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Glossary



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Fraction

a number representing part of a whole; numerator/denominator (number of pieces, intervals, or parts) / (number of pieces, intervals or parts in the whole)

Numerator

In a fraction, something written in the form \frac {a}{b}, the number or expression that makes up the top of the fraction, a,is called the numerator.

Denominator

In a fraction, something written in the form \frac {a}{b}, the number or expression that makes up the bottom of the fraction, b, is called the denominator.

Least Common Denominator (LCD)

the least common multiple of a set of denominators; the least common multiple is the smallest multiple common to all the numbers (in the example below the least common multiple is 20)
The multiples of 4 are: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44,...
The multiples of 5 are: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50,...

Exponent

for positive exponents, the exponent of a number tells us how many times to use the base in a multiplication; for negative exponents, the exponent tell us how many times to divide 1 by the base; an exponent is usually written as a superscript to the right of the base; exponents are also called powers or indices

Scientific Notation

Scientific notation is a special way of writing numbers that is particularly convenient for numbers that are either very large or very small.  A number written in scientific notation will be in the form

a \times 10^b ,

where 1\le a <10 and b is an integer. 

Decimal

when working with decimals, moving the decimal point one place to the right represents division by ten; moving the decimal point one place to the left represents multiplication by 10; in this sense, decimal notation refers to a base-10 positional notation; to convert a decimal to a percent, multiply the decimal number by 100

Percent

percent literally means "divided by 100"; to convert a percent to a decimal, divide the percent number by 100

Order of Operations

The order of operations establishes the order in which different mathematical procedures should be performed. This is often expressed with the acronym PEMDAS (or perhaps BEDMAS, BIDMAS, or BODMAS). PEMDAS, which stands for Parentheses Exponents Multiplication Division Addition Subtraction, reminds us that when simplifying expressions, we should start by simplifying all expressions inside parentheses, then apply exponents, then multiply or divide, and lastly add or subtract. 

  • Level 1 Parentheses, Brackets ("Containers") and Exponents
  • Level 2 Multiplication and Division
  • Level 3 Addition and Subtraction Do the order of operations starting with Level 1, then do Level 2, then do Level 3. It can be good practice to solve the problem from left to right.
    Common Misconception: Multiplication is performed before division. These two operations have equal priority. Do any multiplication or division as you move from left to right in the problem. For example, if we had 1-4+7 , we would simplify it as follows: 1-4+7=-3+7=4.
    Common Misconception: Addition is performed before subtraction. These two operations have equal priority. Do any addition or subtraction as you move from left to right in the problem. For example, if we had 3\div 6\times 9 , we would simplify it as follows: 3\div 6\times 9=2\times 9=18.
    The order of operations gives us a set of general guidelines that we must apply carefully.

Expression

A math statement combining numbers or variables using sums, differences, products, quotients (including fractions), exponents, roots, logarithms, trig functions, parentheses, brackets, functions, or other mathematical operations. Expressions may not contain the equal sign (=) or any type of inequality (< or >)

Absolute Value

The absolute value |a| is the magnitude of a, that is, the number with any negative sign removed. For example, |3|=3 and |-3|=3. Note that any calculation within the absolute value is done first and then the sign ignored eg |5-8| = |-3| = 3. The absolute value of a number also represents the distance of the number from zero.

Factors

The factors of a term or expression are numbers, variables, or combinations of numbers and variables that, when multiplied together, produce that term or expression.  For example, 2 and 3 are factors of 6, and -1, 5, x, and y are factors of -5xy

Term

Each of the quantities connected by a + or - sign in an algebraic expression is called a term.

Commutative Property

For addition: The commutative property of addition states that changing the order in which numbers are added does not affect the final result.  For example, 2+3=3+2.  This is true no matter how many terms are being added together. For multiplication: The commutative property of multiplication states that changing the order in which numbers are multiplied does not affect the final result.  For example, 2\cdot3=3\cdot2.  This is true no matter how many terms are being multiplied together.

Variables

A variable is a symbol, generally a letter, that serves as a placeholder for a number.  The numeric value of a variable may change, depending on the context in which it is used.  For example, in the expression 2x+3, x can take on any value.  If we decide to let x equal 4, for example, then we can get a value for the expression: 2(4)+3=11.
In an equation like 2x+3=7, however, x represents a certain number whose value we just don't know yet.  If we modify the equation to get x by itself on one side, then we can find out that value: x=2.

