Vocabulary Flashcards
domain of a function
the input numbers, the x-values, domain of a function is all real numbers for which the equation produces outputs that are real numbers
range of function
the output values, the y-values or f(x) values
typically, a value of x that must be excluded from the domain of a function makes the denominator _____
zero
in place of an equation, a function with a small finite domain can also be described by _____
a set of ordered pairs
in a function, every x-value has _____ y-values
exactly one y-value. if an x-value has more than one y-value than it is not a function
relation
describes association between two variables. a function is a type of relation
in terms of functions and relations, a circle is ___
a relation that is not a function
relations that are not functions are ___
ellipses, hyperbolas, and parabolas that open side ways
combining functions f(x) and g(x)
…
composition of functions f(x) and g(x)
…
inverse of a function f, denoted by f^(-1)
switch y and x. putting y’s in place of the x’s and x’s in place of the y’s than solve for y
what is something to remember about inverses?
inverses don’t have to be a function
the graphs of inverses are ___
reflection of the original function about the line y=x
if the point with coordinates (a,b) belongs to a function f, then the point with coordinates ___ belongs to the inverse of f
(b,a)
the inverse of any function f can always be made a function by ___
limiting the domain of f. cutting out the parts that keep it from being a function
properties of an even function
f(x)=f(-x)
an even function is symmetric about the y-axis
properties of an odd function
f(-x) = -f(x), an odd function is symmetric about the origin
the sum of even functions is ___
even
the sum of odd functions is ___
odd
the product of an even function and an odd function is ___
odd
linear functions
polynomials in which the largest exponent is 1
graphs of linear equations are always a straight line
general form of the equation for a linear function
Ax + By + C = 0
- A/B = slope of line
- C/B = y intercept
slope intercept
y = mx + b m = slope of line b = y intercept
formula for slope
(y1 -y2)/(x1-x2)
point slope
y - y1 = m( x - x1 )
parallel lines have ___ slopes.
same
the slopes of 2 perpendicular lines are ___
negative reciprocals of one another
distance formula for x-y plane
…
midpoint formula
…
properties of quadratic functions
polynomials where the largest exponent is 2
graph is always a parabola
general form of equation for quadratic functions
y = ax^2 + bx + c
if a is positive, then graph opens up
if a is negative graph opens down
how to calculate x value of vertex of parabola
-b / 2a
zeroes of a quadratic function are ___
the points where the the graph crosses the x-axis, and y = 0
quadratic formula
…
sum of the zeroes of a quadratic equations is ___
-b / a
determinant of a quadratic equation
(b^2) - 4ac
if determinant of quadratic equation equals 0
the two roots equal zero and the graph is tangent to the x-axis
if the determinant of quadratic equation is less than 0
the roots are two complex numbers because there is a negative under the radical and the graph never touches the x-axis
if the determinant of quadratic equation is more than zero
two different real roots and the graph intersects the x-axis in two places
the graphs of polynomial functions are always ________
continuous curves
if the largest exponent of a polynomial function is even then the ends of its graph
point in the same direction ( a parabola or a W)
if the largest exponent of a polynomial function is odd then the ends of its graph
point in opposite directions ( an S or snake graph)
if all the exponents of a polynomial function are even then the polynomial is an ______
even function and symmetric about the y-axis
if all the exponents of a polynomial function are odd then the polynomial is an _______________
odd function and symmetric about the origin
a polynomial of degree n has ___ zeroes
n, same number as degree
when it comes to the zeroes of polynomial functions, imaginary ezroes must ______
occur in pairs
with polynomial functions, real zeroes can occur ____
more than once
Remainder theorem
if a polynomial P(x) is divided by x-r (where r is any constant), then the remainder is P(r)
factor theorem
r is a zero of the polynomial P(x) if and only if x-r is a divisor of P(x)
rational zero (root) theorem
if p/q is a rational zero (reduced to lowest terms) of a polynomial P(x) with integral coefficients, then p is a factor of the constant term and q is is a factor of the leading coefficient
if P(x) is a polynomial with real coefficients, then complex zeroes occur as _____
