in x
Looking at a graph, how can you tell if it is a function?
Vertical line should not intersect graph in more than one place. If it does, then it is not a function.
The output variable and the input variable:
Which one is dependent, and which is independent?
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
The DOMAIN (or INPUT) is on the _____ axis.
The domain is x (x-axis)
The RANGE (or OUTPUT) is on the _____ axis.
The range is f(x) or y (y-axis)
Pythagorean Theorem for a Right Triangle
L2 + H2 = D2
Area of a Circle
(pi) r2
Area of a Triangle
2
Vertical Shift of Function (up/down):
Add or subtract from the function
EG: f(x) → f(x) + 5 will move up 5 units
Horizontal Shift of Function (left/right):
Add or subtract the reverse from x
EG: f(x) → f(x-5) will move right 5 units
Reflect Function Across x-axis:
Multiply function by -1
EG: f(x) → -f(x) will mirror across x-axis
Reflect Function Across y-axis:
Multiply x by -1
EG: f(x) → f(-x) will mirror across y-axis
Vertically Stretch Graph of a Function:
Multiply function by a number greater than 1
EG: f(x) → 3f(x) will vertically stretch the graph
Vertically Shrink Graph of a Function:
Multiply function by a number between 0 and 1
EG: f(x) → 0.5f(x) will vertically shrink the graph
Horizontally Stretch Graph of a Function:
Multiply x by a number between 0 and 1
EG: f(x) → f(0.5x) will horizontally stretch the graph
Horizontally Shrink Graph of a Function:
Multiply x by a number greater than 1
EG: f(x) → f(3x) will horizontally shrink the graph
Odd and Even Functions
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)
How to find the inverse of a function:
- Replace f(x) with y
- Solve for x in terms of y (x on one side, alone)
- Interchange x and y, then replace y with f-1(x)
Is this a function?
X: 3, 2, 4, 6, 8, 12
Y: 3, 3, 7, 12, 4, 8
Yes - passes Vertical Line Test
All Domain values are unique
Is this a function?
X: 3, 2, 4, 3, 8, 12
Y: 3, 3, 7, 12, 4, 8
No - does not pass Vertical Line Test
Domain contains duplicates (3 corresponds to two values in the range- 3 and 12)
Standard Form of a Linear Function
y or f(x) = mx + b
m is the slope
b is the y-intercept
How to calculate slope from coordinates of 2 points on the line:
y2 - y1
M = ———–
x2 - x1
Point-Slope Form
y or f(x) = m(x-x1) + y1
(x-x1) ends up being x
y1 ends up being b or y-intercept
How to find the root of a linear function:
Calculate y = mx + b as
0 = mx + b
Parallel lines have slopes that are ______
Parallel lines have slopes that are EQUAL
EG: m1 = m2
Perpendicular lines have slopes that are ______
Perpendicular lines have slopes that are
NEGATIVELY RECIPROCAL
How to find the point of intersection of 2 lines:
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)
In regression analysis,
r is ________
and r2 is ________
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)
Standard form of a Quadratic Function
y or f(x) = ax2+bx+c
a≠0, if a=0 then it is a horizontal line
Standard Form vs. Vertex Form of a Quadratic Function
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
Finding the Vertex of a Quadratic Function
-b
x = ——
2a
Plug this into the equation to find y
Finding the roots of a Quadratic Function:
The root(s) are at
0 = ax2+bx+c
Use the Quadratic Formula:
-b ± √b2 - 4ac
x= ——————–
2a
What is the Quadratic Formula?
What is it used for?
Quadratic Formula
-b ± √b2 - 4ac
x= ——————–
2a
Quadratic Formula is used to find the roots of a quadratic function
What is the Discriminant and what can it tell you?
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)
What kind of function is this:
f(x) = mx + b
Linear Function
Linear Regression Analysis
What is the correlation coefficient and how is it represented?
correlation coefficient = r
Measures how well the best fitting line fits the data points. Ranges from -1 to 1.
Linear Regression Analysis
What is the coefficient of determination and how is it represented?
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.
What kind of function is this:
f(x) = ax2 + bx + c
Quadratic Function
(a ≠ 0)
The simplest form of a quadratic function is
f(x) = x2
What kind of function is this:
ax4 + ax3 + ax2 + ax + a
Polynomial Function
of degree 4 - quartic polynomial
Standard form of a Polynomial Function
ax4 + ax3 + ax2 + ax + a
(the exponent cannot be negative,
the exponent cannot be a fraction,
x cannot be in the denominator)
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 ________________
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
Polynomial Function
bx4 + ax3 + ax2 + ax + g
What is b?
What is 4?
What is g?
What is bx4?
b is the leading coefficient
4 is the degree/order
g is the constant term
bx4 is the leading term
f(x) = axn
is a monomial function
is a power function
(n > 0
b ≠ 0)
f(x) = axn
if n=0, graph is ________________
if n=1, graph is ________________
if n=2, graph is _________________
if n=3, graph is _________________
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)
Even-exponent Power Functions
xn → n could equal _____
the shape is _______
graph gets ______ the _______ the exponent
When x>1 or xx>1, _______ are ________
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
Odd-exponent Power Functions
xn → n could equal _____
the shape is _______
graph gets ______ the _______ the exponent
When x>1 or xx>1, _______ are ________
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
Intermediate Value Theorem
polynomial functions
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)
Factor Theorem
polynomial functions
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.
(x-c)3 has a _________ of _____
if x=4, the factor of the polynomial is ______
if x = -3, the factor of the polynomial is ______
(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)
(x-c)3 will _____ the x-axis at the x=c
(x-c)2 will _____ the x-axis at the x=c
(x-c)3 will cross the x-axis at the x=c
(x-c)2 will touch the x-axis at the x=c
How do you represent a polynomial factor that does not cross or touch the x-axis anywhere?
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
A polynomial of degree/order “n” can have a maximum of ___ roots
A polynomial of degree/order “n” can have a maximum of n roots
A polynomial of degree/order “n” can have a maximum of ___ turning points
A polynomial of degree/order “n” can have a maximum of n-1 turning points
A quadratic function can have ____ turning points
A cubic function can have ____ turning points
A quartic function can have ____ turning points
A quadratic function can have 1 turning point
A cubic function can have 2 turning points
A quartic function can have 3 turning points
Polynomial Functions
When the absolute value of x is large, end/long-run behavior of the graph will tend to ______
When the absolute value of x is large, end/long-run behavior of the graph will tend to follow the leading term
Every polynomial of a degree of ≥1 with complex coefficients has at least one zero in the complex number system.
This is called _______________
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.
What kind of function is this:
p(x)
f(x)= ———–
q(x)
Rational Function
What kind of function is this:
ax3+bx2+cx+d
f(x)= ————————-
ax4+bx3+cx2+dx+e
Rational Function
What is the domain of a rational function?
The domain of a rational function is the set of all real numbers that are NOT roots of the denominator
(the denominator≠0)
x2-8x+12
- Y-intercept:
- Horizontal Asymptote(s):
- End behavior of graph:
- Degree of numerator/denominator:
x2-8x+12
- Y-intercept: x=0 is not a root of the denominator, so evaluate function at x=0. y=-6
- Horizontal Asymptote(s): oblique asymptote, divide the equation to find it. x+18
- 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
- Degree of numerator/denominator: 3/2
Asymptote of
bn
Horizontal asymptote at
y=a/b
Asymptote of
bN
Horizontal asymptote at
y=0
Asymptote of
bn
Oblique asymptote at
divide the equation to find it
Asymptote of
bn
No line asymptote