Algebraic Expression

An algebraic expression combines constants and variables using addition, subtraction, multiplication, and division.

Variable Terms

A variable term is a term whose value is not fixed, due to the presence of a variable in it.  In other words, a variable term will have one or more variable factors.  In the expression -5xy^2+1000, -5xy^2 is a variable term. 

Constant Term

A constant term is a term whose value is fixed, since it does not have any variables in it (i.e. it does not have any variable factors).  In the expression -5xy^2+1000, 1000 is a constant term. 

Coefficient

A coefficient is the numerical factor of a variable term.  For instance, for the term -5xy^2, the coefficient is -5, and for the term x, the coefficient is 1.

Evaluate

Evaluating an algebraic expression means substituting numeric values for each of the variables in the expression.  If we have the expression 3x-2, and we let x=5, then the value of the expression will be 3(5)-2, or 13.

Equation

A math statement that states the values of two mathematical expressions are equal. The statement includes the equals sign symbol (=).

Like Terms

terms that contain the same powers of variables; like terms are identical except for their numerical factor or coefficient, such as 2xy and -4xy

Distributive Property

For an expression of the form a(b+c), we can say that a(b+c)=ab+ac In words, the distributive property says that if an entire expression is multiplied by some quantity, that quantity multiplies each and every term in the expression.
Examples:

3(2-7)=3(2)+3(-7)=6-21=-15
-4x^2(x^2+3x-2)=(-4x^2)(x^2)+(-4x^2)(3x)+(-4x^2)(-2x)=-4x^4-12x^3+8x^2
(x+3)(2x-1)=(x+3)(2x)+(x+3)(-1)=(2x)(x+3)+(-1)(x+3)=2x(x)+2x(3)-x-3=2x^2+6x-x-3=2x^2+5x-3

Isolating a Variable

Isolating a variable means rearranging and modifying an equation, according to the rules of algebra, so that a variable is all by itself on one side of the equation.  If we started off with the equation x-3=6, we could add 3 to both sides of the equation to end up with x=9.  The variable x is now isolated.

Number Line

The real number line is a horizontal line used to represent the real numbers.  Every point on the line represents a different real number, with values increasing from left to right.  The arrows on either end indicate that the line extends forever in either direction, which allows it to include all values from -\inf to \inf

Inequality

Whereas an equation relates two quantities that are of equal value, an inequality is a way to mathematically relate two quantities that do not have the same value.  An inequality may use one of the following symbols:

  • a\neq b means that a is not equal to b .
  • a>b means that a is greater than b .
  • a<b means that a is less than b .
  • a\ge b means that a is greater than or equal to b .
  • a\le b means that a is less than or equal to b . The set of all numbers that satisfy an inequality (i.e. can be plugged in for the variables to make the inequality true) is called the solution set of that inequality.  Solution sets are often represented by graphs on number lines.
    To write \neq with MathQuill, type \neq and then a space.  To write \le, type \le and then a space, and to write \ge , type \ge and then a space.

Compound Inequality

If we know that a<b and that b<c , then we can use a compound inequality to combine these two inequalities into one statement: a<b<c

Interval Notation

An interval is a set of real numbers that includes all real numbers between a lower bound and an upper bound.  If an interval represents a set of values that a variable, let’s say x for now, is allowed to equal, then it might be expressed with inequality notation: a<x<b , a\le x<b , a<x\le b , or a\le x\le b Intervals can also be expressed using what is called interval notation.  For the intervals written above, we could instead use interval notation to write (a,b) , [a,b) , (a,b] , or [a,b] .  Interval notation thus uses square or round brackets surround the lower and upper bound of the interval, separated by a comma.  A round bracket or parenthesis indicates that the bound or endpoint next to it is not included in the interval, while a square bracket means that that number is included.
Let’s take a look at an example.  If we have -11\le x<4 , we could also write that x lies in the interval [-11,4) .  Here, x can equal -11 , since that has a square bracket next to it, but it can’t equal 4 , since that is by a round parenthesis.

Perimeter

the length around a 2-Dimensional closed, planar figure measured in units; the distance around a two dimensional shape

Area

a measure of 2-Dimensional space inside a closed, planar figure measured in square units; the measure of space or size of a surface

Dimensions

The length and width of a rectangular area. The length is considered to be the longer side length. The width is the shorter side length.

