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CONTENTS 1.1.1 Definition of Absolute Value ........................................................ 3 1.1.3 Lines of Symmetry ........................................................................ 3 1.2.5 Functions ........................................................................................ 4 A.1.1 Non-Commensurate .................................................................... 5 A.1.2 Expressions and Terms ................................................................ 5 A.1.3 Order of Operations..................................................................... 5 A.1.4 Evaluating Expressions and Equations ...................................... 6 A.1.5 Combining Like Terms ................................................................ 6 A.1.6 Simplifying an Expression ........................................................... 7 A.1.8 Using an Equation Mat ................................................................ 7 A.1.9 Checking a Solution ..................................................................... 7 2.1.4 The Slope of a Line ..................................................................... 10 2.2.2 Writing the Equation of a Line from a Graph ......................... 11 2.2.3 x- and y-Intercepts ..................................................................... 11 3.1.2 Laws of Exponents ...................................................................... 13 3.2.1 Using Algebra Tiles to Solve Equations .................................... 14 3.2.2 Multiplying Algebraic Expressions with Tiles ......................... 14 3.2.3 Vocabulary for Expressions ....................................................... 14 3.2.4 Properties of Real Numbers ....................................................... 15 3.3.1 The Distributive Property .......................................................... 16 3.3.2 Linear Equations from Slope and/or Points ............................. 16 3.3.3 Using Generic Rectangles to Multiply....................................... 17 4.1.1 Line of Best Fit ............................................................................ 18 4.1.2 The Equal Values Method .......................................................... 19 4.2.1 Describing Association................................................................ 19 4.2.2 The Substitution Method ............................................................ 20 4.2.3 Systems of Linear Equations ..................................................... 20 4.2.4 Forms of a Linear Function ....................................................... 21 4.2.5 Intersection, Parallel, and Coincide .......................................... 21 5.1.1 The Elimination Method ............................................................ 22 5.1.2 Continuous and Discrete Graphs .............................................. 23 5.3.2 Types of Sequences ..................................................................... 23 6.1.2 Interpreting Slope and y-Intercept on Scatterplots ................. 24 6.1.4 Residuals ...................................................................................... 25 6.2.1 Least Squares Regression Line .................................................. 25 6.2.3 Residual Plots .............................................................................. 25 6.2.4 Correlation Coefficient ............................................................... 25 6.2.5 Non-Linear Models ..................................................................... 26 7.1.1 Graphs with Asymptotes ............................................................ 28 1 7.1.3 Compound Interest ..................................................................... 28 7.1.5 Step Functions ............................................................................. 29 7.2.2 Negative and Fractional Exponents .......................................... 29 7.2.3 Equations for Sequences............................................................. 29 8.1.1 More Vocabulary for Expressions ............................................. 32 8.1.2 Diagonals of Generic Rectangles ............................................... 32 8.1.3 Standard Form of a Quadratic Expression .............................. 32 8.1.4 Factoring Quadratic Expressions .............................................. 32 8.2.2 Zero Product Property ............................................................... 33 8.2.4 Forms of a Quadratic Function ................................................. 33 9.1.1 Zeros and Roots of Quadratics .................................................. 35 9.1.2 The Quadratic Formula ............................................................. 36 9.1.3 Solving a Quadratic Equation ................................................... 