Precalculus

a coinciding system's equation ________ the set of points in common

*is*

A polynomial function has a# _____, the exponents are_______________, and the coefficients are _______________.

0; whole numbers; real numbers

Linear programming can be broken down into seven steps:

1. Define your variables. 2. List the given information. 3. Write linear inequalities (called constraints). 4. Graph the linear inequalities to create a "feasibility region." 5. Find the corner points of the feasibility region. 6. Evaluate the corner points in an expression for the value to be optimized. 7. Answer the question.

parent function

A basic function that has not undergone any transformations.

matrix

A collection of numbers represented in a rectangular format. Rows are horizontal groups of numbers (though the groups could contain names or kinds of information), and columns are vertical groups of numbers.

arithmetic sequence

A collection of numbers separated by commas in which the difference between any two adjacent numbers is constant.

series

A collection of terms in a sequence that are added together to produce a sum.

ordered pair

A combination of x- and y -coordinates that describe a point on a graph.

point-slope form

A form of a linear equation that includes the coordinates of a point on the line and the line's slope

slope-intercept form

A form of a linear equation that includes the slope of the line and the value of the y-intercept.

explicit formula

A formula that allows the direct calculation of all terms of a sequence.

recursive formula

A formula that defines numbers in a sequence in terms of previous numbers in the same sequence.

one-to-one

A function in which each y-value has a single x-value mapped to it.

many-to-one

A function in which two or more x-values map to a given y-value.

transformation

A function or rule that maps an arrangement of points to a new arrangement of those same points.

polynomial function

A function that consists of a polynomial expression, made up of the addition or subtraction of monomial terms.

whole number

A nonnegative integer (0, 1, 2, 3, 4, 5, ...).

intercept

A point at which a graph crosses either the x- or y-axis.

feasible region

A region on a graph that includes all possible solutions to a system of inequalities

commutative property

A rule stating that adding or multiplying two numbers in either order will not change the answer. The commutative property of addition can be written as a + b = b + a. The commutative property of multiplication can be written as a b = b a. Example: The commutative property of addition says that 7 + 4 = 4 + 7.

sequence

A set of numbers written in a particular order.

system of linear equations

A set of two or more equations.

sigma notation

A shorthand notation using the Greek letter sigma to denote the sum of a series.

dilation

A transformation that changes the size of a shape, either by enlarging it or shrinking it.

reflection

A transformation that produces a mirror image of a set of points about a fixed line on a coordinate plane.

rotation

A transformation that rotates a set of points around a single fixed point on the coordinate plane.

translation

A transformation that shifts a set of points horizontally, vertically, or in both directions on a coordinate plane.

zero

A value of a variable that makes a function equal zero.

root

A value of a variable that makes an equation true. Ex: y=(x+2)(x+3) Roots: -2,-3

undefined

A value that cannot be computed.

vertical line test

A vertical line drawn through the graph of a function will intersect the graph in, at most, one place.

function operations

Addition: (f + g)(x) = f(x) + g(x) Subtraction: (f - g)(x) = f(x) - g(x) Multiplication: (f . g)(x) = f(x) . g(x) Division: (f/g)(x) = f(x) / g(x) Composition: (f o g)(x) = f(g(x))

linear equation

An equation in which the variable is of degree one.

monomial

An expression that is a variable, a real number, or a product of variables and real numbers having whole-number exponents.

vertex

An extreme point, either a maximum or a minimum, of the graph of a quadratic function.

linear inequality

An inequality in which the variable is of degree one.

The standard form of a line is ____________________, where A and B represent__________ and C represents the ____________________.

Ax + By = C; coefficients; constant term or value Slope = -A/B y-intercept = B/C

composition of functions

Combining two functions into a single function.

Domain rules

For rational functions and functions with square roots, the domain can be all real numbers except (1) anything that will make the denominator equal to zero (2) anything that will make an expression under a radical less than zero.

complex numbers

Numbers that include both a real term and an imaginary term (i).

end behavior

The behavior of a graph as x approaches positive or negative infinity.

axis of symmetry

The line passing through the vertex that separates a parabola into two identical halves. It's equation is x=-b/2a

synthetic substitution

The method of solving polynomial functions using a table of values instead of direct substitution.

relative extrema

The number of turns in the graph of a function. These are also called local maximum or local minimum values.

When two lines are perpendicular

The result of their slopes' multiplication will be -1 Ex: y= 2x +5 and y=-1/2x a.a = 2 (-1/2) = -1

degree

The sum of all the variables' exponents in a monomial. The degree of a polynomial function is the same as the highest-degree monomial within that function.

partial sum

The sum of the first n terms of a sequence. Denoted by the symbol Sn. Ex: Sn=S1 + S2 + S3 + S4 + ... + Sn

discriminant

The term underneath the radical in the quadratic equation: b^2 - 4ac

If , then the graph of is a __________ reflection of theoriginal function. This means that the graph will be reflected across the__________ axis.

horizontal; vertical

coincide

To overlap, or in the case of lines, to be identical.

evaluate

To substitute a numeric value into a function and solve it.

