Algebra II Final
y-coordinate of vertex
*f(-b/2a)* (plug x into original equation)
Additive Identity Element
0 is the additive identity element because for every number *a , a + 0 = 0 + a = a*
Multiplicative Identity Element
1 is the multiplicative identity element because for every number a, *a⋅1 = 1⋅a = a*
closure
the property of an operation and a set that the performance of the operation on members of the set always yields a member of the set
domain
the set of first coordinates of each ordered pair in a relation
range
the set of second coordinates of each ordered pair in a relation
degree of a term of a polynomial
the sum of the exponents on the variables in a term of a polynomial
Of the three properties, reflexive, symmetric, and transitive that define the relation "is equal to," which one could also apply to "is less than" and "is greater than?"
transitive
∪
union
identity matrix
a square matrix with ones in the main diagonal and zeros everywhere else
row addition
adding a multiple of one row to another row of a matrix
row
all of the numbers in one horizontal line in a matrix
column
all of the numbers in one vertical line in a matrix
standard form of a linear equation
an equation in the form *Ax + By = C* with not both A and B equal to zero
exponential equation
an equation in the form *y=ab^x* where *a ≠ 0, b ≠ 1, b > 0.*
linear equation
an equation in which each term is either a constant or the product of a constant and a single variable
quadratic equation
an equation of the form *ax^2 + bx + c = y* where a, b, and c are real numbers and a ≠ 0
rational equation
an equation that contains one or more rational expressions.
radical expression
an expression that includes a radical.
linear inequality
an inequality in which each term is either a constant or the product of a constant and a single variable
quadratic inequality
an inequality of the form *ax^2 + bx + c < y* where a, b, and c are real numbers and a ≠ 0 , or any similar form with <, ≤, >, or ≥.
entry
another name for an element of a matrix
relation
any set of ordered pairs
Before you can multiply two matrices together, the number of ____________ of the first matrix must equal the number of ____________ of the second matrix.
columns, rows
zero matrix
designated *0m × n*, the *m × n* matrix, all of whose entries are 0
Only matrices of the same __________________ can be added or subtracted.
dimension
A linear ___ is a polynomial equation of the first degree.
expression
Trichotomy Property
for any numbers a = b, either a < b, or a = b, or a > b
Commutative Property of Addition
for any numbers a and b, *a + b = b + a*
Commutative Property of Multiplication
for any numbers a and b, *a⋅b = b⋅a*
Symmetric Property of Equality
for any numbers a and b, if a = b, then b = a
Associative Property of Addition
for any numbers a, b, and c, *a + (b + c) = (a + b) + c*
Distributive Property of Multiplication over Addition
for any numbers a, b, and c, *a(b + c) = (a⋅b) + (a⋅c)*
Associative Property of Multiplication
for any numbers a, b, and c, *a⋅(b⋅c) = (a⋅b)⋅c*
Transitive Property of Equality
for any numbers a, b, and c, if a = b and b = c, then a = c
Multiplicative Inverse Property
for every number a except 0, there is a number *a^-1 = 1/a*, such that *a⋅(a^-1) = (a^-1)⋅a = 1*
Reflexive Property of Equality
for every number a, a = a
Additive Inverse Property
for every number a, there is a number −a, such that *a + (-a) = (-a) + a = 0*
Zero Product Property
if ab = 0, then a = 0 or b = 0
Substitution Property of Equality
if x = y, then y can be substituted for x in any expression
An example of a piecewise function is the greatest _________ function.
integer
row switching
interchanging two rows of a matrix
∈
member of
row multiplication
multiplying a row of a matrix by a nonzero constant
It is closed under addition and multiplication but not closed under subtraction or division.
natural numbers
rational numbers
numbers of the form a {a/b | a,b∈Z, b≠0} 0 and designated with Q
whole numbers
numbers {0, 1, 2, 3, . . .}
When multiplying matrices, multiply the elements in each _______________ of the first matrix times the corresponding elements in each ____________ of the second matrix.
row, column
tangent (tan)
of an angle θ, (0° < θ < 90°), in a right triangle, is the ratio of the side opposite to angle θ to the side adjacent to angle θ
The two restrictions on the value of *b* are that it must be a positive number and not equal ___.