36 9.1.4 Simplifying Square Roots ........................................................... 37 9.2.1 Inequality Symbols...................................................................... 38 9.3.1 Curve Fitting an Exponential Function .................................... 38 9.3.2 Solving One-Variable Inequalities............................................. 39 9.4.1 Graphing Inequalities with Two Variables .............................. 39 10.2.1 Equivalent Equations ............................................................... 41 10.2.2 Solving Equations with Fractions (Fraction Busters) ........... 41 10.2.3 Methods to Solve Single-Variable Equations ......................... 42 10.2.4 Forms of a Quadratic Equation ............................................... 42 10.2.5 The Number System ................................................................. 43 10.2.6 Solving Absolute-Value Equations .......................................... 43 11.1.2 Number of Solutions to a Quadratic Equation ...................... 45 11.2.1 Interquartile Range and Boxplots ........................................... 45 11.2.2 Describing Shape (of a Data Distribution).............................. 46 11.2.3 Describing Spread (of a Data Distribution) ............................ 46 _________________________________________________ _________________________________________________ _________________________________________________ _________________________________________________ _________________________________________________ _________________________________________________ _________________________________________________ _________________________________________________ 2 Name: ________________________________ 1.1.1 Definition of Absolute Value Absolute value, represents the numerical value of a number without regard to its sign. The symbol for absolute value is two vertical bars, | | . Absolute value can represent the distance on a number line between a number and zero. Since a distance is always positive, the absolute value is always either a positive value or zero. The absolute value of a number is never negative. For example, the number −3 is 3 units away from 0, as shown on the number line at right. Therefore, the absolute value of −3 is 3. This is written 3 3 . Likewise, the number 5 is 5 units away from 0. The absolute value of 5 is 5, written 5 5 1.1.2 Families of Relations There are several “families” of special functions that you will study in this course. One of these is called direct variation (also called direct proportion) which is a linear function. The data you gathered in the “Sign on the Dotted Line” lab (in problem 1-9) is an example of a linear relation. Another function is inverse variation (also called inverse proportion). The data collected in the “Hot Tub Design” lab (in problem 1-9) is an example of inverse variation. You also observed an exponential fu nction. The growth of infected people in the “Local Crisis” (in problem 1-9) was exponential. 1.1.3 Lines of Symmetry When a graph or picture can be folded so that both sides of the fold will perfectly match, it is said to have reflective symmetry. The line where the fold would be is called the line of symmetry. Some shapes have more than one line of symmetry. See the examples below. This shape has one line of symmetry. This shape has two lines of symmetry. This shape has eight lines of symmetry. 3 This graph has two lines of symmetry. This shape has no lines of symmetry 1.2.5 Functions A relationship between inputs and outputs is a function if there is no more than one output for each input. We often write a function as y = some expression involving x, where x is the input and y is the output. The following is an example of a function. In the example above the value of y depends on x, so y is also called the dependent variable and x is called the independent variable. Another way to write a function is with the notation “f(x) =” instead of “y =”. The function named “f” has output f(x). The input is x. In the example at right, f(5) = 9. The input is 5 and the output is 9. You read this as, “f of 5 equals 9.” The set of all inputs for which there is an output is called the domain. The set of all possible outputs is called the range. In the example above, notice that you can input any x-value into the equation and get an output. The domain of this function is “all real numbers” because any number can be an input. But the outputs are all greater than or equal to zero. The range is y ≥ 0. x2 + y2 = 1 is not a function because there are two y-values (outputs) for some x-values, as shown below. Prob # 1-32 Date Learning Log Entry Graph Investigation Questions 4 Prob # 1-65 Date Learning Log Entry Functions A.1.1 Non-Commensurate Two measurements are called non-commensurate if no combination of one measurement can equal a combination of