Ratio

A way of comparing one quantity to another; for example, the ratio of

Proportion

A proportion is an equation that equates two ratios

Cross Multiplication

A method used to solve for a missing value in a proportion. The method comes from multiplying both sides of the equation by the product of the two denominators.

Complementary Angles

Two angles whose sum is 90º. One angle is said to be the complement of the other.

Supplementary Angles

Two angles whose sum is 180º. One angle is said to be the supplement of the other.

Straight Line

Supplementary angles that are side by side or that share a vertex form a straight line

Set

A set can be represented by a single letter or name, and the elements of the set can be listed inside braces separated by a commas. For example the set of even numbers between 1 and 11 is {2, 4, 6, 8, 10}.

Union

The union of two sets is the set of all values that belong to either of those two sets, and we denote the union of two sets A and B by writing A\union B .  So if we have A={-4, -2, 3, 5, 6} and B={-3, 0, 3, 4, 5}, then A\union B={-4, -3, -2, 0, 3, 5, 6}.  In this course, we will most often use unions to join together intervals that make up solution sets.

Intersection

The intersection of two sets is the set of all values that belong to both of the sets. The elements must appear in both sets in order to be part of the intersection, and we denote the intersection of two sets A and B by writing A\intersection B So if we have A={-4, -2, 3, 5, 6} and B={-3, 0, 3, 4, 5}, then A\intersection B={3, 5}.

Compound Inequality

A compound inequality connects two or more inequalities by using the words “and” or “or”.  In this course, we’ll limit ourselves to compound inequalities with just two inequalities.

  • If the word “and” is used, then a value must satisfy both of the inequalities in order to be in the solution set.  Example: If we start out with x-1>3 and x+2\le9 , then we can simply to have x>4 and x\le7 , so the solution is 4<x\le 7 , written as a chained inequality.  In interval notation, this is (4,7] .  Some values that would satisfy this compound inequality, then, are 5.6 , 8 , and 7 .  Some values that would not satisfy it are -1 , 4 , and 100 .
  • If the word “or” is used, then a value may satisfy either or both of the inequalities in order to be in the solution set.  Example: Let’s say we begin with 2x<12 or x-5>6 .  We could also write this as 2x<12 \or x-5>6 , where \or, typed as \or in MathQuill, just means “or”.  We can then simplify to have x<6 \or x>11.  In interval notation, we would write this as (-\inf ,6)\union (11,\inf ).  Some values that would satisfy this compound inequality, then, are -1796.45 , 0 , and 83 .  Some values that would not satisfy it are 6 , 8\frac {2}{3} , and 10 .

Coordinate Plane

A coordinate plane, or the Cartesian coordinate plane, is made up of one horizontal number line, called the horizontal axis, and one vertical number line, called the vertical axis. Each axis represents a different variable.  By convention, the variable x is usually shown on the horizontal axis, and the variable y is usually shown on the vertical axis.  Every point on the plane containing these axes can be defined by a number indicating its horizontal position and a number indicating its vertical position.  These numbers, called the point’s coordinates are normally written with ordered pair notation.  If a point has a horizontal coordinate of 3 and a vertical coordinate of -4 , it can be identified by the ordered pair (3,-4).
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Origin

The origin is the point on a coordinate plane where the axes meet one another, giving it the coordinates (0,0).

x-axis

The x-axis is the horizontal number line the Cartesian coordinate plane.  It represents values that the variable x in an equation might equal.  The origin represents an x-value of 0 , positive values lie to the right, and negative values lie to the left.

y-axis

The y-axis is the horizontal number line the Cartesian coordinate plane.  It represents values that the variable y in an equation might equal.  The origin represents a y-value of 0, and values increase going up the axis and decrease moving down.

x-intercept

The x-intercept is the point at which a line (or graph) crosses the x-axis. The coordinates will be (x, 0).

y-intercept

The y-intercept is the point at which a line (or graph) crosses the y-axis. The coordinates will be (0, y).

Slope

The slope of a line is a number that tells us how steep the line is.  We can find this number by calculating how many units the line rises vertically for every unit it moves to the right or to the left.  This is sometimes referred to as “rise over run”.  Two find the slope of the line on which the points (x_1, y_1) and (x_2, y_2) lie, we can use the following formula, in which m represents the slope: m=\frac {y_2-y_1}{x_2-x_1}

Slope-Intercept Form

One way to write the equation of a line is to use slope-intercept form.  If we call the slope of a line m and the y-coordinate of its y-intercept b, then the equation of that line in slope-intercept form will be y=mx+b.