conjugate pairs ( if p + qi is a zero then p - qi is also a zero)
angles measured in a counter clockwise direction from the initial side to the terminal side is said to have a _____ value
positive
angles measured in a clockwise direction from the initial side to the terminal side is said to have a _____ value
negative
sin and cos are always between ___ and ___
-1 and 1
the six trig definitions
…
complimentary angles
add to 90 degrees
sumplementary angles
add to 180 degrees
cofunctions of complementary angles are ___
equal
if a and b are complimentary, then the sina=cosb
radian
the measure of one radius length
one radian equal 360/2pie
arc length equation
radius times angle (in radians)
area of a sector of a circle
(1/2)(radius ^2)( angle in radians)
special triangles
…
periodic functions
where the values eventually repeat over regular intervals
period
smallest positive value in which a periodic function repeats
periods of six trig functions
sin, cos, csc, sec have periods of 2pie
tan and cot have periods of pie
the vertical asymptotes of sec, csc, tan and cot
sec and tan’s occur on pie/2
csc and cot’s occur on pie
general form of a trig function
y = A * trig(Bx + C) + D
amplitude = abs(A)
horizontal translation is -C/B and is called phase shift
period is period of original trig func divided by B
D is amount of vertical translation
reciprocal identities
csc = 1/sin sec = 1/cos cot = 1/ tan
cofunction identities
…
pythagorean identities
…
double angle formulas
…
law of sines
…
law of cosines
…
area of triangle (using sine)
…
basic exponential properties
…
basic log properties
…
logbN is only defined for
positive N
the graphs of all exponential function y = b^x
have roughly the same shape and pass through the point (0,1)
the graphs of all logarithmic functions y = logbX
have roughly same shape and pass through point (1, 0)
the function f is a rational function if and only if
f(x) = p(x)/q(x) , where p(x) and q(x) are both polynomial functions and q(x) is not zero
if p and q are both zero, there is a whole at that x
if only q is zero than there is an asymptote
a point of discontinuities occurs at any value of X that ___
would cause q(x) to become zero
parametric equations
x = x(t) and y = y(t)
piecewise functions
defined by different equations on different parts of the domain
general absolute value function
f(x) = a * abs(x-h) +k vertex at (h,k)
translation of graph
moves graph around plane
accomplished by addition
stretching and shrinking graph
changing scale of graph
accomplished by multiplication
ellipses
the oval graphs, the set of points whose distances from two given points (foci) sum to a constant
ellipse equation
…
how to figure out wither ellipse is x or y orientated
which ever has the bigger constant underneath, usually defined as a while smaller constant is b
center of ellipse
h,k
vertices of ellipse
endpoints of major axis
(h-a, k) (h +a, k) if on x axis
(h, k-a) (h, k +a) if on y axis
foci of ellipse
on major axis, at distance of c from center, c = sqr( a^2 -b^2)
if the constants in the denominators equal each other in an ellipse equation, then
its the equation of a circle
equation of circle
(x-h)^2 + (y-k)^2 = r^2
hyberbola
set of points whose distances from two fixed points (focci) differ by a constant, the twin parabola graph
equation of hyberbola
…
how to figure out which directions the graph of a hyperbola points
if its x - y, its on the x-axis
if its y - x, than its on the y-axis
vertices of hyperbola
(h-a, k) (h+a,k)
or (h, k-a) (h, k+a) if on y axis
transverse axis
segment joining vertices of hyperbola
conjugate axis
segment with length 2b, perpendicular to transverse axis , intersects it at midpoint
equation of asymptotes of hyperbola
y-k = +or - (b/a)(x-h) if on x axis
if on y axis, its a/b instead of b/a
eccentricity of ellipse or hyperbola
c/a
c^2 = a^2 + b^2
polar cordinates
distance from origin and angle between x axis and ray
r, angle
relationships between polar and regular coordinates
x = r * cos(ang)
y = r * sin(ang)
x^2 + y^2 = r^2
volume of prism
v = Bh
surface area of prism
(perimeter of base)h + 2B
vol of rectangular prism
lwh
SA of rect solid
2lw + 2lh + 2wh
vol of cube
s^3
SA of cube
6s^2
vol of pyramid
(1/3)Bh
SA of pyramid
(1/2)(base perimeter)(slant height) + B
vol of cylinder
(r^2)(h)(pie)
SA of cylinder
2(pie)(r)(h) + 2(pie)(r^2)
vol of cone
(1/3)(pie)(r^2)(h)
SA of cone
(pie)(r)(slant height) + (pie)(r^2)
vol of sphere
(4/3)(pie)(r^3)
SA of sphere
4(pie)(r^2)
formula for length of diagonal of rectangular solid
sqroot of ((l^2)(w^2)(h^2))
distance formula for 3D plane
…
equation for sphere in 3D plane
…
know how to use venn diagram for counting problems
…
factorial
n!