one
The base of the logarithm function cannot be equal to ________ and must be _____________.
one, positive
The absolute value function is a ___ linear function.
piecewise
A linear function, in the form *f(x)=mx+b* is a ___ function.
polynomial
Closure means that whenever you add or subtract two polynomials, you get a ____.
polynomial
The composition of a polynomial function and another polynomial function will be a ___ function.
polynomial
If the discriminant, *b² - 4ac*, is ___, a quadratic will have two real roots, two points of intersection with the x -axis.
positive
A term is a constant, a variable, or a ___ of numbers and variables.
product
Quotients of polynomials are called ___ expressions.
rational
A variable is a letter that stands for a(n) ___ number.
real
When we combine complex numbers, we combine the ___ parts, then combine the imaginary parts.
real
irrational numbers
real numbers which cannot be written as the ratio of two integers; designated with ǭ (Q with line over it)
It is closed under addition and subtraction, multiplication, and division, with the exception of division by 0 which is not defined.
real numbers, rational numbers
The graph of the inverse of a function may be found by ___ over the line y=x
reflecting
Bernoulli trial
repeated trials of an experiment with two possible outcomes
The solutions to a quadratic equation are called the ___ or x -intercepts.
roots
The vertical asymptotes are found from the ____ of the denominator of a rational equation.
roots
elementary row operation
row switching, row multiplication, or row addition
The reciprocal of the cosine is the ___.
secant
The ratio of the leg opposite to an angle to the hypotenuse of a right triangle is called the ___ of the angle.
sine
If an inequality contains the symbols, ≤ or ≥ it would be graphed as a ___ line.
solid
x-intercept
solutions to the equation
If the leading coefficient of a polynomial function is negative, then the left end of the graph ____ points down.
sometimes
The range of a polynomial function is ____ all real numbers.
sometimes
The ___ deviation is the square root of the variance in a set of numerical data.
standard
If the variable of a radical function is multiplied by a number, the graph of the function will be ___ and enlarged by the value of that number.
stretched
⊂
subset of: { } = set notation
Each number in the triangle is the ___ of the two numbers directly above it in the previous row.
sum
The ratio of the leg opposite to an angle to the leg adjacent to the angle in a right triangle is called the ___ of the angle.
tangent
deviation
the absolute value of the difference between the member and the mean of the data in a set of numerical data
variance
the average of the sum of the squared differences of the mean from each element in a set of numerical data
mean
the average value of a set of numerical data, found by adding all the values and dividing by the number of elements in the set
binomial coefficient
the coefficient of each term of the polynomial expansion of (a+b)^n
conjugate
the conjugate of a complex number *a + bi* is the complex number *a - bi*
degree of a polynomial
the degree of the term with the highest degree in a polynomial
absolute value
the distance of a number from zero
discriminant
the expression, *b^2 - 4ac*, which tells the nature of the roots of a quadratic equation of the form *ax^2 +bx + c = 0*
factorial
the factorial of a counting number n, written n!, is the product of the counting number, n, and all of the counting numbers less than n
quadratic formula
the formula, *x = -b ± √(b² - 4ac)/2a*, used to find the solutions to a quadratic equation of the form *ax^2 + bx + c = 0*
quadratic formula
the formula, *x = -b ± √(b² - 4ac)/2a*, used to find the solutions to a quadratic equation of the form *ax² + bx + c = 0*
tangent
the function f(x)= tanx is defined for real numbers x as the quotient, where that quotient is defined, of the y-coordinate and the x-coordinate of a point on the unit circle centered at the origin such that the length of the arc from the point (1, 0) to that point is x radians, measured counterclockwise
cosine
the function f(x)=cosx is defined for all real numbers x as the value of the x-coordinate of a point on the unit circle centered at the origin such that the length of the arc from the point (1, 0) to that point is x radians, measured counterclockwise
sine
the function f(x)=sinx defined for all real numbers x as the value of the y-coordinate of a point on the unit circle centered at the origin such that the length of the arc from the point (1, 0) to that point is x radians, measured counterclockwise
logarithm
the logarithm *logb x* for a base *b* and a number *x* is defined to be the number *y*, so that *b^y=x*
opposite of a matrix
the matrix composed of the opposites of each entry in a matrix
dimension