Point-Slope Form

If a line passes through the point (x_1,y_1) and has a slope of m, then we can write the equation for this line in point-slope form as follows: y-y_1=m(x-x_1)

Standard Form

The standard form for the equation of a line can be written as follows: Ax+By=C, where A, B, and C are constants. A, B, and C must be integers, and A and B are not both zero.  Although the equation of a vertical line cannot be written in either slope-intercept form or point-slope form, it can be written in the form x=C.  Some sources write the general form of a line as Ax+By+C=0.

Parallel Lines

Two lines are called parallel if their slopes are equal.  Parallel lines that are not identical will never intersect one another.  The symbol || is sometimes used to show that two lines are parallel to one another.

Perpendicular Lines

Two lines are called perpendicular if they form a 90^{\circ} angle.  This will happen if their slopes are the negative reciprocals of one another.  Put more mathematically, if a line has a slope of m, then every line perpendicular to it will have a slope of -\frac {1}{m}.

Horizontal Lines

Horizontal lines have no rise. Therefore the numerator of the fraction that makes up the slope will be zero. This gives horizontal lines a slope of zero. Horizontal lines are of the form y=A, where A is a constant.

Vertical Lines

Vertical lines have no run. Therefore the denominator of the fraction that makes up the slope will be zero. Since the denominator of a fraction cannot equal zero, vertical lines have no slope or have a slope that is undefined. Vertical lines are of the form x=B, where B is a constant. Vertical lines do not represent functions.

Linear Inequality

The solution set of a linear inequality consists of a boundary line whose points may or may not be included in the solution of all points on one side of the boundary line.

Boundary Line

The line that is graphed for a linear inequality because it splits the graph into two regions.

  • A dashed or dotted boundary line is used for < or > linear inequalities.
  • A solid boundary line is used for \le or \ge linear inequalities.

Independent Variable

In an equation, an independent variable is a variable whose value does not depend on the value of other variables.  In algebra, this is often the variable x.  The independent variable generally appears as part of an expression on one side of the equation, as with y=4x^3-7x^2+11.

Dependent Variable

In an equation, a dependent variable is a variable whose value depends on the the values of one or more other variables.  In algebra, this is usually the variable y.  The dependent variable is often isolated on one side of an equation, as with y=4x^3-7x^2+11

Function

A function relates an independent variable, the input of the function, to a dependent variable, the output of the function so that for each value of the independent variable, there is only one value of the dependent variable.  If we denote a function as y=f(x) , then y is the dependent variable, x is the independent variable, f is the name of the function, and f(x) is the value of the function at x.
There are many types of functions - polynomial functions (which include a wide variety of functions), rational functions, absolute value functions, square root functions, and more.

Domain

The domain of a function is the set of all inputs, or values of the independent variable, for which the function has an output.  If some value x is in the domain of a function, then the function is said to be defined at x.  If x is not in the domain, then the function is said to be undefined at x.

Range

The range of a function is the set of all values that the function can take on.  In other words, it is the set of all outputs that the function can produce.

Vertical Line Test

We can use the vertical line test to determine whether or not a graph represents a function.  Since a function can only have one output for each input, or one value of the dependent variable for each value of the independent variable, its graph can only cross each horizontal position on the coordinate plane at most one time.  To perform the vertical line test on a graph, take a straight object, like a pencil or a ruler, position it so that it is parallel with the vertical axis of the graph, and then move it horizontally from one side of the graph to the other.  If at any position the vertical line intersects the graph more than once, the graph does not represent a function. Please note that use of the vertical line test assumes that the independent variable, x, is represented on the horizontal axis and that the dependent variable, y, is represented on the vertical axis.

Point of Intersection

A point of intersection is a point at which two or more lines or curves meet.  In other words, it is a point shared by these graphs.  The coordinates of this point will satisfy the equation of every line going through the point.

Interest, Principal, and Balance

When a person decides to invest money in a bank account, the amount of money he or she initially invests is called the principal.  Once this money is invested, a certain percent of that money will be added into the account at designated times.  The money that is added is known as interest, and the total amount of money in the account at any given time is called the balance.  The percent of the account balance that is earned in interest is called the interest rate

Polynomial

A polynomial is a mathematical expression in which the variables have only whole number exponents (0, 1, 2, 3, ...). The expression is made up of variables and/or constants that are combined with addition, subtraction, and multiplication.  This excludes any expressions that have variables in the denominator and any variables appearing within a radical. Examples: -3x+4, 6y^4+7y^2-8y+13, 5xy^2z-11

Monomial

Polynomials that have only one term like 2x.

