Chapter 1-7
x*-m/n
((x*1/n)*m)*-1
what is the reciprocal way of rewriting the expression x^-m/n
((x^-1)^m)^1/n
(b*m)(b*n)
b*m+n
b*m/b*n
b*m-n
quotient of powers
b*m/b*n
domain of f(x)=-x^8
set of all real numbers
what do you know about the points that make up a function
they can form a horizontal line, but not verticle
what is the nth root way of rewriting the expression x^-m/n
((x^1/n)^m)^-1
what is the mth power way of rewriting the expression x^-m/n
((x^m)^1/n)^-1
the graph of a direct variation function goes through
(0,0)
what point does every power function f(x)=x^n pass through
(0,0) the origin
(ab)*m
(a*m)(b*m)
power of a quotient
(a/b)*m
power of a quotient postulate
(a/b)^m = a^m/b^m
power of a product
(ab)*m
power of a product postulate
(ab)^m = a^m b^m
product of powers
(b*m)(b*n)
power of a power
(b*m)*n
powers of powers postulate
(b^m)^n = b^m*n
i*2
-1
explicit formula
an=a1+(n-1)d
domain of f(x)=-x^9
set of all real numbers
A u B -or/union
the union of sets A and B. Draw all on one graph and leave as is
what is the vertex
(h,k)
what is the vertex of y-k = a(x-h)*2
(h,k)
w varies directly as a variable t
(t,w) w=dependent t=independent
what is a perfect square trinomial
(the square of a binomial) x*2 + 2hx + h*2
y-k=a(x-h)*2
(vertex form for the equation of a parabola) h greater than 0 moves right, h less than 0 moves left, k greater than 0 moves up, k less than 0 moves down
binomial square theorem
(x + y)*2 = x*2 + 2xy + y*2
x*m/n
(x*1/n)*m
discrete domain
not all numbers between
in the direct variation function y=kx what does the constant of variation represent
the slope
natural numbers
1,2,3
(a/b)*m
a*m/b*m
generic notation for is y a function of x
(x, y)
integers
-2,-1,0,1,2
every negative real number has
0 real nth roots when n is even 1 real nth root when n is odd
b*2-4ac is less than 0
0 x intercepts, two non real solutions
whole numbers
0,1,2,3
real numbers
0,1,27,2.34,pie, square root of 5 (all rational and irrationals)
b*2-4ac is equal to 0
1 x intercept, 1 real solution
steps for graphing a linear inequality in the coordinate plane
1-graph the appropriate boundary line (greater than is dashed, greater than or equal to is solid) 2-test (0,0) and see if it satisfies the inequality 3-shade the 1/2 plane that satisfies the inequality
how do we use the linear combination method when there are three equations in standard forms
1-take any pair of equations and eliminate 1 of the variables 2-tale another pair of equation and eliminate the same variable
what is the four step algorithm for using variation functions to predict values
1-write an equation that describes the variation 2-find the constant of variation 3-rewrite the variation functions using the constant of variation 4-evaluate the function for the desired value of the independent variable
how many points determine a line
2
every positive real number has
2 real nth roots when n is even 1 real nth root when n is odd
b*2-4ac is greater than 0
2 x intercepts, 2 real solutions
how many data points do you need in order to fit a quadratic model to data
3 non colinear points, no 2 on the same vertical line
annual compounding interest formula
A(t) = P(1+r/n)^nt a=total amount p=amount invested r=rate compounded t=time in years n=number of times compounded per year
standard form for the equation of a linear function
Ax+By=C
quadratic formula theorem
X=-b+or- the square root of b*2-4ac/2a
what is the complex conjugate of a + bi
a - bi
what is a linear function
a function with an equation in the form of y=mx+b and graphs into a line
what is a scatter plot
a graph of all the ordered pairs in a set of data
what is the range of ax*2 + bx + c
a greater than 0=up, y greater than or equal to k a less than 0=down, y less than or equal to k
what is the range of y-k = a(x-h)*2
a greater than 0=up, y greater than or equal to k a less than 0=down, y less than or equal to k
what is the graph of Ax+By=C when A and B are not both zero
a line
what type of function can always be used to model a constant increase or decrease situation
a linear equation f(x)=mx+b
what is the result of fitting a line to data