the other. For example, your algebra tiles are called non-commensurate because no combination of unit squares will ever be exactly equal to a combination of x-tiles (although at times they may appear close in comparison). In the same way, in the example below, no combination of x-tiles will ever be exactly equal to a combination of y-tiles. A.1.2 Expressions and Terms A mathematical expression is a combination of numbers, variables, and operation symbols. Addition and subtraction separate expressions into parts called terms. For example, 4x2 − 3x+ 6 is an expression. It has three terms: 4x2, 3x, and 6. A more complex expression is 2x + 3(5 − 2x) + 8, which has three terms: 2x, 3(5 − 2x), and 8. However, the term 3(5 − 2x) also has an expression inside the parentheses: 5 – 2x. 5 and 2x are terms of this inside expression. A.1.3 Order of Operations Mathematicians have agreed on an order of operations for simplifying expressions. 5 (10 3 2) 22 Original expression: (10 3 2) 22 Circle expressions that are grouped within parentheses or by a fraction bar: 13 3 2 2 13 3 2 2 6 6 Simplify within circled terms using the order of operations: Evaluate exponents. (10 3 2) 22 Multiply and divide from left to right. (10 6) 22 Combine terms by adding and subtracting from left to right. (4) 22 6 13 3 3 2 13 9 2 6 6 4 2 4 (4) 22 6 Circle the remaining terms: 2 4 Simplify within circled terms using the order of operations as above. 4 2 2 6 2 16 2 620 A.1.4 Evaluating Expressions and Equations The word evaluate indicates that the value of an expression should be calculated when a variable is replaced by a numerical value. For example, when you evaluate the expression x2 + 4x – 3 for x = 5, the result is: (5)2 + 4(5) –3 25 + 20 – 3 42 When you evaluate the equation y = 3x2 – 5x + 2 for x = 4, the result is: y = 3(4)2 – 5(4) + 2 y = 3(16) – 5(4) + 2 y = 48 – 20 + 2 y = 30 In each case remember to follow the Order of Operations. A.1.5 Combining Like Terms Combining tiles that have the same area to write a simpler expression is called combining like terms. See the example shown at right. When you are not working with actual tiles, it can help to picture the tiles in your mind. You can use these images to combine the terms that are the same. Here are two examples: 6 Example 1: 2x2 + xy + y2 + x + 3 + x2 + 3xy + 2 ⇒ 3x2 + 4xy + y2 + x + 5 Example 2: 3x2 − 2x + 7 − 5x2 + 3x − 2 ⇒ −2x2 + x + 5 Remember that addition and subtraction separate expressions into terms. A.1.6 Simplifying an Expression In addition to combing like terms, the following two ways to simplify an expression are common. Flip tiles and move them from the negative region to the positive region (that is, find the opposite). For example, the two unit tiles in the “–” region can be flipped and placed in the “+” region Remove an equal number of opposite tiles that are in the same region. For example, the positive and negative tile to the right can be removed. A.1.8 Using an Equation Mat An equation mat can help you visually represent an equation with algebra tiles. For example, the equation 2x − 1 − (−x + 3) = 6 − 2x can be represented by the equation mat at right. (Note that there are other possible ways to represent this equation correctly on the equation mat.) A.1.9 Checking a Solution To check a solution to an equation, substitute the solution into the equation and verify that it makes the two sides of the equation equal. For example, to verify that x = 10 is a solution to the equation 3(x −5) = 15, substitute 10 into the equation for x and then verify that the two sides of the equation are equal. As shown at right, x = 10 is a solution to the equation 3(x −5) = 15. 7 What happens if your answer is incorrect? To investigate this, test any solution that is not correct. For example, try substituting x = 2 into the same equation. The result shows that x = 2 is not a solution to this equation. Prob # Date Learning Log Entry A-5 Variables (And Algebra Tiles) A-16 Finding Perimeter and Combining Like Terms 8 Prob # Date Learning Log Entry A-23 Meaning of Minue A-29 Representing Expressions on an Expression Mat A-41 Using Zeroes to Simplify 9 Prob # Date Learning Log Entry A-52 “Legal” Moves for Simplifying and Comparing Expressions A-91 Solutions of an Equation 2.1.4 The Slope of a Line The slope of a line is the ratio of the vertical distance to the horizontal distance of a slope triangle formed by two points on a line. The vertical part of the triangle is called y (Δy ) (read “change in y”), while the horizontal part of the triangle is called x (Δx ) (read “change in x”). It indicates both how steep the line is and its direction, upward or downward, left to right. Note that “Δ ” is the Greek letter “delta” that is often used to represent a difference or a change. Note that lines pointing upward from left to right have positive slope, while lines pointing downward from left to right have negative slope. A horizontal line has zero slope, while a vertical line has undefined slope. To calculate the slope of a line, pick two points on the line, draw a slope triangle (as shown in the example above), determine Δy and Δx, and then write the slope ratio. You can verify that your slope correctly resulted in a negative or positive value based on its direction. In the example above, Δy = 2 and Δx = 5, so the slope is 2 . 