Chapter 1

Prerequisites

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Summary

The rational numbers and irrational numbers make up the set of real numbers.. Algebraic expressions are composed of constants and variables that are combined using addition, subtraction, multiplication, and division.. They take on a numerical value when evaluated by replacing variables with constants Rational numbers may be written as fractions or terminating or repeating decimals

Key terms

distributive property

the product of a factor times a sum is the sum of the factor times each term in the sum; in symbols, a ⋅ ( b + c ) = a ⋅ b + a ⋅ c

irrational numbers

the set of all numbers that are not rational; they cannot be written as either a terminating or repeating decimal; they cannot be expressed as a fraction of two integers

rational numbers

the set of all numbers of the form m n , where m and n are integers and n ≠ 0. Any rational number may be written as a fraction or a terminating or repeating decimal

order of operations

a set of rules governing how mathematical expressions are to be evaluated, assigning priorities to operations

algebraic expression

constants and variables combined using addition, subtraction, multiplication, and division

real numbers

the sets of rational numbers and irrational numbers taken together

constant

a quantity that does not change value

variable

a quantity that may change value

Chapter 2

Equations and Inequalities

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Summary

We can locate, or plot, points in the Cartesian coordinate system using ordered pairs, which are defined as displacement from the x- axis and displacement from the y- axis The distance formula is derived from the Pythagorean Theorem and is used to find the length of a line segment Finding the x- and y- intercepts can define the graph of a line. The midpoint formula provides a method of finding the coordinates of the midpoint dividing the sum of the x -coordinates and the sum of the y -coordinates of the endpoints by 2

Key terms

distance formula

a formula that can be used to find the length of a line segment if the endpoints are known

ordered pair

a pair of numbers indicating horizontal displacement and vertical displacement from the origin; also known as a coordinate pair, ( x , y )

Pythagorean Theorem

a theorem that states the relationship among the lengths of the sides of a right triangle, used to solve right triangle problems

midpoint formula

a formula to find the point that divides a line segment into two parts of equal length

intercepts

the points at which the graph of an equation crosses the x -axis and the y -axis

Cartesian coordinate system

a grid system designed with perpendicular axes invented by René Descartes

area

in square units, the area formula used in this section is used to find the area of any two-dimensional rectangular region: A = L W

completing the square

a process for solving quadratic equations in which terms are added to or subtracted from both sides of the equation in order to make one side a perfect square

Chapter 3

Functions

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Summary

A function is a specific type of relation in which each domain value, or input, leads to exactly one range value, or output Function notation is a shorthand method for relating the input to the output in the form y = f ( x ) In tabular form, a function can be represented by rows or columns that relate to input and output values To evaluate a function, we determine an output value for a corresponding input value.

Key terms

function

a relation in which each input value yields a unique output value

output

each object or value in the range that is produced when an input value is entered into a function

input

each object or value in a domain that relates to another object or value by a relationship known as a function

range

the set of output values that result from the input values in a relation

domain

the set of all possible input values for a relation

decreasing function

a function is decreasing in some open interval if f ( b ) < f ( a ) for any two input values a and b in the given interval where b > a

horizontal line test

a method of testing whether a function is one-to-one by determining whether any horizontal line intersects the graph more than once

Chapter 4

Linear Functions

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Summary

An increasing linear function results in a graph that slants upward from left to right and has a positive slope. A decreasing linear function results in a graph that slants downward from left to right and has a negative slope. A constant linear function results in a graph that is a horizontal line The slope of a linear function can be calculated by dividing the difference between y -values by the difference in corresponding x -values of any two points on the line

Key terms

linear function

a function with a constant rate of change that is a polynomial of degree 1, and whose graph is a straight line

horizontal line

a line defined by f ( x ) = b , where b is a real number. The slope of a horizontal line is 0

slope

the ratio of the change in output values to the change in input values; a measure of the steepness of a line

decreasing linear function

a function with a negative slope: If f ( x ) = m x + b , then m < 0

increasing linear function

a function with a positive slope: If f ( x ) = m x + b , then m > 0

perpendicular lines

two lines that intersect at right angles and have slopes that are negative reciprocals of each other

vertical line

a line defined by x = a , where a is a real number. The slope of a vertical line is undefined

Chapter 5

Polynomial and Rational Functions

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Summary

A polynomial function of degree two is called a quadratic function The axis of symmetry is the vertical line passing through the vertex. The zeros, or x - intercepts, are the points at which the parabola crosses the x - axis. The y - intercept is the point at which the parabola crosses the y - axis

