Glossary A absolute value function An absolute value function is a function that can be written in the form f(x) x where x is any number. Additive Identity The number is the additive identity because when is added to any number, the sum is that number. s 1 6 6 p 1 p alternate exterior angles Alternate exterior angles are angles formed when a line (transversal) intersects two other lines. These angles are on opposite sides of the transversal and are outside the other two lines. Angles 1 and 2 are alternate exterior angles. 1 Additive Inverse Two numbers are additive inverses if their sum is the additive identity, or. s a 1 2a 219 1 19 adjacent angles Adjacent angles are two angles that share a common vertex and share a common side. Angles BAC and CAD are adjacent angles. The angles share the vertex A and the side AC. alternate interior angles 2 Alternate interior angles are angles formed when a line (transversal) intersects two other lines. These angles are on opposite sides of the transversal and are between the other two lines. B A C D 2 1 Angles 1 and 2 are alternate interior angles. angle of rotation The angle of rotation is the amount of rotation, in degrees, about a fixed point. Glossary G-1
Angle-Angle (AA) Similarity Theorem The Angle-Angle (AA) Similarity Theorem states that if two angles of one triangle are congruent to the corresponding angles of another triangle, then the triangles are similar. Angle-Angle-Side (AAS) Congruence Theorem The Angle-Angle-Side (AAS) Congruence Theorem states that if two angles and a non-included side of one triangle are congruent to the corresponding angles and the corresponding non-included side of a second triangle, then the triangles are congruent. Associative Property of Addition The Associative Property of Addition states that changing the grouping of the terms in an addition problem does not change the sum. For any numbers a, b, and c, (a 1 b) 1 c a 1 (b 1 c). s 4 1 (3 1 2) (4 1 3) 1 2 4 1 7 1 2 9 9 (2 x 1 x) 1 x 2 x 1 (x 1 x) Angle-Side-Angle (ASA) Congruence Theorem The Angle-Side-Angle (ASA) Congruence Theorem states that if two angles and the included side of one triangle are congruent to the corresponding two angles and the included side of another triangle, then the triangles are congruent. antipodes The antipodes of a sphere are the endpoints of a diameter. Points A and B are the antipodes of the sphere shown. 7x 1 x 2 x 1 1x 12 x 12 x Associative Property of Multiplication The Associative Property of Multiplication states that changing the grouping of the terms in a multiplication problem does not change the product. For any numbers a, b, and c, (a 3 b) 3 c a 3 (b 3 c) s (12 3 ) 3 2 12 3 ( 3 2) 6 3 2 12 3 1 12 12 A diameter B 1 ( 1 3 1 2 ) ( 1 1 3 ) 1 2 1 1 6 1 1 1 2 associations The patterns or relationships identified in scatter plots of a two-variable data set are called associations. 1 3 base 1 3 B The base of a power is the expression that is used as a factor in the repeated multiplication. s 2 3 2 3 2 3 2 8 8 1 base base G-2 Glossary
break-even point When one line represents the cost of an item and the other line represents the income from selling the item, the point of intersection is also called the break-even point. categorical data C Categorical data are data that each fit into exactly one of several different groups, or categories. Categorical data are also called qualitative data. s Animals: lions, tigers, bears, etc. U.S. Cities: Los Angeles, Atlanta, New York City, Dodge City, etc. center of a sphere The center of a sphere is a fixed point in space that is at an equal distance from every point on the sphere. The center and radius of the sphere are labeled. center radius characteristic In an expression of the form a 3 1 n, the variable n is called the characteristic. 6.1 3 1 61, characteristic circumference The distance around a circle or a cylinder is known as the circumference. The circumference is calculated by the following formula: C p (d). closure A set of numbers is said to be closed under an operation if the result of the operation on two numbers in the set is a defined value also in the set. clustered When the points on a scatter plot are clustered, the data values are arranged in such a way that as you look at the graph from left to right, you can imagine a line going through the scatter plot with most of the points being clustered close to the line. coincidental lines Coincidental lines are lines that have equivalent linear equations and overlap at every point when they are graphed. center of dilation The point from which a dilation is generated is called the center of dilation. E E A A D D C C The center of dilation is point D. B B Commutative Property of Addition The Commutative Property of Addition states that changing the order of two or more terms in an addition problem does not change the sum. For any numbers a and b, a 1 b b 1 a. s 1 2.12 1 2.12 1 1 3.12 3.12 1 9 1 7 1 7 1 1 1 9 9 9 9 9 13 9 13 9 Glossary G-3
Commutative Property of Multiplication The Commutative Property of Multiplication states that changing the order of two or more factors in a multiplication problem does not change the product. For any numbers a and b, a 3 b b 3 a. s 7.2.9 3.9 3 7.2 648 118 1 36 413 4.248 4.248 congruent angles Congruent angles are two or more angles that have equal measures. congruent line segments Line segments that have the same length are called congruent line segments. consistent system A system of equations that has one or many solutions is called a consistent system. 1 3 2 3 3 2 3 1 3 3 3 8 3 8 6 1 12 2 6 1 12 2 cone A cone is a three-dimensional figure with a circular or elliptical base and one vertex. constant function When the y-value does not change or remains constant, the function is called a constant function. constant interval When a function is constant for some values of the independent variable, it is said to have a constant interval. continuous graph A continuous graph is a graph with no breaks in it. converse The converse of a theorem is created when the if-then parts of that theorem are exchanged. Triangle Inequality Theorem: If a polygon is a triangle, then the sum of any two of its side lengths is always greater than the length of the third side. Converse of the Pythagorean Theorem Converse of Triangle Inequality Theorem: If you have three side lengths, and the sum of any two of the side lengths is greater than the third side length, then the side lengths can form a triangle. The Converse of the Pythagorean Theorem states: If the lengths of the sides of a triangle satisfy the equation a 2 1 b 2 c 2, then the triangle is a right triangle..6 in. 1 in..6 2.8 2 1 2.36.64 1.6 in..8 in. 1 in..8 in. G-4 Glossary