Binomial

Polynomials that have exactly two terms like 2x+5.

Trinomial

Polynomials that have exactly three terms like 2x^2-3x+7

Degree of a Polynomial

To find the degree of a polynomial, we first need to find the degree of each of its terms.  The degree of a term is the sum of the powers of all of the variables in that term.  So, for instance, 7x^2 has a degree of 2, and -4xy^2z has a degree of 4.  The degree of a polynomial is equal to the degree of its highest-degree term.  Examples: 6y^4+7y^2-8y+13 has a degree of 4 or, in other words, is a fourth degree polynomial.

Standard Form of a Polynomial

A polynomial is said to be in standard form when it is written with its highest-degree term first, with the rest of the terms written in order of descending degree.  For example, 6y^4+7y^2-8y+13 is written in standard form, whereas 7y^2-8y+13+6y^4 is not.

Factoring

When we factor a polynomial, we break it down so that it is written as a product of factors.  In this course, we will say that a polynomial is fully factored or completely factored when it is written as the product of factors with integer coefficients and none of its factors can be factored any further.  For example, (x-3)(x+1)(2x-5) is fully factored, whereas (3x+1)(x^2-1) is not because we could instead write it as (3x+1)(x-1)(x+1) .

Linear Factor

A linear factor is a factor of degree 1.  In other words, it can be written in the form ax+b , where a and b are constants and a is not equal to 0 .  For the quadratic polynomial 2x^2-5x-18 , the linear factors are 2x-9 and x+2 .

Greatest Common Factor

The greatest common factor, or GCF, of two or more numbers is the largest positive integer that divides evenly into all of the numbers (i.e. leaves a remainder of 0).  For example, the GCF of 60 and 18 is 6.
The GCF of two or more polynomials is the largest polynomial of the highest degree that can be factored out of all of the polynomials in question.  For example, the GCF of 3x^2-11x-20 and 6x^2+11x+4 is 3x+4 .

Factoring the Difference of Two Squares

When we have a difference of two squares, like x^2-4 or 9x^2-25 , the expression can be factored according to a certain pattern.  In general, a^2-b^2=(a-b)(a+b) .  For the two examples already mentioned, then, we could factor to have x^2-4=(x-2)(x+2) and 9x^2-25=(3x-5)(3x+5) .

Factoring by Grouping

Factoring by grouping is a technique that can be used to factor certain polynomials that can be written with more than 3 terms.  Let’s look at a couple of examples: x^3+2x^2-5x-10=(x^3+2x^2)-(5x+10)=x^2(x+2)-5(x+2)=(x+2)(x^2-5) If we start out with a polynomial that only has three terms, we can often expand the middle term and then factor by grouping. x^2+3x-40=x^2-5x+8x-40=(x^2-5x)+(8x-40)=x(x-5)+8(x-5)=(x-5)(x+8)

Quadratic Equation

A quadratic equation is an equation of the form ax^2+bx+c=0 , where a , b , and c are constants and a is not equal to 0 .

Plus or Minus (\pm)

The plus or minus sign \pm is used to indicate that the value to its can be either added or subtracted to the value to its left.  For example, 8\pm 3 is a way of writing 8+3 or 8-3 using just one expression.  To write this symbol with MathQuill, type \pm and then a space. 

Completing the Square

Completing the square is a technique used to convert an expression of the form ax^2+bx+c , where a , b , and c are constants, to an expression of the form a(x-h)^2+k , where h and k are both constants. 

Quadratic Formula

The quadratic formula is a formula that can be used to find solutions to quadratic equations.  If a quadratic equation is written in the form ax^2+bx+c=0 , then the quadratic formula states that its solutions will be: x=\frac {-b\pm \sqrt{b^2-4ac}}{2a}

Root or Zero

A root of a polynomial, also called a zero of a polynomial, is a value of x that makes the polynomial equal to 0.  For a polynomial equation, if a root is a real number, then this tells us the x-coordinate of one of the graphs’ x-intercepts.  Solutions to the quadratic formula, for example, tell us the roots of quadratic equations. Let’s say that we have the equation y=x^2+4x+3.  We can factor this to be y=(x+3)(x+1) .  If we set y=0 , then the values that solve the quadratic equation 0=(x+3)(x+1) are the roots of the polynomial equation, x=-3 and x=-1