Key terms

rational function

a function that can be written as the ratio of two polynomials

axis of symmetry

a vertical line drawn through the vertex of a parabola, that opens up or down, around which the parabola is symmetric; it is defined by x = - b 2 a

polynomial function

a function that consists of either zero or the sum of a finite number of non-zero terms, each of which is a product of a number, called the coefficient of the term, and a…

vertex

the point at which a parabola changes direction, corresponding to the minimum or maximum value of the quadratic function

zeros

in a given function, the values of x at which y = 0 , also called roots

degree

the highest power of the variable that occurs in a polynomial

y - intercept

the point at which the parabola crosses the y - axis

parabola

a U-shaped curve that can open either up or down

Chapter 6

Exponential and Logarithmic Functions

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Summary

An exponential function is defined as a function with a positive constant other than 1 raised to a variable exponent A function is evaluated by solving at a specific value The value of an account at any time t can be calculated using the compound interest formula when the principal, annual interest rate, and compounding periods are known The initial investment of an account can be found using the compound interest formula when the value of the account, annual interest rate, compounding periods, and life span of the account are known

Key terms

logarithm

the exponent to which b must be raised to get x ; written y = log b ( x )

function

evaluated by solving at a specific value

exponential function

defined as a function with a positive constant other than 1 raised to a variable exponent

compound interest

interest earned on the total balance, not just the principal

logistic growth model

a function of the form f ( x ) = c 1 + a e - b x where c 1 + a is the initial value, c is the carrying capacity, or limiting value, and b is a constant determined by the rate of…

Newton’s Law of Cooling

the scientific formula for temperature as a function of time as an object’s temperature is equalized with the ambient temperature

power rule for logarithms

a rule of logarithms that states that the log of a power is equal to the product of the exponent and the log of its base

order of magnitude

the power of ten, when a number is expressed in scientific notation, with one non-zero digit to the left of the decimal

Chapter 7

The Unit Circle: Sine and Cosine Functions

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Summary

In addition to degrees, the measure of an angle can be described in radians To convert between degrees and radians, use the proportion θ 180 = θ R π An angle is formed from the union of two rays, by keeping the initial side fixed and rotating the terminal side. An angle is in standard position if its vertex is at the origin and its initial side lies along the positive x -axis.

Key terms

cosine function

the x -value of the point on a unit circle corresponding to a given angle

sine function

the y -value of the point on a unit circle corresponding to a given angle

unit circle

a circle with a center at ( 0 , 0 ) and radius 1

radian

the measure of a central angle of a circle that intercepts an arc equal in length to the radius of that circle

standard position

the position of an angle having the vertex at the origin and the initial side along the positive x -axis

ray

one point on a line and all points extending in one direction from that point; one side of an angle

coterminal angles

description of positive and negative angles in standard position sharing the same terminal side

degree

a unit of measure describing the size of an angle as one-360th of a full revolution of a circle

Chapter 8

Periodic Functions

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Summary

Periodic functions repeat after a given value. In the general formula for a sinusoidal function, the period is P = 2 π | B | In the general formula for a sinusoidal function, | A | represents amplitude.

Key terms

periodic function

a function f ( x ) that satisfies f ( x + P ) = f ( x ) for a specific constant P and any value of x

function cos x

even, so its graph is symmetric about the y -axis

function sin x

odd, so its graph is symmetric about the origin

sinusoidal function

any function that can be expressed in the form f ( x ) = A sin ( B x - C ) + D or f ( x ) = A cos ( B x - C ) + D

amplitude

the vertical height of a function; the constant A appearing in the definition of a sinusoidal function

inverse cosine function

the function cos - 1 x , which is the inverse of the cosine function and the angle that has a cosine equal to a given number

inverse sine function

the function sin - 1 x , which is the inverse of the sine function and the angle that has a sine equal to a given number

Chapter 9

Trigonometric Identities and Equations

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Summary

There are multiple ways to represent a trigonometric expression. Verifying the identities illustrates how expressions can be rewritten to simplify a problem Graphing both sides of an identity will verify it Simplifying one side of the equation to equal the other side is another method for verifying an identity

Key terms

even-odd identities

set of equations involving trigonometric functions such that if f ( - x ) = - f ( x ) , the identity is odd, and if f ( - x ) = f ( x ) , the identity is even

quotient identities

pair of identities based on the fact that tangent is the ratio of sine and cosine, and cotangent is the ratio of cosine and sine

double-angle formulas

identities derived from the sum formulas for sine, cosine, and tangent in which the angles are equal

product-to-sum formula

a trigonometric identity that allows the writing of a product of trigonometric functions as a sum or difference of trigonometric functions

sum-to-product formula

a trigonometric identity that allows, by using substitution, the writing of a sum of trigonometric functions as a product of trigonometric functions

half-angle formulas

identities derived from the reduction formulas and used to determine half-angle values of trigonometric functions

reduction formulas

identities derived from the double-angle formulas and used to reduce the power of a trigonometric function

Pythagorean identities

set of equations involving trigonometric functions based on the right triangle properties

Chapter 10

Further Applications of Trigonometry

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Summary

The Law of Sines can be used to solve oblique triangles, which are non-right triangles The ambiguous case arises when an oblique triangle can have different outcomes According to the Law of Sines, the ratio of the measurement of one of the angles to the length of its opposite side equals the other two ratios of angle measure to opposite side The Law of Sines can be used to solve triangles with given criteria