coplanar lines Coplanar lines are two or more lines that are located in the same plane. corresponding angles Corresponding angles are angles that have the same relative positions in geometric figures A E D dependent variable (response variable) The dependent variable is the variable whose value is determined by an independent variable. It can also be called the response variable because it is the variable that responds to what occurs to the explanatory variable. diagonal of a square A diagonal of a square is a line segment connecting opposite vertices of the square. B C F Angle B and Angle E are corresponding angles. diameter of a sphere A segment drawn between two points on the sphere that passes through the center is a diameter. corresponding sides Corresponding sides are sides that have the same relative positions in geometric figures. A diameter B A E D The diameter of the sphere is labeled. B C F Sides AB and DE are corresponding sides. cubic function A cubic function is function that can be written in the form f(x) a 3 x 3 1 a 2 x 2 1 a 1 x 1 a. cylinder A cylinder is a three-dimensional object with two parallel, congruent, circular bases. decreasing function D When the value of a dependent variable decreases as the independent variable increases, the function is said to be a decreasing function. dependent system A dependent system is a system of equations that has infinitely many solutions. dilation factor The dilation factor is the ratio of the distance of the image from the center of dilation to the distance of the original figure from the center of dilation. This is also known as the scale factor. dilations Dilations are transformations that produce images that are the same shape as the original image, but not the same size. E' E D' D A A' C B C' B' Pentagon ABCDE has been dilated by a scale factor of 2 about point D to create pentagon A B C D E. Glossary G-
direct variation When a linear equation is written in y-intercept form, the variables x and y show direct variation. Direct variation is the relationship between two quantities x and y such that the two variables have a constant ratio. If Melissa earns $8.2 per hour, then the amount she earns is in direct variation with the number of hours she works. The amount $8.2 is a constant. first differences F First differences are the values determined by subtracting consecutive y-values in a table when the x-values are consecutive integers. When the first differences are equal, the points represented by the ordered pairs in the table will form a straight line. x y discrete graph A discrete graph is a graph of isolated points. 1 2 2 34 3 4 9 11 domain The domain of a function is the set of all inputs of the function. ellipsis E An ellipsis is three periods that mean and so on. enlargement When the scale factor is greater than one, the image is called an enlargement. estimation When you do not need to know the exact value of an expression, you can use estimation to determine the approximate value. The first differences are 9 and 11, so the points represented by these ordered pairs will not form a straight line. frequency The frequency of a variable is the number of times it appears in a data set. Number Rolled Tally Frequency 2 IIII II 7 The number 2 was rolled 7 times, so its frequency was 7. function A function maps each input to one and only one output. exponent The exponent of a power is the number of times that the base is used as a factor in the repeated multiplication. s 2 3 2 3 2 3 2 8 8 4 8 3 8 3 8 3 8 496 exponent extrapolating exponent Extrapolating is predicting values that fall outside the plotted values on a scatter plot. great circle G A great circle is the circumference of the sphere at the sphere's widest part. Point A is the center of the sphere. It is also the center of the great circle. great circle A G-6 Glossary
height of a cone H The height of a cone is the length of a line segment drawn from the vertex to the base of the cone. The line segment is perpendicular to the base. image The new figure created from a transformation is called the image. I height of a cylinder The height of a cylinder is the length of a line segment drawn from one base to the other base. This line segment is perpendicular to the other base. Image y C A B A B C hemisphere x A hemisphere is half of a sphere bounded by a great circle. A hemisphere is shown. hemisphere included angle An included angle is an angle formed by two consecutive sides of a figure. In triangle ABC, angle A is the included angle formed by consecutive sides AB and AC. A hypotenuse The hypotenuse of a right triangle is the side opposite the right angle. s hypotenuse hypotenuse C B included side An included side is a line segment between two consecutive angles of a figure. In triangle ABC, AB is the included side formed by consecutive angles A and B. C A B Glossary G-7
income Income is the amount of money earned from sales. inconsistent systems A system of equations with no solution is called an inconsistent system. increasing function When both values of a function increase, the function is said to be an increasing function. independent system An independent system is a system of equations that has only one solution. independent variable (explanatory variable) The independent variable, also called the explanatory variable, is the variable whose value is not determined by another variable. input The first coordinate of an ordered pair in a relation is the input. interval of decrease When a function is decreasing for some values of the independent variable, it is said to have an interval of decrease. interval of increase When a function is increasing for some values of the independent variable, it is said to have an interval of increase. inverse operation An inverse operation is an operation that undoes another operation. s Addition and subtraction are inverse operations: 31 1 2 2 2 31. Multiplication and division are inverse operations: 31 3 2 4 2 31. irrational number An irrational number is a number that cannot be written as a, where a and b are integers. b s The numbers numbers. 2,.313113111..., and p are irrational interpolating Interpolating is predicting values that will fall within the plotted values on a scatter plot. intersecting lines Intersecting lines are lines in a plane that intersect, or cross each other. leg L A leg of a right triangle is either of the two shorter sides. Together, the two legs form the right angle of a right triangle. s leg leg leg leg G-8 Glossary