a linear model that can be used to estimate the relation in the data
what is a arithmetic sequence
a list of numbers in which the order matters and which has a constant difference between successive terms
what is a geometric sequence
a list of numbers in which the order matters and which has a constant multiplier between successive terms
definition of a complex number
a number in the form a + bi, where a and b are real numbers and i = the square root of -1, a is called the real part and bi is called the imaginary part
what is the curve of a direct variation function called
a parabola
sequence
a pattern (1,3,6,10,15)
what does pitch mean
a ratio
what is a system
a set of conditions (usually expressed as equations or inequalities) joined by the word AND
what is a consistent system of equations
a system which has 1 or more solutions
what is an inconsistent system of equations
a system which has no solutions
rational numbers
a/b when a and b a re both integers, and b does not equal 0
recursive formula
a1=__first term___ an=a(n-1)+d
what is the solution to the simplest quadratic equation x*2 = k
abesolute value of x*2 = square root of k, x = square root of k or x = negative square root of k
what is the piecewise algebraic definition for the absolute value of a number x?
absolute value of x = x for x greater than or equal to 0, -x for x less than 0
how to complete the square of a binomial on x*2 + bx
add (1/2 b)*2 to both sides
how do we add or subtract complex numbers
add or subtract the real parts and add or subtract the imaginary parts, then put in a + bi form
range for the absolute value of x
all nonnegative real numbers
continuous domain
all numbers between
find the domain of a relation
all numbers of the independent variable (x)
domain for the abesolute value of x
all real numbers
domain of the direct variation function y=kx*2 when k is greater than 0
all real numbers
domain of the direct variation function y=kx*2 when k is less than 0
all real numbers
what is the domain and range of an oblique line
all real numbers
what is the domain of ax*2 + bx + c
all real numbers
what is the domain of y-k = a(x-h)*2
all real numbers
find the range of a relation
all the numbers of the dependent variable (y)
correlation coefficent
an r value of ____ would indicate a ___strong/weak, positive/negative___ correlation between the __independent variable__ and the ___dependent variable___
in a geometric sequence what can you tell about the sequence when r is less than 0
as n increases gn alternates between positive and negative numbers
in a geometric sequence what can you tell about the sequence when r is greater than 0 and less than 1
as n increases, gn decreases
in a geometric sequence what can you tell about the sequence when r is greater than 1
as n increases, gn increases
statement for when something is not a function
at least one value of the domain corresponds to more than one value of the range
what is the general quadratic expression
ax*2 + bx + c
what is the general quadratic equation
ax*2 + bx + c = 0
discriminant of the quadratic equation
b*2-4ac
(b*m)*n
b*mXn
negative exponent theorem
b^-n = 1/b^n
product of powers postulate
b^m * b^n = b^m+n
quotient of powers postulate
b^m/b^n = b^m-n
zero exponent theorm
b^n/b^n = b^n-n = b^0 = 1
what is the y intercept of ax*2 + bx + c
c
rewrite d=k/w without fractions
d=kw^-1
the _______ variable is a function of the _________ variable
dependent, independent
y=kx
direct variation, k is the slope, points make an oblique line, domain/range = all real numbers, goes through origin, ordered pain (1,k), do not have asymptotes, k + slope = line going up, k - slope = line going down
which direction does the parabola open when k is less than 0
down
term of a sequence
each number in a sequence 1st=1 2nd=3 3rd=6 4th=10 5th=15
statement for when something is a function
each value of the domain corresponds to exactly one value of the range
what is the general quadratic function
f(x) = ax*2 + bx + c
recursive formula for a geometric sequence
g1=_____ gn = rg(n-1) for integers greater than or equal to 2
in order to write a linear equation in slope-intecept form we need to...