the number of rows and the number of columns in a matrix
conditional probability
the probability of an event K occurring, given that event L has or will occur, written P(K|L) and reads "the probability of K given L"
composition of functions
the process of combining two functions using the output of one function as the input of another function
real numbers
the rational numbers together with the irrational numbers; designated with R
secant (sec)
the reciprocal of the cosine, in other words, secx=1/cosx
cosecant (csc)
the reciprocal of the sine, in other words, cscx=1/sinx
cotangent (cot)
the reciprocal of the tangent, in other words, cotx=1/tanx
hypotenuse
the side opposite the right angle in a right triangle
standard deviation
the square root of the variance in a set of numerical data
expected value
the sum of the values of a variable multiplied by the probability of that value occurring
Pythagorean Identity
trigonometric identity derived from the Pythagorean Theorem
independent
two events K and L are independent if P(K)=P(K|L)
inverse
two matrices are inverses if their product is the identity matrix
compound event
two or more events occurring at the same time
sine (sin)
of an angle θ, (0° < θ < 90°), in a right triangle, is the ratio of the side opposite to angle θ to the hypotenuse
e
a mathematical constant approximately equal to 2.71828
Cramer's rule
a method of calculating the solution to a system of linear equations by finding the quotients of determinants
radical equation
an equation that contains a variable within a radical expression.
rational inequality
an inequality involving a rational expression
oblique asymptote
an oblique line that a graph approaches as the value of a variable gets extremely large or extremely small
The quadratic formula can be used to find the roots of ___ quadratic equation.
any
solution
any value or values for a variable or variables that make an equation or inequality true
If the monomial is not zero, the product of a monomial and a polynomial will have ___ the polynomial.
as many terms as
A(n) _____ is a line that a graph approaches as the value of a variable gets extremely large or extremely small.
asymptote
In the equation f(x)=acos(2πb(x+c))+d, changing the parameter ______ changes the period to that value.
b
For an exponential function *f(x)= ab^cx*, changing the value for *c* will change the ____ *b* to *b^c*
base
In an exponential function of the form *f(x)= ab^cx*, the number *b* is called the ____.
base
If *f(x)=3^x* and *g(x)=7^x*, then graph of *f(x)* will be ___ the graph of *g(x)* when x > 0
below
In most cases, the product of a monomial and a binomial is a ___.
binomial
The coefficients of the terms on the right side of a ___ equation are the numbers in the triangle.
binomial
Will the border line of the graph of y<-x2+4x+5 be a solid line or a broken line?
broken
In the equation f(x)=acos(2πb(x+c))+d, changing the parameter ______ shifts the graph horizontally that many units.
c
To stretch the graph of *f(x)=a+clogb(dx+g)* change the parameter_____.
c
In general, the composition of functions is not ___.
commutative
A ___ is the intersection of a plane with one or both nappes of a double cone.
conic section
The numerators of any rational roots of a polynomial will be factors of the ___.
constant term
The reciprocal of the sine is the ___.
cosecant
The ratio of the leg adjacent to an angle to the hypotenuse of a right triangle is called the ___ of the angle.
cosine
The reciprocal of the tangent is the ___.
cotangent
In the equation f(x)=acos(2πb(x+c))+d, changing the parameter ______ shifts the graph vertically that many units.
d
To shift the graph of *f(x)=a+clogb(dx+g)* vertically, change parameters ___ or ___.
d, a
If an inequality contains the less than symbol (<), its graph would be a ___ line.
dashed
The quadratic ___ calculates the roots of a quadratic equation and indicates the nature of its graph.
formula
To shift the graph of *f(x)= a+clogb(dx+g)* horizontally, change the parameter____.
g
Any point that is on a ___ is a solution.
graph
You can determine if the inverse of a polynomial function is a function by using the ____ line test on the polynomial function.
horizontal
The side opposite the right angle in a right triangle is called the ___.
hypotenuse
There will ____ be a horizontal asymptote if the degree of the numerator is greater than the degree of the denominator.
never
If the value under the square root sign in the quadratic equation is negative, there are ___ x -intercepts.
no
The graphs of a line and parabola could intersect at one point, two points, or ___.
no points
Every linear function with a ___ slope will have an inverse function.
nonzero
The set of polynomials is ___ closed under division.
not
∉
not a member of
No matter what the base, an exponential function of the form *f(x)=b^x* always goes through the point ___.