5 10 2.2.2 Writing the Equation of a Line from a Graph One of the ways to write the equation of a line directly from a graph is to find the slope of the line (m) and the y-intercept (b). These values can then be substituted into the general slope-intercept form of a line: y = mx + b. For example, the slope of the line at right is m = 1 , while the y-intercept is (0, 2). By 3 substituting m = 1 and b = 2 into y = mx + b, the 3 equation of the line is: 2.2.3 x- and y-Intercepts Recall that the x-intercept of a line is the point where the graph crosses the x-axis (where y = 0). To find the x-intercept, substitute 0 for y and solve for x. The coordinates of the x-intercept are (x, 0). Similarly, the y-intercept of a line is the point where the graph crosses the y-axis, which happens when x = 0. To find the y-intercept, substitute 0 forx and solve for y. The coordinates of the y-intercept are (0, y). Example: The graph of 2x + 3y = 6 is a line, as shown above right. To calculate the x-intercept, Let y 0;2 x 3(0) 6 2x 6 x3 x-intercept: (3, 0) Prob # 2-30 Date To calculate the y-intercept, Let x 0;2(0) 3x 6 3x 6 x2 y-intercept: (0, 2) Learning Log Entry Multiple Representations Web for Linear Relationships TABLE EQUATION GRAPH SITUATION 11 Prob # Date Learning Log Entry 2-40 y = mx + b 2-58 Rates of Change and Slope What does it mean if the line is steeper? Less steep? What does a positive slope mean? What about a negative slope? What does a line with slope zero look like? What does a zero slope mean in your situation? Why does a vertical line have undefined slope? 2-81 Multiple Representations Web for Linear Relationships TABLE EQUATION GRAPH SITUATION 12 Prob # Date 2-89 Learning Log Entry Finding the Equation of a Line Through Two Points 5 5 -5 -5 3.1.2 Laws of Exponents In the expression x3, x is the base and 3 is the exponent. x3 = x · x · x The patterns that you have been using during this section of the book are called the laws of exponents. Here are the basic rules with examples: 13 3.2.1 Using Algebra Tiles to Solve Equations Algebra tiles are a physical and visual representation of an equation. For example, the equation 2x + (−1) − (−x) −3 = 6 − 2x can be represented by the Equation Mat below. An Equation Mat can be used to represent the process of solving an equation. The “legal” moves on an Equation Mat correspond with the mathematical properties used to algebraically solve an equation. “Legal” Tile Move Group tiles which are alike together. Flip all tiles from subtraction region to addition region Flip everything on both sides Corresponding Algebra Combine like terms Change subtraction to “adding the opposite” Multiply (or divide) both sides by -1 Remove zero pairs (pairs of tiles that are opposites) within a region of the mat Place or remove the same tiles from both sides Arrange the tiles into equal size groups A number plus its opposite equals 0 Add or subtract the same value from both sides Divide both sides by the same value 3.2.2 Multiplying Algebraic Expressions with Tiles The area of a rectangle can be written two different ways. It can be written as a product of its width and length or as a sum of its parts. For example, the area of the shaded rectangle at right can be written two ways: 3.2.3 Vocabulary for Expressions A mathematical expression is a combination of numbers, variables, and operation symbols. Addition and subtraction separate expressions into parts called terms. For example, 4x2 − 3x + 6 is an expression. It has three terms: 4x2, 3x, and 6. The coefficients are 4 and 3. 6 is called a constant term. A one-variable polynomial is an expression which only has terms of the form: (any real number) x(whole number) For example, 4x2 − 3x1 + 6x0 is a polynomials, so the simplified form, 4x2 − 3x + 6 is a polynomial. 14 1 The function f ( x) 7 x5 2.5 x3 x 7 is a polynomial function. 2 The following are not polynomials: 2x − 3, 1 x 2 2 and x2 A binomial is a polynomial with only two terms, for example, x3 − 0.5x and 2x + 5. 3.2.4 Properties of Real Numbers The legal tiles moves have formal mathematical names, called the properties of real numbers. The Commutative Property states that when adding or multiplying two or more numbers or terms, order is not important. That is: a + b = b + a For example, 2 + 7 = 7 + 2 a·b=b·a For example, 3 · 5 = 5 · 3 However, subtraction and division are not commutative, as shown below. 