Function

a special kind of relation in which each value of the input variable is paired with exactly one value of the ouput variable.

distributive property

a(b+c)=ab+ac

a. __________ the graph of __________ by a factor of c when c > 1. b. __________ the graph of __________ to c times as tall when c < 1. c. __________ the graph of __________ to times as wide when c > 1. d. __________ the graph of __________ by a factor of c when c > 1.

a. stretches; vertically b. shrinks; vertically c. shrinks; horizontally d. stretches; horizontally

When using the linear combination method to solve a system of linear equations, the four options are: a. Eliminate the _____ terms with __________. b. Eliminate the _____ terms with __________. c. Eliminate the _____ terms with __________. d. Eliminate the _____ terms with __________. y;

a. x; addition b. x; subtraction c. y; addition d. subtraction

To find the partial sum of an arithmetic sequence, add _____ and _____together and multiply the sum by _____.

a1; an; n/2

The degree of any monomial is calculated by __________ the __________ ofits __________.

adding; exponents; variables

The commutative property holds for both the __________ and __________ offunctions, but not for the __________ and __________ of functions.

addition; multiplication; subtraction; division

When two lines coincide, the system of linear equations has_________________________ solution(s).

an infinite number of

There are three important matrices involving a system of equations. They arematrix A, the __________ matrix, matrix X, the __________ matrix, and matrix C,the __________ matrix.

coefficient; variable; constant

The standard form of a __________ number is , where a is called the__________ part and b is called the __________ part.

complex; real; imaginary

The distributive property (does/does not) work for functions. Symbolically, this is written as ____________________.

does not; f(x+y) # f(x) + f(y)

Parallel lines have slopes that are __________. Two lines with slopes m1 and m2 are perpendicular if and only if their slopes are ____________________; that is, so that m1 • m2 = __________.

equal/negative reciprocal/-1

If n is __________, the left side will match the right side.

even

absolute value function

f(x) = |x|. The magnitude of x.

Linear programming

finding the minimum or maximum value of a certain quantity, such as profit or expense, given a set of constraints to which the results must adhere.

A _________ line has a slope of zero, while a __________ line has a slope that is undefined.

horizontal; vertical

If b^2 - 4ac >0, but is not a perfect square, the solutions of the quadratic equation will be two distinct (unequal) _______ numbers. If b^2 - 4ac >0, and is a perfect square, the solutions of the quadratic equation will be two distinct (unequal) _______ numbers. If b^2 - 4ac =0, the solutions of the quadratic equation will be _____. In this case, the root is repeated, or has a multiplicity equal to two. This root is also called a double root. If b^2 - 4ac <0, the solutions of the quadratic equation will be two ________ numbers.

irrational; rational; equal; complex;

A relation

is simply a pairing of one set of information with another set of information.

The greatest integer function

is written f(x) = [[x]], and is defined to be the greatest integer less than or equal to x. Ex: [[-2.3]]=-3 [[-2]]=-2 [[4.5]]=4

The axis of symmetry is always reported as a _______ equation. It is incorrect to simply give a value (number) rather than an equation. (Ex: x=1)

linear

The slope-intercept form of a line is , where _____ represents the slope and _____ represents the _____-intercept.

m; b; y

In the point-slope form of a line, , the slope is represented by_____, the x-coordinate of a point is represented by _____, and the y-coordinateof the point is represented by _____.

m; x1; y1

To solve the matrix equation , multiply both sides of the equation by the____________________, which is also called _______________, and is writtenas _____.

matrix reciprocal of A; A-inverse; A^-1

For mathematicians, linear programming is finding the __________ or__________ value of a certain quantity, such as profit or expense, given a set of__________ to which the results must adhere.

minimum; maximum; constraints

If a polynomial function has degree n, its graph will have at most __________relative minimum or maximum values.

n − 1

When two lines don't intersect, the system of linear equations has __________solution(s).

no

If n is __________, the left side will be the opposite of the right side.

odd

When two lines intersect at a single point, the system of linear equations has__________ solution(s).

one

When you add a finite number of terms in a sequence, you are finding the__________ of a __________.

partial sum; series

In a recursive formula, after the first term, each term is dependent on the terms__________ it.

preceding

The method of square roots can be applied to equations in which you canisolate the ____________________, setting it equal to the____________________.

quadratic term; constant term

A monomial is an expression that is a _______________, a __________, or aproduct of _______________ and __________ having _______________exponents.

real number; variable; real numbers; variables; whole-number

If the leading coefficient is negative, the graph will point down on the__________ side.

right

If the leading coefficient is positive, the graph will point up on the __________side.

right

A matrix is a system of __________ and __________ that contains __________.

rows; columns; numbers

When evaluating a polynomial function using synthetic substitution, it is first necessary to write the polynomial in __________ form.

standard

The slope of a linear function tells

the dependent variable changes for every one unit increase in the independent variable.

Slope

the ratio of vertical change to horizontal change between any two points on a line

Parallel lines have slopes that are (the same/different) and y-intercepts thatare (the same/different).

the same; different

Lines that coincide have slopes that are (the same/different) and y-interceptsthat are (the same/different).

the same; the same

A sequence is a function whose domain is

the set of all natural numbers (counting numbers: 1, 2, 3, 4, 5,...).

The range of a function

the set of all possible values of the dependent variable.

The domain of a function

the set of all values of the independent variable.

functional notation.

the way a function is written: y=f(x) -y is dependent variable -f is function name -x is independent variable

A quadratic function a function whose largest exponent (degree) is ____. The graph of a quadratic function is a ________.

two; parabola

A line with a positive slope slants __________ and to the right. A line with a negative slope slants __________ and to the right. A __________ line has zero slope. A __________ line has undefined slope.

up/down/horizontal/vertical

The slope of a line can be interpreted as describing a ______, since it represents the rate at which the ball's position changes over time.

velocity

If , then the graph of is a __________ reflection of the original function. This means that the graph will be reflected across the__________ axis.

vertical; horizontal