line of best fit A line of best fit is a line that is as close to as many points as possible, but doesn't have to go through all the points. linear combinations method (elimination) The linear combinations method is a process to solve a system of equations by adding two equations to each other, resulting in an equation with one variable. line of best fit y 9 8 linear function A function whose graph is a straight line is a linear function. Monthly Sales (in dollars) 7 6 4 3 2 y x 1 1 2 3 4 6 7 8 9 Average Monthly Temperature (in F) x The function f(x) x 1 1 is a linear function. linear association A linear association occurs when the points on the scatter plot seem to form a line. linear graph A linear graph is a graph that is a line or a series of collinear points. Trinkets Produced (trinkets) y 2 18 16 14 12 1 8 6 4 2 1 2 3 Production of Trinkets x 4 6 7 8 9 1 Time (hours) linear pair of angles A linear pair of angles, or a linear pair, consists of two adjacent angles that form a straight line. Angles 1 and 2 are a linear pair. linear regression 2 Linear regression means to model the relationship of two variables in a data set by drawing a line of best fit. 1 linear regression equation The linear regression equation is the equation for the line of best fit. Glossary G-9
mantissa M In the expression a 3 1 n, the variable a is called the mantissa. In scientific notation, the mantissa is greater than or equal to 1 and less than 1. 6.1 3 1 61, N negative association On a scatter plot, if the dependent variable decreases as the independent variable increases, then the two variables have a negative association. mantissa 6 Hot Chocolate Sales matching list In the Stroop Test, when the color of the ink matches each color word in a list, the list is called a matching list. Multiplicative Identity The number 1 is the multiplicative identity because when 1 is multiplied by any number, the product is that number. s a 1 a 43 1 43 Multiplicative Inverse Two numbers are multiplicative inverses if their product is the multiplicative identity, or 1. s 9 1 9 1 a 1 a 1 Monthly Sales ($) 4 4 3 3 2 2 1 1 1 3 7 9 11 Average Monthly Temperature ( F) As the average monthly temperature increases, the monthly sales of hot chocolate decreases. So, there is a negative association between average monthly temperature and hot chocolate sales. negative reciprocal When the product of two numbers is 21, the numbers are negative reciprocals of one another. s The negative reciprocal of 3 7 is 27 3 : 3 7 3 27 3 _ 221 21 21 The negative reciprocal of is 21 : 1 3 _ 21 2 21 G-1 Glossary
non-linear association (non-linear relationship) Data that have a pattern that is not linear are said to have a non-linear association, or non-linear relationship. Non-linear relationships graphed on a scatter plot have the data points in the shape of a curve. outlier An outlier for two-variable data is a point that varies greatly from the overall pattern of the data. Temperature of the first 16 days of April Height of Ball Tossed into Air 62 Height Degrees Fahrenheit 4 46 38 outlier 3 1 6 11 Date in April 16 nonlinear graph Time A nonlinear graph is a graph that is not a line and therefore not a series of collinear points. non-matching list In the Stroop Test, when the color of the ink does not match each color word in a list, the list is called a non-matching list. output The second coordinate of an ordered pair in a relation is the output. parallel lines P Parallel lines are lines that lie in the same plane and do not intersect no matter how far they extend. The symbol for parallel is i. The symbol for not parallel is i. O order of magnitude The order of magnitude is an estimate of size expressed as a power of ten. The Earth s mass has an order of magnitude of about 1 24 kilograms. per Per means for each or for every. Glossary G-11
perpendicular lines Perpendicular lines are lines that intersect at a right angle. The symbol for perpendicular is. The symbol for not perpendicular is. point of intersection In a system of linear equations, a point of intersection indicates a solution to both equations. point of rotation The point of rotation is the point around which a figure is rotated. The point of rotation can be a point on the figure, inside the figure, or outside the figure. y pi Pi is the ratio of the circumference of a circle to its diameter. Generally, p is approximately 3.14, or 22 7. x piecewise function A piecewise function is a function whose equation changes for different parts, or pieces, of the domain. The figure has been rotated 9 counterclockwise. The point of rotation is the origin (, ). y Distance from Home (in mi) 1 7 2 point-slope form The point-slope form of a linear equation that passes through the point ( x 1, y 1 ) and has a slope m is m( x 2 x 1 ) ( y 2 y 1 ). 6 12 18 24 Time (in min) The graph shows a piecewise function. plane A plane extends infinitely in all directions in two dimensions and has no thickness. x G-12 Glossary
positive association On a scatter plot, the two variables have a positive association if, as the independent variable increases, the dependent variable also increases. Monthly Sales ($) 6 4 4 3 3 2 2 1 Ice Cream Cone Sales pre-image The original figure in a transformation is called the pre-image. Product Rule of Powers The Product Rule of Powers states that when multiplying powers with the same base, keep the base and add the exponents. s a m a n a m 1 n 2a 8 2a 7 2a 1 profit The profit is the amount of money that you have left after you subtract the costs from the amount of money that you receive. 1 1 3 7 9 11 Average Monthly Temperature ( F) proof A proof is a series of steps used to prove the validity of a theorem. As the average monthly temperature increases, the monthly sales of ice cream cones also increases. So, there is a positive association between average monthly temperature and ice cream cone sales. postulate A postulate is a mathematical statement that cannot be proved but is considered true. Pythagorean Theorem The Pythagorean Theorem states: If a and b are the lengths of the legs of a right triangle, and c is the length of the hypotenuse, then a 2 1 b 2 c 2..6 in..8 in. 1 in..6 2 1.8 2 1 2.36 1.64 1 power A power consists of a base and an exponent. base 6 2 exponent power Power to a Power Rule The Power to a Power rule states that to simplify a power to a power, keep the base and multiply the exponents. s Pythagorean triple Any set of three positive integers a, b, and c that satisfies the equation a 2 1 b 2 c 2 is a Pythagorean triple. 3, 4 and is a Pythagorean triple: 3 2 1 4 2 2 9 1 16 2 (a m ) n a mn (8a 4 ) 2 8a 8 Glossary G-13