Rational Expression

A rational expression is the ratio of one polynomial to another.  Thus, a rational expression is anything that can be written as \frac {p}{q} , where p and q are both polynomials and q is not 0

Critical Value

When solving a polynomial inequality or a rational inequality, critical numbers or critical values are values that divide the number line into test intervals, intervals in which we know the inequality will be either true for every value contained in the interval or not true for any value in it.
We can find the critical numbers for a polynomial inequality by simply finding the x-coordinates of the x-intercepts.  If we rearrange a rational inequality so that one side of the inequality is equal to 0 and the other side is just a rational expression, the critical numbers will be values of x that make either the numerator or the denominator of the rational expression equal to 0 .

Square Root

If we have some number a, then a square root of a is any number b such that b^2=a.  In other words, multiplying b by itself gives a.  We can denote the square root of a quantity a by using a radical sign over a : \sqrt a.  By convention, \sqrt a denotes the positive square root of a.  For example, 3\cdot3=9, and -3\cdot-3=9 as well, but so \sqrt 9=3.  To indicate the negative square root, -3 in this case, we must write -\sqrt 9

nth Root

The nth root of a number a is some number b that, when raised to the nth power, equals a : If b^n=a , then b is an nth root of a.  For example, since 3^4=81, 3 is a fourth root of 81.  We can denote the nth root of a quantity a using a radical sign with a

Product Rule for Radicals

\sqrt{xy}=\sqrt{x}\cdot \sqrt{y} Note: \sqrt{x+y}\neq \sqrt{x}+\sqrt{y} For example, \sqrt{25}=\sqrt{16+9}\neq \sqrt{16}+\sqrt{9}

Imaginary Numbers

An imaginary number is any number that can be written in the form bi , where b is a real number and i=\sqrt{-1} .  We can find imaginary numbers by taking the square roots of negative numbers.  Since the imaginary number i is defined by the equation i^2=-1 , we can write all square roots of negative numbers in terms of i .  For example, \sqrt{-25}=-5i , and \sqrt {-3} = i\sqrt 3 .  The set of imaginary numbers lies completely separate from the set of real numbers.

Complex Numbers

A complex number is any number that can be written in the form a+bi , where a and b are real numbers and i=\sqrt{-1} .  If a=0 , this will give us an imaginary number, and if b=0 , this will give us a real number.  The set of complex numbers full encompasses the real numbers and the complex numbers, but it also has members that are neither real nor imaginary.  In other words, all real numbers are complex, and all imaginary numbers are complex.  Some complex numbers, however, have both imaginary and real parts, preventing them from falling into the imaginary category or the real category.  3-7i has a real part (3) and an imaginary part (-7i) , so it is a complex number. Although all imaginary numbers and all real numbers are complex numbers, when a number is specifically referred to as a complex number, this sometimes means that it has both real and imaginary parts (i.e. it is neither real nor imaginary).

Complex Plane

We can use the complex plane to visualize complex numbers, just as we can use the number line to visualize real numbers.  The complex plane is formed by two perpendicular axes.  The horizontal one represents the real numbers, just like a real number line, and the vertical one represents the imaginary numbers.  Any complex number of the form a+bi can be plotted on the complex plane by letting its horizontal coordinate be equal to its real part, a , and its vertical coordinate be equal to the coefficient of its imaginary part , b.  We can find the number 5-11i at the point (5,-11) , and we can find the number 8+2i at the point (8,2) .

Complex Conjugates

Complex conjugates are complex numbers whose real parts are equal and whose imaginary parts have opposite signs.  So if one complex number is a+bi , then its complex conjugate is a-bi .
Examples:

  • The complex conjugate of -7+6i is -7-6i
  • The complex conjugate of 2i is -2i.  (Remember, all imaginary numbers are complex numbers!)
  • The complex conjugate of -9 is -9 .  (Remember, all real numbers are complex numbers!)

Direct Variation

In a problem involving direct variation, as one quantity increases, the other quantity also increases. Also, if one quantity decreases, then the other quantity decreases.

Indirect Variation

In a problem involving indirect variation, as one quantity increases, the other quantity decreases. Also, if one quantity decreases, then the other quantity increases.

Right Triangle

A right triangle is a triangle with one right angle, or one angle that measures 90^{\circ}.
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Hypotenuse

The hypotenuse is the longest side of a right triangle.  It lies opposite the right angle. 