7 − 2 ≠ 2 − 7 since 5 ≠ −5 50 ÷ 10 ≠ 10 ÷ 50 since 5 ≠ 0.2 The Associative Property states that when adding or multiplying three or more numbers or terms together, grouping is not important. That is: (a + b) + c = a + (b + c) For example, (5 + 2) + 6 = 5 + (2 + 6) (a · b) · c = a ·(b · c) For example, (5 · 2) · 6 = 5 · (2 · 6) However, subtraction and division are not associative, as shown below. (5 − 2) − 3 ≠ 5 −(2 − 3) since 0 ≠ 6 (20 ÷ 4) ÷ 2 ≠ 20 ÷(4 ÷ 2) since 2.5 ≠ 10 The Identity Property of Addition states that adding zero to any expression gives the same expression. That is: a+0=a For example, 6 + 0 = 6 The Identity Property of Multiplication states that multiplying any expression by one gives the same expression. That is: 1·a=a For example,1 · 6 = 6 The Additive Inverse Property states that for every number a there is a number – a such that a + – a = 0. A common name used for the additive inverse is the opposite. For example, 3 (3) 0 and 5 5 0 The Multiplicative Inverse Property states that for every nonzero number a there is a number 1 such that 1 1 1 . A common name used for the multiplicative inverse is a a the reciprocal. That is, 1 a is the reciprocal of a. For example, 6 1 6 15 3.3.1 The Distributive Property The Distributive Property states that for any three terms a, b, and c: a(b + c) = ab + ac That is, when a multiplies a group of terms, such as (b + c), then it multiplies each term of the group. For example, when multiplying 2x(3x + 4y), the 2x multiplies both the 3x and the 4y.This can be shown with algebra tiles or in a generic rectangle (see below). 3.3.2 Linear Equations from Slope and/or Points If you know the slope, m, and y-intercept, (0, b), of a line, you can write the equation of the line as y = mx + b. You can also find the equation of a line when you know the slope and one point on the line. To do so, rewrite y = mx + b with the known slope and substitute the coordinates of the known point for x and y. Then solve for b and write the new equation. For example, find the equation of the line with a slope of –4 that passes through the point (5, 30). Rewrite y = mx + b as y = –4x + b. Substituting (5, 30) into the equation results in 30 = –4(5) + b. Solve the equation to find b = 50. Since you now know the slope and y-intercept of the line, you can write the equation of the line as y = –4x + 50. Similarly you can write the equation of the line when you know two points. First use the two points to find the slope. Then substitute the known slope and either of the known points into y = mx + b. Solve for b and write the new equation. For example, find the equation of the line through (3, 7) and (9, 9). The slope is y x 2 6 1 . Substituting m = 3 1 and 3 (x, y) = (3, 7) into y = mx + b results in 7 1 3 (3) b . Then solve the equation to find b = 6. Since you now know the slope and y-intercept, you can write the equation of the line as y 1 3 xb. 16 3.3.3 Using Generic Rectangles to Multiply A generic rectangle can be used to find products because it helps to organize the different areas that make up the total rectangle. For example, to multiply (2x + 5)(x + 3), a generic rectangle can be set up and completed as shown below. Notice that each product in the generic rectangle represents the area of that part of the rectangle. Note that while a generic rectangle helps organize the problem, its size and scale are not important. Some students find it helpful to write the dimensions on the rectangle twice, that is, on both pairs of opposite sides. Prob # Date Learning Log Entry 3-18 Zero and Negative Exponents 3-47 Area as a Product and as a Sum 17 Prob # 3-106 Date Learning Log Entry Summary of Solving Equations 4.1.1 Line of Best Fit A line of best fit is a straight line that represents, in general, the data on a scatterplot, as shown in the diagram. This line does not need to touch any of the actual data points, nor does it need to go through the origin. The line of best fit is a model of numerical two-variable data that helps describe the data in order to make predictions for other data. To write the equation of a line of best fit, find the coordinates of any two convenient points on the line (they could be lattice points where the gridlines intersect, or they could be data points, or the origin, or a combination). Then write the equation of the line that goes through these two points. 18 4.1.2 The Equal Values Method The equal values method is a method to find the solution to a system of equations. For example, solve the system of equations below: 2x + y = 5 y=x–1 Put both equations into y = mx + b form. The two equations are now y = −2x + 5 and y = x − 1. 2x y 5 y x 1 y 2 x 5 Take the two expressions that equal y and set them equal to each other. Then solve this new equation to find x. See the example at right Once you know x, substitute your solution for x into either original equation to find y. In this example, the second equation is used. Check your solution by evaluating for x and y in both of the original equations. 