quadratic function Q A quadratic function is a function that can be written in the form f(x) ax 2 1 bx 1 c. Quotient Rule of Powers The Quotient Rule of Powers states that when dividing powers with the same base, keep the base and subtract the exponents. s m a = a m n a n radius of a cylinder 6b 6b 9 4 R 6b The radius of a cylinder is the distance from the center of the base to any point on the base. radius of a sphere A segment drawn from the center of the sphere to a point on the sphere is called a radius. range The range of a function is the set of all outputs of the function. rate of change A rate of change is the amount that the dependent value changes for every amount that the independent value changes. Distance from Home (in min) y 1 7 2 independent value dependent value 6 12 18 24 x Time (in min) The rate of change shown in this graph is mi, or 6 min miles per hour. rational numbers Rational numbers are numbers that can be written as a b, where a and b are integers, but b is not equal to. s 4, 1, 2 2 3,.67, and 22 are examples of rational numbers. 7 rate A rate is a ratio in which the two quantities being compared are measured in different units. The speed 6 miles in 2 hours is a rate: 6 mi 3 mi 2 h 1 h. real numbers The real numbers consist of all rational numbers and irrational numbers. Real numbers can be represented on the real number line. s The numbers 3, 1.2, 11, and 13 shown below are real numbers. 4 3 1.2 11 4 13 4 3 2 1 1 2 3 4 G-14 Glossary
reciprocal When the product of two numbers is 1, the numbers are reciprocals of one another. s The reciprocal of 3 is 7 : 7 3 3 3 7 21 7 3 21 1 The reciprocal of is 1 : 1 1 1 reduction When the scale factor is less than one, the image is called a reduction. reflection A reflection is a transformation that flips a figure over a reflection line. y Reflexive Property of Equality The Reflexive Property of Equality states that for any real number a, a a. 7 7 relation A relation is any set of ordered pairs, or the mapping between a set of inputs and a set of outputs. relative frequency A relative frequency is the ratio or percent of occurrences within a category to the total of the category. right angle A right angle has a measure of 9 and is indicated by a square in a right triangle. right angle x refection line A reflection line is a line that acts as a mirror so that corresponding points are the same distance from the mirror. y x right circular cylinder A right circular cylinder is a cylinder in which the bases are circles and they are aligned one directly above the other. right triangle A right triangle is a triangle with exactly one right angle. s The x-axis is the reflection line. rise The rise is the vertical distance between two points on a line. Glossary G-1
rise run The ratio rise run is a representation of the rate of change. y Distance from Home (in min) 1 7 2 run rise S same-side exterior angles Same-side exterior angles are angles formed when a line (transversal) intersects two other lines. These angles are on the same side of the transversal and are outside the other two lines. 1 2 Angles 1 and 2 are same-side exterior angles. 6 12 18 24 x Time (in min) The rate of change shown in this graph is mi, or 6 min miles per hour. rotation A rotation is a transformation that turns a figure about a fixed point for a given angle. y same-side interior angles Same-side interior angles are angles formed when a line (transversal) intersects two other lines. These angles are on the same side of the transversal and are between the other two lines. 1 2 x Angles 1 and 2 are same-side interior angles. The figure has been rotated 9 counterclockwise about the point (, ). run The run is the horizontal distance between two points on a line. scale factor The scale factor is the ratio of the distance of the image from the center of dilation to the distance of the original figure from the center of dilation. This is also known as the dilation factor. A 4 cm B cm 3 cm C Q 1. cm R 2 cm P 2. cm The scale factor from triangle ABC to triangle PQR is 1 2. G-16 Glossary
scatter plot A scatter plot is a graph of a collection of ordered pairs that allows exploration of the relationship between the points. Monthly Sales ($) 6 4 4 3 3 2 2 1 1 Hot Chocolate Sales 1 3 7 9 11 Average Monthly Temperature ( F) scientific notation Scientific notation is a way to express a very large or very small number as the product of a number greater than or equal to 1 and less than 1 and a power of 1. The number 1,34,, is written in scientific notation as 1.34 3 1 9. Side-Angle-Side (SAS) Similarity Theorem The Side-Angle-Side (SAS) Similarity Theorem states that if two pairs of corresponding sides of two triangles are proportional and the included angles are congruent, then the triangles are similar. Side-Side-Side (SSS) Congruence Theorem The Side-Side-Side (SSS) Congruence Theorem states that if three sides of one triangle are congruent to the corresponding sides of another triangle, then the triangles are congruent. Side-Side-Side (SSS) Similarity Theorem The Side-Side-Side (SSS) Similarity Theorem states that if three pairs of corresponding sides of two triangles are proportional, then the triangles are similar. similar triangles Similar triangles are triangles that have the same shape. skew lines Skew lines, or non-coplanar lines, are lines that are not located in the same plane. sequence A sequence is a pattern involving an ordered arrangement of numbers, geometric figures, letters, or other objects. Side-Angle-Side (SAS) Congruence Theorem The Side-Angle-Side (SAS) Congruence Theorem states that if two sides and the included angle of one triangle are congruent to the corresponding two sides and the included angle of a second triangle, then the triangles are congruent. Glossary G-17
slope Slope is another mathematical term for rate of change. y Distance from Home (in min) 1 7 rise run 2 sphere A sphere is defined as the set of all points in three dimensions that are equidistant from a given point called the center. Point C is the center of the sphere, and r is the radius of the sphere. C r 6 12 18 24 Time (in min) The slope of the line is, or 6 6. slope-intercept form The slope-intercept form of a linear equation is y mx 1 b, where m is the slope of the line and b is the y-intercept. solution of a linear system The solution of a linear system is an ordered pair (x, y) that is a solution to both equations in the system. Graphically, the solution is the point of intersection, the point at which two or more lines cross. y x 1 y 22x 1 8 The solution to this system of equations is (1, 6). x standard form The standard form of a linear equation is Ax 1 By C, where A, B, and C are constants and A and B are not both zero. Stroop Test The Stroop Test is used to study a person s perception of words and colors by using lists of color words (red, green, black, and blue) that are written in one of the four colors. substitution method The substitution method is a process of solving a system of equations by substituting a variable in one equation with an equivalent expression. supplementary angles Two angles are supplementary angles if the sum of their angle measures is equal to 18. 1 2 Angles 1 and 2 are supplementary angles. G-18 Glossary