Pythagorean Theorem

The Pythagorean Theorem states that if a right triangle’s sides have lengths a, b, and c, where c is the longest side (the hypotenuse), then a, b, and c are related to one another according to the equation a^2+b^2=c^2.
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Distance Formula

We can use the distance formula to find the distance between any two points on the coordinate plane.  If our points are (x_1, y_1) and (x_2, y_2) and we call the distance between them d, then we can say that d=\sqrt {(x_2-x_1)^2+(y_2-y_1)^2} . The distance formula treats the straight line connecting these two points as the hypotenuse of a right triangle, and it uses the Pythagorean Theorem to find the length of this line.

Circle

An equation of the form (x-h)^2+(y-k)^2=r^2 where (h, k) is the center of the circle and r is the radius of the circle. Note: The equation of a circle will always have equal coefficients for the x^2 and y^2 terms.

Ellipses

An equation of the form \frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1 This equation differs from a circle in that there are different coefficients or amounts of the x^2 and y^2 terms.

Hyperbolas

An equation of the form \frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1 is a hyperbola that opens to the left and to the right. An equation of the form \frac{(y-k)^2}{b^2}-\frac{(x-h)^2}{a^2}=1 is a hyperbola that opens to the left and to the right. In both equations, the centers of the hyperbola are (h,k). This equation differs from a circle in that there are different coefficients or amounts of the x^2 and y^2 terms. This equation differs from a circle and an ellipse in that there is a subtraction sign between the x^2 and y^2 terms. 

Parabolas

Parabolas have either an x^2 or a y^2 term, but not both.
Parabolas of the form y= or that have an x^2 term open up or down. This includes parabolas of the form y=ax^2+bx+c. Parabolas of the form x= or that have a y^2 term open left or right. This includes parabolas of the form x=ay^2+by+c.

Vertex

The vertex of a parabola is the point at which the parabola itself intersects its axis of symmetry, the line that divides it in half.  For parabolas that open upward, the vertex is the lowest point on the curve, the minimum.  For parabolas that open downward, the vertex is the highest point on the curve, the maximum.

Vertex Form

If the coordinates of the vertex of a parabola are (h,k) , then the equation of the parabola can be written in vertex form as:

y-k=a(x-h)^2

where a is a constant.  If a>0 , the parabola opens upward, and if a<0 , it opens downward.

Composite Function

Given two functions, we can create a composite function by using the output of one of the functions as the input of the other function. If we have two functions named f and g , then we could combine them to create the composite functions f(g(x)) and g(f(x)) .  These can also be denoted as (f\circ g)(x) and (g\circ f)(x) , respectively.  For example, if f(x)=3x-2 and g(x)=x^3 , then (f\circ g)(x)=3x^3-2 , and (g\circ f)(x)=(3x-2)^3 .

Inverse Function

We can think of the inverse function of a function f as the function, denoted f^{-1} that does the opposite operations that f does.  In other words, f^{-1} undoes what f does to the independent variable (usually x ). 

More formally, a function g is the inverse of a function f if f(g(x)) = x for all values in the domain of g and g(f(x)) = x for all values in the domain of f .  If all of that is true, then g(x) = f^{-1}(x) , and f(x) = g^{-1}(x)

We can find the inverse of a function f graphically by reflecting the graph of f in the line y=x .  This is because we can find the inverse of a function by switching its x- and y-coordinates.

Logarithmic Function

A logarithmic function is a function that can be written in the form

f(x)=\log_ax

Here, a is the base of the logarithmic function, and a is positive number not equal to 1.  The domain of a function of this form is x>0.  The equation y=\log_ax is true if and only if x=a^y

Logarithm

A logarithm is an expression of the form \log_ab, and it represents the number that a would need to be taken to the power of in order to equal b.  For example, log_39 is equal to 2

Product property

The product property of logarithms can be stated as the following:

\log_aAB=\log_aA+\log_bB

Quotient property

The quotient property of logarithms can be stated as the following:

\log_a\frac{A}{B}=\log_aA-\log_aB

Power Property

The power property of logarithms can be stated as the following:

\log_aA^n=n\log_aA

One-to-One Properties of Exponents and Logs

a^x=a^y if and only if x=y
\log_ax=\log_ay if and only if x=y

Inverse Properties of Exponents and Logs

a^{\log_ax}=x
\log_aa^x=x