get the y value alone
explicit formula for a geometric sequence
gn = g1(r)^n-1 for integers greater than or equal to 1
equation for free falling objects
h = -1/2 g t*2 + v t + h g=acceleration due to gravity (9.8 m/sec, 32ft/sec) t=time v=initial velocity h= initial height
formula for the amount of time pasturized milk will stay fresh
h(t) = 349 * 10^-.02t
when the rate of change for a linear situation is zero, what type of line do we have
horizontal
y=k/x
hyperbola, inverse variation, vertical asymptote x=0, horizontal asymptote y=0, domain=all real numbers except 0, range=all real numbers except 0, there is no y or x intercepts, graphs into a hyperbola, symmetry line y=x and y=-x
all square roots of negative numbers are multiples of
i
negative exponent
if b does not equal 0, b*-n = 1/b*n
zero exponent
if b does not equal 0, b*0 = 1
how do we know if a system is inconsistent
if the statement has no solutions
parabola congruency
if they have the same a value they are congruent
y=k/x*2
inverse variation, vertical asymptote x=0, horizontal asymptote y=0, domain=all real numbers except 0, range k greater than 0=all real positive numbers except 0, range k less than 0=all real negative numbers except 0, symmetry is y axis (x=0)
what can you conclude about the slope of the direct variation function y=kx
it is constant slope k
what do you know about the rate of change for the direct variation function y=kx*2
it is not constant (the graph is not linear, there is not one slope
in an arithmetic sequence what is true about the difference between any term and the preceding term
it will be the same for every adjacent pair in that sequence. the constant difference
slope of a direct variation function y=kx
k
definition of a direct variation function
k is a non zero constant, called the constant variation, and n is a positive number (y=kx*n)
what is the constant of variation in the definition of direct variation above
k-nonzero constant
what is the graph of the solution to an inequality
linear
how to create a quadratic model for data using your calculator
lists and spreadsheets, enter data, statistics, stat calc., quadratic
the graph of an absolute value
makes a v with 2 straight lines
does the direct variation function have a min or max point when k is less than 0
maximum
does the direct variation function have a min or max point when k is greater than 0
minimum
how do we handle dividing by non real complex numbers
multiply both the numerator and the denominator by the complex conjugate of the denominator. Divide each term in the numerator by the real number result
how to clear fractions
multiply each side of the equation by the least common denominator
what is the range of every power function
n is even = the set of nonnegative real numbers n is odd = the set of all real numbers
parallel lines
no intersection implies the system is inconsistent, no solutions
can you raise a positive number to a power so that the result is negative?
no they just get closer to 0
vertical line test for functions
no vertical line intersects the graph of a function in more than 1 point
what kind of numbers have square roots that are irrational numbers
non perfect squares, square roots that are irrational
range of the direct variation function y=kx*2 when k is greater than 0
nonnegative real numbers
range of the direct variation function y=kx*2 when k is less than 0
nonpositive real numbers
y=kx*2
not linear, graph is a parabola, direct variation, does through origin, ordered pair (1,k), doesn't have one slope, never has asymptotes, ling of symmetry x=0, domain = all real numbers, range k greater than 0= all numbers greater than or equal to 0, range k less than 0= all numbers less than or equal to 0
joint variation
one quantity varies directly as the product of 2 or more independent variables (no division)
the slopes of perpendicular lines are
opposite reciprocals
what do you know if a is less than 0
parabola opens down, graph has a maximum y value (k), the range is less than or equal to k (y is less than or equal to k)
what do you know if a is greater than 0
parabola opens up, y coordinate of the vertex is the minimum y value (k), the range is greater than or equal to k (y greater than or equal to k)
what kinds of numbers have square roots that are rational numbers
perfect square integers (1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144)
how to find the constant of variation for a particular situation
plug know values into y=kx*n and solve for k
what is another term used for the slope of a linear function
rate of change
if a,b,c are rational numbers and b*2-4ac is a perfect square then then solutions will be