(0, 1)
Place a check mark in the box if the expression is a term.
-7xy, 5x
A determinant will have a reciprocal, and a matrix will have an inverse if the determinant is not ___.
0
The local grocery store has sorted apples by size. The distribution is approximately normal. The large apples have a mean diameter of 10 cm with a standard deviation equal to 0.5 cm. In a bin of 100 large apples, how many can be expected to have a diameter greater than 11.5 cm, three standard deviations above the mean?
0
The possible number of intersection points of two different ellipses range from ___ to as many as four.
0
No matter what the base, an exponential function of the form *f(x)=b^x* always goes through the point ( ___, *b*) where *b* is the base.
1
The equation f(x)=3cos(2πx) is used to model the motion of a weight attached to the end of a spring. The period is measured in seconds. What is the period of the motion of the weight?
1 second
On a certain standardized test used for college entrance purposes, the mean score was 21 and the standard deviation was 5. The distribution was approximately normal. Carlos scored 31, two standard deviations above the mean. A total of 1.5 million people across the country took the test at the same time as Carlos. How many people had scores lower than Carlos?
1,465,500 people
To find the solution area of the graph of an inequality, chose a point ___ the curve and determine if it is a part of the solution.
not on
A constant is a real ___.
number
The absolute value of a positive number is the ___ itself.
number
If the coefficient in front of the x² in a quadratic equation is negative, the parabola will curve up.
False
integers
numbers {0, +1, −1, +2, −2, . . .} and designated with Z
natural numbers
numbers {1, 2, 3, 4, . . .} and designated with N
If the degree of the numerator is exactly one greater than the degree of the denominator, there will be a(n) ____ asymptote.
oblique
cosine (cos)
of an angle θ, (0° < θ < 90°), in a right triangle, is the ratio of the side adjacent to angle θ to the hypotenuse
A flashlight battery manufacturer makes a model of battery whose mean shelf life is three years and four months, with a standard deviation of three months. The distribution is approximately normal. One production run of batteries in the factory was 25,000 batteries. How many of those batteries can be expected to last between three years and one month and three years and seven months?
17,050 batteries
The number *e* is approximately equal to ___.
2.71828
The equation f(x)=3cos(2πx) is used to model the motion of a weight attached to the end of a spring. How many units are there between the highest and lowest points in the motion of the weight?
6 units
The equation g(x)=3sin(π4x)+3 is used to model the motion of a rocking chair. The front edge of the rockers on the chair start out at 3 cm above the floor. The person in the chair starts by rocking back, which raises the front edge 3 more centimeters. When the chair rocks forward, the front edge of the rocker touches the floor. The period is measured in seconds. What is the period of the rocker?
8 seconds
At the 2012 Summer Olympic Games in London, in the Men's Shot Put qualifying round, the distances ranged from 17.58 meters to 21.36 meters with a mean distance of 19.64 meters and a standard deviation equal to 0.92 meters. If the distribution were truly a normal distribution, what percent of the athletes would have had a distance greater than 20.56 meters, one standard deviation above the mean? There were 40 athletes in the competition. How many athletes does that represent?
About 6 athletes
Multiplying polynomials is done by applying the ___ Property when necessary.
Distributive
Factoring should always be used to solve quadratic equations.
False
horizontal line test
If every horizontal line intersects a graph of a function just once, then the function is one-to-one and therefore an inverse function exists.
vertical line test
If every vertical line intersects a graph just once, the graph is of a function.
The triangle of numbers used to find the pattern for any power of binomials is called ___ Triangle.
Pascal's
A trigonometric identity derived from the Pythagorean Theorem is called a ___ Identity.
Pythagorean
_____ number roots of a polynomial are the points where the graph of the related polynomial function crosses the x -axis.
Real
Describe how the graph of *f(x)= 4/x* will differ from the graph of *g(x)= 1/x*
The graph will be "stretched" and enlarged by a factor of four.