Symmetric Property of Equality The Symmetric Property of Equality states that for any real numbers a and b, if a b, then b a If 1 2., then. 1 2 system of linear equations When two or more linear equations define a relationship between quantities, they form a system of linear equations. y x 1 y 22x 1 8 transformation A transformation is the mapping, or movement, of all the points of a figure in a plane according to a common operation. s Translations, reflections, and rotations are examples of transformations. translation A translation is a transformation that slides each point of a figure the same distance and direction. y T theorem A theorem is a mathematical statement that can be proven using definitions, postulates, and other theorems. x The Pythagorean Theorem states: If a and b are the lengths of the legs of a right triangle and c is the length of the hypotenuse, then a 2 1 b 2 c 2. term A term of a sequence is an individual number, figure, or letter in the sequence. 2, 7, 12, 17, 22, 27, 32,. transversal A transversal is a line that intersects two or more lines at distinct points. A transversal is said to cut the lines. term The Transitive Property of Equality The Transitive Property of Equality states that for any real numbers a, b, and c, if a b and b c, then a c If 1 2., and. %, then 1 2 % trend line transversal When a line of best fit is used, the line and its equation are often referred to as a model of the data, or a trend line. Glossary G-19
Triangle Sum Theorem The Triangle Sum Theorem states that the sum of the measures of the three interior angles of a triangle is equal to 18. two-step equation A two-step equation requires that two inverse operations be performed in order to isolate the variable. two-variable data set When collecting information on two separate characteristics for the same person or thing to determine if there is a relationship between them, those characteristics can be described as a two-variable data set. If you collect data about your classmates heights and shoe sizes, then the data you collect would be called a two-variable data set. two-way table A two-way table displays categorical data that shows the number of data points that fall into each group for two variables. One variable is divided into rows, and the other is divided into columns. April May June 7 th graders 8 th graders unit rate U A unit rate is a comparison of two measurements in which the denominator has a value of one unit. The speed 6 miles in 2 hours can be written as a unit rate: 6 mi 3 mi 2 h 1 h. The unit rate is 3 mi, or 3 miles per hour. 1 h Venn diagram V A Venn diagram uses circles to show how elements among sets of numbers or objects are related. Factors of 18 18 9 1 2 3 6 Factors of 3 3 1 1 G-2 Glossary
vertical angles Vertical angles are two non-adjacent angles that are formed by two intersecting lines. y-intercept Y The y-intercept is the y-coordinate of the point where a graph crosses the y-axis. The y-intercept can also be written in the form (, y). 2 3 1 4 Z Zero Power Rule Angles 1 and 2 are vertical angles. Angles 3 and 4 are vertical angles. vertical line test The vertical line test is a visual method of determining whether a relation represented as a graph is a function. To apply the vertical line test, consider all of the vertical lines that could be drawn on the graph of a relation. If any of the vertical lines intersect the graph of the relation at more than one point, then the relation is not a function. The Zero Power Rule states that the zero power of any number except for is 1. s a 1 6 1 y x The line drawn at x 3 crosses two points on the graph, so the relation is not a function. Glossary G-21
Index Symbols { } (brackets used to represent sets), 286 > (congruence relationship), 442... (ellipsis), 44 (is approximately equal to), 11 (is equal to), 3, 442, 78 ± (is not equal to), 78 2 (negative symbol), 691 (parallel), 27 ( ) (parentheses), 691 (perpendicular), 26 (absolute value), 18 π (pi), 281, 291, 292, 29 (triangle), 492 (no) A Absolute value functions, 18 19, 114 Addition Property of Equality, 12 Additive identity, 28, 289, 31 Additive inverse, 286 288, 32, 34 Adjacent angles description of, 36 determined by two intersecting lines, 36, 43 on maps, 39 4 Alternate exterior angles definition of, 47 formed by two lines and a transversal, measuring, 6 on maps, 1, 3 4 relationships of the measures of, 7, 61 in transversals, 47 Alternate interior angles definition of, 46 formed by two lines and a transversal, measuring, on maps, 2 4 relationships of the measures of, 7, 61 in transversals, 46 Angle-Angle-Side (AAS) Congruence Theorem, 464 469 Angle-Angle (AA) Similarity Theorem, 2, 8, 14 16 Angle of rotation, 49 Angle relationships congruent, 7, 61 formed by line relationships intersected by a transversal, 4 61 (See also Corresponding angles) alternate exterior angles, 47, 1, 3 7, 61 alternate interior angles, 46, 2, 7, 61 measures of, solving for unknown, 8 6 same-side exterior angles, 48, 1, 3 7, 61 same-side interior angles, 47 48, 1, 3 7, 61 formed by two intersecting lines, 3 44 adjacent angles, 36, 43 congruent angles, 37 linear pair of angles, 38, 42 protractor used to draw, 37 38, 43 44 supplementary angles, 38, 42 unknown measures of, solving for, 41 42 vertical angles, 37, 42 not congruent, 7, 61 not supplementary, 7, 61 supplementary, 7, 61 Angles adjacent, 36, 43 alternate exterior, 47, 1, 3 7, 61 alternate interior, 46, 2, 7, 61 congruent, 37 corresponding, 48 49, 1, 3 7, 61 linear pair of, 38, 42 same-side exterior, 48, 1, 3 7, 61 same-side interior, 47 48, 1, 3 7, 61 supplementary, 38, 42 vertical, 37, 42 Angle-Side-Angle (ASA) Congruence Theorem, 462 463, 467 469 Antipodes, 776 Arc markers, 46 Associations, 89 812 Associative Property of Addition, 33 Associative Property of Multiplication, 33 B Bar notation, 293 Base of a cone, 762 763, 766, 773 of a