ex: 5! = 5 * 4 * 3 * 2 *1 = 120
use to find number of ways objects can be ordered
permutations
(n!) / ((n-r)!)
use when having to only find partial number of ways to order objects out of the total set
Ex: ways to order first second and third out of set of 20, in this case n=20 and r=3
combinations
(n!) / (n-r)!(r!)
number of ways of choosing r of n objects
value of i
i = sqroot (-1)
powers of i
i^1 = i i^2 = -1 i^3 = -i i^4 = 1 i^5 = i pattern repeats from there
imaginary numbers
form of bi, where b= real number
sq root of negative number
i times the sqroot of the abs val of the number
complex number
a + bi, where a and b are real numbers
to find the quotient of 2 complex numbers….
multiply the denominator and numerator by the conjugate of the denominator
how to graphically represent complex numbers
x = real part (a) y = imaginary part (bi)
the modulus of a complex number is
the sq of its distance to the origin
the product of an imaginary number and its conjugate is
the square of the modulus
distance of complex number
sq root of ((a^2) + (b^2))
two matrices are equal if
they are the same size and their corresponding entries are equal
row matrix
a matrix with only one row
column matrix
a matrix with only one column
square matrix
a matrix with equal amount of rows and columns
scalar multiplication
when each number in a matrix is multiplied by a constant
two matrices can be subtracted or added together
if they are the same size
matrices A and B can be multiplied if
the # of rows of A = # pf columns of B
or vice versa
how to multiply matrices
multiply rows of A times columns of B and add values
determinant of 2 by 2 matrix
…
determinant of 3 by 3
…
sequence
a function with a domain consisting of natural numbers
the one with comas
ex: 1, 2, 3, 4, 5
series
sum of the terms of a sequence
the one with addition or subtraction marks
ex: 1 + 2 + 3 + 4 + 5
infinite series or sequence
has …. at the end
idea is that it goes on forever
finite series or sequence
only goes to a specific term
recursion formula
formula of a sequence/ series where every term is expressed with respect of the term before it
ex: insert picture pg153
series can be abbreviated by
using Greek letter sigma # on bottom the start # on top the end
arithmetic sequences
each term differs from previous term by a constant value
addition/subtraction sequence
formula for arithmetic sequence
…
formula of sum of arithmetic series
…
arithmetic mean
one term falling between two given terms of an arithmetic sequence
geometric sequence
ration of any two consecutive terms is constant r
multiply/ divide sequence
formula for geometric sequence
…
formula for sum of geometric series
…
geometric mean
one term falling between two given terms of a geometric sequence
if r is less than 1, than sum of geometric series as it approaches infinity equals
…
vector
a vector in a plane is defined to be an ordered pair of real numbers
vector in space defined as an ordered triple of real numbers
resultant vector
when two vectors are added together
Ex: vector U plus vector V has resultant of (u1 +v1, u2 + v2)
vector - V has ____ magnitude and _______ direction of vector V
equal , opposite
dot product
insert picture
dot product is a real number
2 vectors are perpendicular, their dot product equal
zero
measures of center
summarize a data set using a single “typical” value
the 3 on the SAT: mean, median, mode
mean
sum of all data values divided by number of values
median
order values from greatest to lest and find one in the middle
mode
value that repeats the most in the set
range
measure of the spread of the value set
highest value - lowest value
standard deviation
the average difference between individual data values and their mean
formula for standard deviation
…
regression
a technique for analyzing the relationship between two variables
probability of an event happening is a number defined to be the
number of ways the event can happen successfully divided by the total number of ways the event can happen
sample space of experiment
set of all outcomes of an experiment
odds in favor of an event happening are defined to be
the probability of the event happening divided by the probability of the event not happening
independent events
events that have no effect on one another
the probability of both events A and B happening is the same as the Probability of A times the probability of B
dependent events
events that are not dependent so
probability of both does not equal probability of A times probability of B
to find probability of “at least one”
find 1 - probability of “none”
mutually exclusive events
events where the probability of both happening is zero
probability of event A happening or event B happening or both happening equals
P(A) + P(B) - P(both)
generally in probability, “and” means ______ and “or” means _________
and means multiply and or means add