Describe how the graph of *f(x)= 1/x+3* will differ from the graph of *g(x)= 1/x*
The graph will be shifted to the left three units.
Describe how the graph of *f(x)= 1/x+5* will differ from the graph of *g(x)= 1/x*
The graph will be shifted up five units.
If the parabola intersects the x -axis at two points, the axis of symmetry will be halfway between the two points of intersection.
True
Seth claims that the equation h(x)=3cos(π4(x-2))+3 will model the behavior of a rocking chair just as well as g(x)=3sin(π4x)+3. Use a graphic utility to determine if Seth was correct.
Yes
In the equation f(x)=acos(2πb(x+c))+d, changing the parameter ______ changes the distance between the maximum and minimum values.
a
If abc = 0, then ______.
a = 0 OR b = 0 OR c = 0
exponential function
a function of the form *f(x)= ab^cx* where *a*, *b*, and *c* are real numbers, *b > 0*, *b ≠ 1*
logarithmic function
a function of the form *f(x)= logb^x*
regression model
a function that estimates the relationship among two variables
standard normal curve
a graph of the standard normal distribution
horizontal asymptote
a horizontal line that a graph approaches as the value of a variable gets extremely large or extremely small
asymptote
a line that a graph approaches as the value of a variable gets extremely large or extremely small.
standard normal distribution
a normal distribution in which the mean is 0 and the standard deviation is 1
determinant
a number calculated from the entries in a square matrix that gives information about the matrix, including the nature of the solutions to a related system of linear equations
element
a number in a matrix
complex number
a number of the form *a + bi* where *a* and *b* are real numbers and *i² = -1*; the set of complex numbers is designated by *C*
imaginary number
a number of the form *bi* where *b* is a real number, and *i² = -1*
vector
a quantity that has both length and direction
synthetic division
a shortcut method of dividing a polynomial by a linear polynomial by using only the coefficients of the terms of the polynomials
normal distribution
a statistical distribution of data which is symmetrical and in which the mean, median, and mode are equal
Pascal's Triangle
a triangular array of numbers such that each row is the coefficients of the terms of the expanded form of the powers of a binomial of the form *(a + b)^n*
radian
a unit of angle measure such that an angle with center at the center of a unit circle, with measure one radian produces an arc with arc length one
root of a polynomial
a value for the variable of a polynomial that makes the polynomial equal to zero
vertical asymptote
a vertical line that a graph approaches as the value of a variable gets extremely large or extremely small
If *f(x)=0.5^x* and *g(x)=0.3^x*, then the graph of *f(x)* will be _____ the graph of *g(x)* when x < 0
above
When finding the solution to a system of equations, it is important to find ___ solutions.
all
If the highest exponent of a polynomial function is odd, then the range of the function is ____ all real numbers.
always
The domain of a polynomial function is ____ all real numbers.
always
The set of polynomials is ___ closed under multiplication.
always
The ___ is the absolute value of the difference between the member and the mean of the data in a set of numerical data.
deviation
When we compose functions, we must make sure that the output of the first function is part of the ___ of the second function.
domain
If a negative number is added to a radical function, its graph will shift ___ by the value of that number.
down
To find the value of y using the value of x, use ___ equation of the system.
either
The quadratic formula will give us the coordinates of the points of intersection of a line and a quadratic only when the value of the discriminant, *b² - 4ac*, is ___.
either positive or zero
The inverse of an exponential function tells us the ___ to which the base of an exponential function is raised to give us the value x.
exponent
To find the roots of a polynomial, it is often useful to find the ____ of the polynomial.
factors
counting principle
if a single event has m choices and a second event has n choices, the total number of possible outcomes for the compound event is m×n outcomes
Complex numbers are represented on a Cartesian coordinate system with a horizontal real axis and a vertical ___ axis.
imaginary
law of cosines
in a triangle with vertices A, B, and C and sides of lengths a, b, and c opposite the angle with the same letter name, c^2 = a^2 + b^2 - 2abcosC
law of sines
in a triangle with vertices A, B, and C and sides of lengths a, b, and c opposite the angle with the same letter name, sinA/a = sinB/b = sinC/c
base of an exponential function
in an exponential function of the form *f(x)= ab^cx*, the number *b*
The inverse of a function can be found by ___ the numbers in each ordered pair of the function.