cylinder, 71, 73 definition of, 689 expanded notation and, 689 vs. exponents, 692 identifying, 69 692 of powers, 689 692 Base 1 system, 69 Binary math, 69 Brackets used to represent sets ( ), 286 Break-even point, 99 66 Byte (B), 694 C Categorical (qualitative) data definition of, 91 relative frequency of, 96, 99 two-way tables used to organize, 91 Center of a dilation, 48 482, 484 48, 487, 489, 2 Center of a sphere, 776 Characteristics in scientific notation, 713 Circumference of a cylinder, 71 of a sphere, 776, 778 779 Closed sets, 286 289, 31 Closure, 31 Clustered data values, 89 Coefficients correlation, in linear regression equation, 868 definition of, 9 in linear combinations method, 62 in two-step equations, 9 Coincidental lines, 28 33 definition of, 28 line relationships and, 28 29 on maps, 31 33 Collinear points, 61 Commutative Property of Addition, 32 Commutative Property of Multiplication, 33 Composite figures, determining area of, 363 364 Cone, 761 773 base of, 762 763, 766, 773 characteristics of, 764 766 definition of, 762 height of, 763 764 radius of, 763 doubling effect and, 789 vertex of, 762 763 volume of, 766 773 Congruence Angle-Angle-Side (AAS) Congruence Theorem, 464 469 Angle-Side-Angle (ASA) Congruence Theorem, 462 463, 467 469 congruent triangles, 441 449 included angle, 44 4, 49, 462, 466 included side, 462 46 markers used to denote, 46 Side-Angle-Side (SAS) Congruence Theorem, 44 4, 47 49, 462 Side-Side-Side (SSS) Congruence Theorem, 42 43, 47 49, 462 symbol (>), 442 symbols used to express, 442 understanding, 442 Congruent angle relationships alternate exterior angles, 7, 61 alternate interior angles, 7, 61 corresponding angles, 7, 61 Congruent angles, 442 determined by two intersecting lines, 37 identifying, 37 on maps, 39 4 See also Congruent triangles Index I-1
Congruent line segments, 442 Congruent triangles, 441 449 Angle-Angle-Side (AAS) Congruence Theorem and, 464 469 Angle-Side-Angle (ASA) Congruence Theorem and, 462 463, 467 469 corresponding angles of, 44 446, 449 corresponding sides of, 443 444, 449 included angles in, 44 4, 49, 462, 466 included sides in, 462 46 Side-Angle-Side (SAS) Congruence Theorem and, 44 4, 47 49, 462 Side-Side-Side (SSS) Congruence Theorem and, 42 43, 47 49, 462 statements of, 446 448 Consistent system, 62 Constant definition of, 9 in linear combinations method, 62 in two-step equations, 9 Constant function, 249 Constant interval, 23 Constant quantities, 94 Contexts as functions, analyzing, 71 72 rate of change determined with, 163 17, 187 y-intercepts determined with, 191, 193 Continuous graphs, 6 61 Converse, 336 Converse of the Pythagorean Theorem definition of, 336 unknown side lengths in right triangles determined with, 33 348 Coordinate plane dilations of triangles on, 484 487 graphs of equations on, in line relationships, 64 76 reflection of triangles on, 436 438, 44 rotation of geometric figures on, 47 414 similar triangles on, 9 1 translation of triangles on, 43 432, 439 Coordinate points on graphs, 6, 88, 137 139 Coplanar lines definition of, 28 in Euclidean geometry, 29 3 line relationships and, 28, 3 on maps, 31 33 Correlation, 878 88 coefficient in linear regression equation, 868 definition of, 878 experiment performed in, 874 878 lines of best fit and, 873 88 negative, 787 879 positive, 878 879 Corresponding angles of congruent triangles, 44 446, 449 definition of, 44, 48 formed by two lines and a transversal, measuring, 6 on maps, 1, 3 4 measures of relationships of, 7, 61 solving for unknown, 6 in transversals, 49 Corresponding sides of congruent triangles, 443 444, 449 definition of, 444 Cube or cubic functions, 113 114 Cylinder, 747 79 base of, 71, 73 characteristics of, 749 71 circumference of, 71 definition of, 748 height of, 749, 71 72, 74 7, 77 79 prototypes of, comparing, 7 76 radius of, 748 749, 77 78, 763 right circular, 748 volume of, 72 79 D Data displays and analysis, 79 828 See also Scatter plots; Two-variable data sets Decimal notation, 716, 718 Decimals expressions involving, estimating values of, 11 fractions converted to, 292 repeating, 292 294 terminating, 292 293 Decreasing function, 21 Dependent system, 62 Dependent variables in absolute value functions, 19 in cubic functions, 112 definition of, 88 in linear functions, 89 9 in quadratic functions, 11 on scatter plots, 88, 81 812, 891 893, 896, 913 in variable quantities, 94 Diagonal of a square definition of, 31 determining length of, 38 39 Diagonals in three-dimensional solids determining length of, formula for given the diagonal measurements of three perpendicular sides, 373 37 given the lengths of three perpendicular edges, 368 372 Pythagorean Theorem and, 36 37 drawing, 368 Diagonals in two-dimensional figures, determining length of composite figures, 363 364 Pythagorean Theorem and, 37 364 rectangle and a square, 38 39 trapezoids, 36 362 Diameter antipodes of, 776 of a sphere, 776 777, 779 782 Digital storage, 694 Dilation factor (scale factor), 48, 482, 1, 12, 14 1 Dilations center of, 48 482, 484 48, 487, 489, 2 definition of, 48 scale factor (dilation factor) in, 48, 482, 484 487, 492, 2, 1, 12, 14 1 of triangles, 479 49 on a coordinate plane, 484 487 enlargements and, 48, 488 489, 1, 12, 14 1 ratios and, 482 483, 489 reductions and, 482 483, 488 489, 1, 12, 14 1 similar triangles resulting from, 1 1 Diorama, 46 Direct variation, 194 Discrete graphs, 6 61 Division Property of Equality, 12 Domain definition of, 69, 132 of functions, 69, 73, 19, 111, 113, 28 29, 261 of problem situations, 96, 132, 148, 247 248, 2, 22 Doubling effect, 77 79, 789 E Elimination. See Linear combinations method (elimination) Ellipsis (...), 44 Enlargments, 48, 488 489, 1, 12, 14 1 Equality properties, 12 Equals sign (), 3 Equations analyzing, 77 as functions, 77, 11, 112 graphs of, on coordinate plane, 64 76 with infinite solutions, 13 23 with no solutions, 21 23 rate of change determined with, 171 183, 187 to represent problem situations, 9, 97, 13 y-intercepts determined with, 192 193 See also Linear equations; Two-step equations Estimation definition of, 11 of point of intersection, 623 in problem situations, 11 12 of value of square roots, 31 of values of expressions involving decimals, 11 Expanded notation, 689, 697 I-2 Index