Key terms

Law of Sines

states that the ratio of the measurement of one angle of a triangle to the length of its opposite side is equal to the remaining two ratios of angle measure to opposite side; any…

ambiguous case

a scenario in which more than one triangle is a valid solution for a given oblique SSA triangle

oblique triangle

any triangle that is not a right triangle

cardioid

a member of the limaçon family of curves, named for its resemblance to a heart; its equation is given as r = a ± b cos θ and r = a ± b sin θ , where a b = 1

De Moivre’s Theorem

formula used to find the n th power or n th roots of a complex number; states that, for a positive integer n , z n is found by raising the modulus to the n th power and…

Law of Cosines

states that the square of any side of a triangle is equal to the sum of the squares of the other two sides minus twice the product of the other two sides and the cosine of the…

rose curve

a polar equation resembling a flower, given by the equations r = a cos n θ and r = a sin n θ ; when n is even there are 2 n petals, and the curve is highly symmetrical; when n is…

unit vector

a vector that begins at the origin and has magnitude of 1; the horizontal unit vector runs along the x -axis and is defined as i = 〈 1 , 0 〉 the vertical unit vector runs along…

Chapter 11

Systems of Equations and Inequalities

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Summary

A system of linear equations consists of two or more equations made up of two or more variables such that all equations in the system are considered simultaneously The solution to a system of linear equations in two variables is any ordered pair that satisfies each equation independently One method of solving a system of linear equations in two variables is by graphing. Another method of solving a system of linear equations is by substitution.

Key terms

system

considered simultaneously

system of linear equations

a set of two or more equations in two or more variables that must be considered simultaneously

addition method

an algebraic technique used to solve systems of linear equations in which the equations are added in a way that eliminates one variable, allowing the resulting equation to be…

consistent system

a system for which there is a single solution to all equations in the system and it is an independent system, or if there are an infinite number of solutions and it is a…

dependent system

a system of linear equations in which the two equations represent the same line; there are an infinite number of solutions to a dependent system

determinant

a number calculated using the entries of a square matrix that determines such information as whether there is a solution to a system of equations

feasible region

the solution to a system of nonlinear inequalities that is the region of the graph where the shaded regions of each inequality intersect

substitution method

an algebraic technique used to solve systems of linear equations in which one of the two equations is solved for one variable and then substituted into the second equation to…

Chapter 12

Analytic Geometry

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Summary

When given the coordinates of the foci and vertices of an ellipse, we can write the equation of the ellipse in standard form When given an equation for an ellipse centered at the origin in standard form, we can identify its vertices, co-vertices, foci, and the lengths and positions of the major and minor axes in order to graph the ellipse Real-world situations can be modeled using the standard equations of ellipses and then evaluated to find key features, such as lengths of axes and distance between foci A hyperbola is the set of all points ( x , y ) in a plane such that the difference of the distances between ( x , y ) and the foci is a positive constant

Key terms

hyperbola

the set of all points ( x , y ) in a plane such that the difference of the distances between ( x , y ) and the foci is a positive constant

ellipse

the set of all points ( x , y ) in a plane such that the sum of their distances from two fixed points is a constant

angle of rotation

an acute angle formed by a set of axes rotated from the Cartesian plane where, if cot ( 2 θ ) > 0 , then θ is between ( 0° , 45° ) ; if cot ( 2 θ ) < 0 , then θ is between ( 45°…

directrix

a line perpendicular to the axis of symmetry of a parabola; a line such that the ratio of the distance between the points on the conic and the focus to the distance to the…

eccentricity

the ratio of the distances from a point P on the graph to the focus F and to the directrix D represented by e = P F P D , where e is a positive real number

focus (of an ellipse)

one of the two fixed points on the major axis of an ellipse such that the sum of the distances from these points to any point ( x , y ) on the ellipse is a constant

parabola

the set of all points ( x , y ) in a plane that are the same distance from a fixed line, called the directrix, and a fixed point (the focus) not on the directrix

Chapter 13

Sequences, Probability, and Counting Theory

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Summary

Explicit formulas define each term of a sequence using the position of the term An explicit formula for the n th term of a sequence can be written by analyzing the pattern of several terms Recursive formulas define each term of a sequence using previous terms Recursive formulas must state the initial term, or terms, of a sequence

Key terms

sequence

a function whose domain is a subset of the positive integers

probability

a number from 0 to 1 indicating the likelihood of an event

explicit formula

a formula that defines each term of a sequence in terms of its position in the sequence

recursive formula

a formula that defines each term of a sequence using previous term(s)

factorial

a mathematical operation that can be defined recursively

factorial of n

the product of all integers from 1 to n

arithmetic sequence

a sequence in which the difference between any two consecutive terms is a constant