interchanging
The denominators of any rational roots of a polynomial will be factors of the ____.
leading coefficient
If a positive number is added to the variable of a radical function, its graph will shift to the ___ by the value of that number.
left
We expect to see a ___ for the graph of a composition of a function and its inverse function, if the domain of each is all real numbers.
line
The number, y so that b^y=x is called the ____ of x.
logarithm
natural logarithm
logarithm with base *e*, and indicated by *lnx*
common logarithm
logarithm with base 10, and indicated by log x
The ___ is the average value of a set of numerical data, found by adding all the values and dividing by the number of elements in the set.
mean
periodic motion
motion repeated in equal intervals of time
If the discriminant is ___, a quadratic will have no real number roots and will not intersect the x -axis at all.
negative
If the highest exponent of a polynomial function is even, then the range of the function is ____ all real numbers.
never
If the leading coefficient of a polynomial function is positive, then the right end of the graph ____ points down.
never
The ___ is the average of the sum of the squared differences of the mean from each element in a set of numerical data.
variance
You can determine if the inverse of a polynomial function is a function by using the ____ line test on the inverse.
vertical
If the determinant of the coefficient matrix is equal to zero, Cramer's Rule ___ work.
will not
A function composed with its inverse function will always equal ___.
x
What are the restrictions on the domain of *g(x)= 1/x*?
x ≠ 0
The value of a polynomial at x=1 is the remainder when the polynomial is divided by ____.
x-1
The greater the base, the closer the graph is to the ___ when x>1.
x-axis
y = x^2 + 4x - 5
x-coordinate of vertex: -2 y-coordinate of vertex: -9 x intercepts: x=-5, x=1
x-coordinate of vertex
x=-b/2a
The graph of the inverse of a polynomial function is the reflection of the graph of the polynomial function over the line ___.
y = x
For an exponential function *f(x)= ab^cx*, changing the value for *a* will change the ____ to the value of *a*
y-intercept
Is the point (0,0) part of the solution of y < -x^2 + 4x + 5?
yes
If the discriminant is ___, there will be one real number root and the vertex of the quadratic will be on the x -axis.
zero
Roots of a polynomial are the values that make the polynomial equal to ___.
zero
linear system
a set of two or more linear equations or inequalities considered together
rational expression
a quotient of two polynomials; an expression of the form *P(x)/Q(x)* where *P(x) and Q(x)* are polynomials and *Q(x) ≠ 0.*
matrix
a rectangular array of numbers
___ is the highest power or the sum of the powers (exponents) on the variables of a termin a polynomial.
Degree
x + 4 = x - 2/ x
Solution: *x = -1 or x = -2* Extraneous solution: *none*
√4x = x - 3
Solution: *x = 9* Extraneous solution: *x = 1*
transpose
a related matrix formed by making the rows of a matrix into columns and the columns into rows
inverse
a relation found by interchanging the domain and range values in each ordered pair of a relation
term
a constant, variable, or the product of a constant and variables
graph
a diagram showing the relations among numbers
piecewise function
a function defined differently on different intervals of the function's domain
asymptote
a line that a graph gets closer to as the value of a variable gets extremely large or extremely small.
slope-intercept form
a linear equation of the form where m is the slope and b is the y-intercept
polynomial
a mathematical expression consisting of constants and variables, combined with the operations of addition and multiplication
linear equation
a mathematical statement that two linear expressions, or a linear expression and a constant, are equal
linear inequality
a mathematical statement that two linear expressions, or a linear expression and a constant, are not equal
augmented matrix
a matrix formed by adding a column of the constant terms to the coefficient matrix of a linear system
coefficient matrix
a matrix formed from the coefficients of the variables of a system of linear equations
square matrix
a matrix which has equal numbers of rows and columns
Gaussian elimination
a method of solving a system of equations or inequalities by systematically adding or subtracting a multiple of one equation or inequality to a multiple of another equation to eliminate a variable
scalar
a number which can be used as an element of a matrix
linear expression
a polynomial of the first degree
monomial
a polynomial with one term
trinomial
a polynomial with three terms
binomial
a polynomial with two terms
function
a relation in which each domain element is paired with exactly one range element