Experiments in lines of best fit chain connected at the shoulder, 864 871 comparing results of, 861 measuring the height of jumps, 874 878 Stroop Test, 83 861 Experiments to display and interpret data, 818 82 Explanatory variables. See Independent variables Exponential notation, 717 718 Exponents, 68 744 vs. base, 692 definition of, 689 expanded notation and, 717 718 identifying, 69 691 key on graphing calculator, 689 negative, 73 71 powers of 1 in, 712 prime factorization and, 688 69 zero, 73 71 See also Powers; Scientific notation Extrapolating, 836 F First differences, 19 161 Formal method for determining rate of change, 11 12, 179 Fractions converting to decimals, 292 decimals representing, 293 294 irrational numbers and, 3 Frequency definition of, 92 recorded on two-way tables, 92 relative, 96, 98 99 Frequency bar graph, 912 914, 916 917 Functions, 41 12 absolute value, 18 19, 114 analyzing, 23 24 constant function, 249 constant interval, 23 contexts as, 71 72 cubic, 113 114 decreasing function, 21 defining, 67 72 domain of, 69, 73, 19, 111, 113, 28 29, 261 equations as, 77, 11, 112 graphical representations of, 66 increasing function, 247 interval of decrease, 23 24 interval of increase, 23 24 linear, 8 12 mapping used to represent, 69 71 ordered pairs as, 68 71 piecewise function, 2 27 quadratic, 111, 114 range of, 69, 73, 88, 19, 111, 113, 28 29 recognizing, 78 83 relations as, 7, 7, 78, 83 sequences and, 43 4, 73 tables used to represent, 7 vertical line test for, 7 76 G Geometric figures reflection of, 41 421 over axes, 416 419 over horizontal and vertical lines, 419 42 of parallelogram, 417 419 rotation of, 47 414 angle of, 49 definition of, 48 49 of geometric figures on the coordinate plane, 47 414 of parallelogram, 411 412 point of, 49, 414 of trapezoid, 412 translation of, 38 394 horizontal and vertical, 388 394 parallelogram, 387 388 trapezoid, 392 393 Gigabyte (GB), 694 Graphing calculator to explore extremely large or small numbers, 712 exponent key on, 689 linear regression equation determined with, 866 87 linear systems analyzed with, 67 678 linear systems solved with, 669 674 piecewise function graphed with, 267 27 scatter plots created with, 84 8 slope of a line calculated with, 18 183 Graphs analyzing, 62 66, 74 76 break in, symbol for, 876 characteristics of, 6 66 collinear points on, 61 continuous, 6 61 coordinate points on, 6, 88 selecting, 137 139 of data created from tables, 9, 13, 1 data values from a line drawn on, 88 discrete, 6 61 of equations on coordinate plane in line relationships, 64 76 linear equations representing, 66, 68, 74 parallel lines in, 64 68, 7 76 perpendicular lines in, 69 7 in slope-intercept form, 64 6 equivalent ratios represented by points on, 88 linear, 61, 88 9 linear systems solved with, 97 62, 661 nonlinear, 61, 18 114 of piecewise functions, 26 27 point of intersection on, 16 rate of change determined with, 129 143, 187 relations modeled on, 74 76, 88 9, 9, 19, 111, 113 to represent problem situations, 63 64, 88, 9, 97, 13 rise in, 134 136 run in, 134 136 scatter plots, 74 7 slope-intercept form used to graph a line, 196 197 translation of linear functions represented by, 43 44 zero on, axis label used to represent, 833 Great circle, 776 H Height of a cone, 763 764 of a cylinder, 749, 71 72, 74 7, 77 79 Hemisphere, 777, 784 Human wrist chain experiment, 818 82 Hypotenuse converse of the Pythagorean Theorem and, 344 determining length of, 314 31, 319, 323, 327 33, 344, 33 identifying, 314 Pythagorean Theorem and, 33, 336, 33 I Images definition of, 388 enlargements, 48, 488 489, 1, 12, 14 1 reductions, 482 483, 488 489, 1, 12, 14 1 Included angles, 44 4, 49, 462, 466 Included sides, 462 46 Income, 99 66 Inconsistent system, 619 Increasing function, 247 Independent system, 62 Independent variables in absolute value functions, 19 in cubic functions, 112 definition of, 88 in linear functions, 89 9 in quadratic functions, 11 on scatter plots, 88, 81 812, 891 893, 896, 913 in variable quantities, 94 Informal method for determining rate of change, 11, 16, 179 Input-output definition of, 69 mapping, 69, 74 relationship between, in linear functions, 148 tables, for linear functions, 87 89 Integers, 286 288 Interpolating, 836 Intersecting lines coincidental lines and line relationships in, 28 29 coplanar lines and line relationships in, 28, 3 definition of, 26 on maps, 31 33 Index I-3
parallel lines and line relationships in, 27 perpendicular lines and line relationships in, 26 27 skew lines and line relationships in, 28, 3 Interval of decrease, 23 24 Interval of increase, 23 24 Inverse operations definition of, 9 in two-step equations, 9 1 Irrational numbers, 291 296 Is approximately equal to ( ), 11 Is equal to (), 78 Is not equal to (±), 78 Isosceles right triangles, 31, 343 Isosceles trapezoid, 361 362 K Kilobyte (K), 694 L Legs definition of, 314 determining length of, 323, 34, 348, 367 371 Pythagorean Theorem and, 33, 336, 34, 367 371 Length contraction, 441 Linear associations, 89 812 Linear combinations method (elimination) coefficients in, 62 constants in, 62 definition of, 62 linear systems solved with, 647 661 summary of, 661 Linear equations, 1 4 analyzing, 127 214 direct variation in, 194 with infinite solutions, 13 23 of lines of best fit, 83 with no solutions, 21 23 for parallel lines in line relationships, 66, 7 for perpendicular lines in line relationships, 74 7 point-slope form of, 21 23 slope-intercept form of, 179, 623 624, 626, 628 in line relationships, 64 6 writing, 199 21 special case of, 192 standard form of, 23 27, 622, 628 translation of linear functions represented by, 44 4 two-step, 9 12 writing and solving, 3 36 See also System of linear equations; Two-step equations Linear functions, 8 12 characteristics of, 89 9 definition of, 148 graphing, 88 9 input-output tables for, 86 87 investigating, 26 29 multiple representations of, 21 28 standard form of, 22 244 translation of, 39 4 equations used to represent, 44 4 graphs used to represent, 43 44 horizontal and vertical, 396, 4 left or right, 399 41 tables used to represent, 42 Linear graphs, 61 Linear pair of angles definition of, 38 determined by two intersecting lines, 38, 42 on maps, 39 4 Linear regression, definition of, 866 Linear regression equation, 866 871 correlation coefficient and, 868 definition of, 866 experiment performed in, 864 871 graphing calculator used to determine, 866 87 lines of best fit and, 866 871 predicting values in, 836 Linear relationships on scatter plots, 89, 897 898 Linear systems, 9 644 of equations to represent a problem context, 66 678 graphs used to solve, 97 62 See also Solution of a linear system Line relationships, 2 34 angle relationships formed by, 3 61 coincidental, 28 29, 31 33 coplanar, 29 33 graphs of equations on coordinate plane, 64 76 intersecting, 26 33 on maps, 31 34 parallel, 27, 31 34, 2, 9 6 perpendicular, 26 27, 31 34 possible, in Euclidean geometry, 