rational
symmetry if n is even
reflection symmetric over the y axis
symmetry if n is odd
rotation symmetry at a 180 degree rotation around the origin
range of f(x)=x^8
set of al nonnegative real numbers y greater than or equal to 0
range of f(x)=-x^8
set of all nonpositive real numbers y less than or equal to 0
domain of f(x)=x^8
set of all real numbers
domain of f(x)=x^9
set of all real numbers
range of f(x)=-x^9
set of all real numbers
range of f(x)=x^9
set of all real numbers
domain of direct variation y=kx
set of real numbers
range of direct variation y=kx
set of real numbers
what translation does the notation T(h,k) indicate
slides a figure h units horizontally, and k units vertically
vertical line
slope is undefined
horizontal line
slope is zero
i
square root of -1
irrational numbers
square root of 2, any square root of a non perfect square, non ending numbers
index
subscript
how do we find the y intercept
substitute 0 for x
how do we find the x intercept
substitute 0 for y
the substitution method for solving systems of equations uses what property of equaltiy
substitution property of equality
what does the word quadratic refer to
sum of possibly constant, possibly term with a variable to the first power, and definitely a term with a variable to the second power with no higher power
linear programming theorem
the feasible region of every linear programming problem is convex, and the maximum or minimum quantity is determined at one of the vertices of this feasible region
what does the phrase "in terms of" mean
the first variable is isolated, and the other side of the sentence includes the "in terms of" variable
A n B -and
the intersection of sets A and B. Draw 2 separate lines and then find what lies on both
what is the solution set for a system
the intersection of the relations (all common ordered pairs)
describe the solution set of a compound sentence which uses the work and to connect two inequalities
the intersection of the solution sets to the individual sentences. Values common to both sets
what does a line graphed in a plane do to the plane
the line separates the plane into 2 distinct regions called half planes, the line is called the boundary line
subscript
the number written below/to the right of a variable tn
what is a piece-wise linear function
the rate of change is constant, but then changes to another constant rate
what is the domain of every power function
the set of all real numbers
define what is meant by a lattice of points
the solution set consists of points whose coordinates are integers are called lattice points
what does it mean when the linear combination method results in a sentence like 0=12
the statement is inconsistent and has no solutions
what does it mean when the linear combination method results in a sentence like 12=12
the system is consistent and has infinitely many solutions
2 intersections
the system is consistent with 2 solutions
same equation
the system is consistent with infinite solutions
no intersections
the system is inconsistent, no solutions
subscripted variable
the variable and its subscript (tn)
which direction does the parabola open when k is greater than 0
up
how do we multiply complex numbers
use the distributive property, then simplify
explicit formula
used to calculate any term in a sequence by substituting a value for n t(n)=n(n+1)/2
how are complex numbers formed
when a real number and an imaginary number are added
what does the term combined variation refer to
when direct and inverse variations occur together
when is a mathematical sentence called an inequality
when it contains one of the symbols (greater than, less than, greater than or equal to, less than or equal to, or not equal)
exponent theorem
when x is greater than or equal to 0 and n is an integer greater than 1, 1/x^n is an nth root of x
when solving inequalities when are we required to reverse the inequality sign
when you multiply or divide by a negative
negative rational exponents
x*-m/n
what is the simplest quadratic expression
x*2
rational exponents
x*m/n
equation for the axis of symmetry of the parabola
x=0
what is the axis of symmetry of y-k = a(x-h)*2
x=h
what is the line of symmetry
x=h
point slope form
y-y1=m(x-x1)
definition of slope
y2-y1/x2-x1
a hyperbola is the graph of what function
y=k/x
inverse
y=k/x*n (if x is multiplied by nonzero constant c, y is divided by c*n)
definition of inverse variation function
y=k/x*n k does not = 0 and n is greater than 0
direct
y=kx*n (if x is multiplied by c, y is multiplied by c*n)
is y a function of x
yes, the graph of an oblique line passes the vertical line test