29 skew, 28, 3 33 Lines of best fit, 829 88 analyzing, 843 82 correlation and, 878 88 definition of, 833 drawing, 831 841 equations of, 83 linear regression equation and, 866 871 ordered pairs in, 833, 83 836, 87, 89, 86 867, 87 scatter plots used in, 833, 838, 846 82, 87, 89, 866, 877 879 slope in, 83 836, 839, 88 89, 869, 871, 877 878 trend line (model of data) and, 833 y-intercept in, 836, 839 84, 87, 89 See also Experiments in lines of best fit Lines on graphs data values from, 88 passing through points determined with y-intercept, 2 slope-intercept form used to graph, 196 197 See also Line relationships; Slopes Line transformations, 77 8 parallel lines in, 78 79, 83 8 perpendicular lines in, 79 8 reflecting lines in, 81 82 rotating lines in, 79 8 translating lines in, 78 79 triangle relationships in, 83 84 Lucky numbers, 283 M Mantissa, 713, 71, 718, 721, 72 726 Mapping definition of, 68 functions represented by, 69 71 between inputs and outputs, 69, 74 ordered pairs shown by, 68 69 Matching lists in Stroop Test, 84 8, 87, 861 Measurements converting, 284 converting, in linear regression equation, 869 87 of distance, 429 units of, comparing, 167 Megabyte (MB), 694 Millimeter (mm), 717, 72 Model of data (trend line), 833 Morse code, 647 Mueller-Lyer illusion, 461 Multiple representations analyzing problem situations with, 217 224 of linear functions, 21 28 Multiplication Property of Equality, 12 Multiplicative identity, 32 Multiplicative inverse, 32 N Natural numbers (counting numbers), 28 Necker Cube, 383 Negative associations, 89 812 Negative Exponent Rule identifying, 729 738 for negative exponent in the denominator, 77, 71 for negative exponent in the numerator, 77, 71 in simplifying powers, 7 71, 73, 73 summary of, 71 Negative exponents, 73 71 Negatively correlated, 787 879 Negative rate of change, 13 Negative reciprocals, 74 Nonlinear functions absolute value functions, 18 19 cubic functions, 113 graphs of, 18 114 quadratic functions, 111 tables of values for, 18 114 Nonlinear graphs, 61 Non-linear relationships definition of, 894 graphed on scatter plots, 89 898 Non-matching lists in Stroop Test, 84 86, 89, 861 I-4 Index
Not congruent angle relationships alternate exterior angles, 7, 61 alternate interior angles, 7, 61 corresponding angles, 7, 61 Not supplementary angle relationships same-side exterior angles, 7, 61 same-side interior angles, 7, 61 Number line, rational numbers compared and order with, 284 Numbers irrational, 291 296 less than one, powers representing, 7 lucky, 283 natural (counting numbers), 28 pi (π), 281, 291 292, 29 pronic, 283 rational, 283 29 real, 298 34 repunit, 283 types of, 28 288 whole, 28 O One-to-one correspondence, 41 Optical illusions, 383 Ordered pairs analyzing, 68 71 in lines of best fit, 833, 83 836, 87, 89, 86 867, 87 mapping used to show, 68 69 of non-linear data on scatter plots, 891 893 rate of change determined with, 12, 17 161 set of, 68 69, 79, 81 as solution of a linear system, 611 x- and y-coordinates in, 68 Order of magnitude, 712 Outliers, 813 814 Output. See Input-output P Parallel lines algebraically solving a linear system containing, 627 definition of, 27 in line relationships, 27 equations written for, 66, 7 slopes of, 64 68, 76 verifying, 68 in line transformations translating lines related to, 78 79 triangle relationships in, 83 84 Triangle Sum Theorem and, 8 on maps, 31 34, 2 measures of angles formed by, solving for unknown, 9 6 slope used to determine, 613 614, 619 symbol for ( ), 27 Parallelogram, 387 388 Perpendicular lines definition of, 26 intersections of, 618 in line relationships, 26 27 equations written for, 74 7 reciprocals of, 74 slopes of, 69 74, 76 verifying, 74 in line transformations, rotating lines related to, 79 8 on maps, 31 34 slope used to determine, 617, 619 62 symbol for ( ), 26 Per used in rate of change, 131 Pi (π), 281, 291 292, 29 Piecewise functions, 2 27 algebraic representation of, 26 263 definition of, 261 graphing from a diagram, 264 266 from a table, 26 263 using a graphing calculator, 267 27 Plane, 26 Point of intersection coordinates of, 616 definition of, 61 estimating, 623 identifying, 613 as solution of a linear system, 611, 617, 619 in system of linear equations, 61 66 Point of rotation, 49, 414 Points collinear, 61 coordinate, on graphs, 6, 88, 137 139 passing through lines on graphs determined with y-intercept, 2 Pythagorean Theorem used to determine distance between, 349 36 See also Point of intersection Point-slope form, 21 23 Positive associations, 89 812 Positively correlated, 878 879 Postulate, 33 Powers base of, 689 692 definition of, 689 equal sign used in, 78 expanded notation and, 689, 697 exponent of, 689 692 multiplying and dividing, 693 71 negative sign used in, 691 of numbers in scientific notation, 728 numbers written as, 73 77 parentheses in, 691 power to a power, 697 698 products and, 696 697 properties of correct use of, 739 identifying, 729 738 simplifying, 79 71 quotients and, 698 699 representing numbers less than one, 7 rules for operating with, developing, 693 71 of 1, 712 working with, 69 692 See also Simplifying powers Power to a power, 697 698 Power to a Power Rule identifying, 729 738 in scientific notation, 728 in simplifying powers, 698, 737 738 summary of, 7 Pre-image, 388 Prime factorization, exponents and, 688 69 Problem situations analyzing with multiple representations, 217 224 domain of, 96, 247 248, 2, 22 equations used to represent, 9, 97, 13 estimation used in, 11 12 graphs used to represent, 63 64, 88, 9, 97, 13 intercepts in, 22, 223 224, 24 241, 244, 246, 248, 2, 27, 263 range of, 96, 247 248, 2, 22 sentences used to represent, 92 93, 97, 1 slope in, 22, 224, 246, 248, 2 tables used to represent, 92 97, 12 13 variable quantities in, 94 9 Product Rule of Powers identifying, 729 738 in scientific notation, 72 722 in simplifying powers, 696 697, 7, 79 summary of, 7 Profits calculating, 168, 99 66 definition, 99 Pronic numbers, 283 Proof, 33 Properties Associative Property of Addition, 33 Associative Property of Multiplication, 33 Commutative Property of Addition, 32 Commutative Property of Multiplication, 33 of equality, 12 identifying, 33 of powers correct use of, 739 identifying, 729 738 simplifying, 79 71 Reflexive Property of Equality, 33 Symmetric Property of Equality, 33 Transitive Property of Equality, 33 Properties of Equality, 12 Pythagorean Theorem definition of, 33 distance between two points determined with, 349 36 lengths determined with of diagonals in three-dimensional solids, 36 37 of diagonals in two-dimensional figures, 37 364 Index I-