2x y 5 y x 1 2(2) 1 5 1 2 1 55 11 4.2.1 Describing Association An association (relationship) between two numerical variables can be described by its form, direction, strength, and outliers. The shape of the pattern is called the form of the association: linear or non-linear. The form can be made of clusters of data. If one variable increases as the other variable increases, the direction is said to be a positive association. If one variable increases as the other variable decreases, there is said to be a negative association. If there is no apparent pattern in the scatterplot, then the variables have no association. Strength is a description of how much scatter there is in the data away from the line of best fit. See some examples below. An outlier is a piece of data that does not seem to fit into the overall pattern. There is one obvious outlier in the association graphed at right. 19 4.2.2 The Substitution Method The Substitution Method is a way to change two equations with two variables into one equation with one variable. It is convenient to use when only one equation is solved for a variable. For example, to solve the system: Use substitution to rewrite the two equations as one. In other words, replace x in the second equation with (−3y + 1) from the first equation to get 4(−3y + 1) − 3y = −11. This equation can then be solved to find y. In this case, y = 1. To find the point of intersection, substitute the value you found into either original equation to find the other value. In the example, substitute y = 1 into x = −3y + 1 and write the answer for x and y as an ordered pair. To check the solution, substitute x = −2 and y = 1 into both of the original equations. 4.2.3 Systems of Linear Equations A system of linear equations is a set of two or more linear equations that are given together, such as the example at right: y 2x y 3x 5 If the equations come from a real-world context, then each variable will represent some type of quantity in both equations. For example, in the system of equations above, y could stand for a number of dollars in both equations and x could stand for the number of weeks. (1,2) To represent a system of equations graphically, you can simply graph each equation on the same set of axes. The graph may or may not have a point of intersection, as shown circled at right. Also notice that the point of intersection lies on both graphs in the system of equations. This means that the point of intersection is a solution to both equations in the system. For example, the point of intersection of the two lines graphed above is (1, 2). This point of intersection makes both equations true, as shown at right. y 2x (2) 2(1) 22 y 3x 5 (2) 3(1) 5 2 3 5 22 The point of intersection makes both equations true; therefore the point of intersection is a solution to both equations. For this reason, the point of intersection is sometimes called asolution to the system of equations. 20 4.2.4 Forms of a Linear Function There are three main forms of a linear function: slope-intercept form, standard form, and point-slope form. Study the examples below. Slope-Intercept form: y = mx + b. The slope is m, and the y-intercept is (0, b). Standard form: ax + by = c Point-Slope form: y − k = m(x − h). The slope is m, and (h, k) is a point on the line. For example, if the slope is –7 and the point (20, –10) is on the line, the equation of the line can be written y − (−10) = −7(x − 20) or y + 10 = −7(x − 20). 4.2.5 Intersection, Parallel, and Coincide When two lines lie on the same flat surface (called a plane), they may intersect (cross each other) once, an infinite number of times, or never. For example, if the two lines are parallel, then they never intersect. The system of these two lines does not have a solution. Examine the graph of two parallel lines at right. Notice that the distance between the two lines is constant and that they have the same slope but different y-intercepts.. However, what if the two lines lie exactly on top of each other? When this happens, we say that the two lines coincide. When you look at two lines that coincide, they appear to be one line. Since these two lines intersect each other at all points along the line, coinciding lines have an infinite number of intersections. The system has an infinite number of solutions. Both lines have the same slope and y-intercept. While some systems contain lines that are parallel and others coincide, the most common case for a system of equations is when the two lines intersect once, as shown at right. The system has one solution, namely, the point where the lines intersect, (x, y). Prob # 4-48 Date Learning Log Entry Solutions to Two-Variable Equations 21 Prob # 4-80 Date Learning Log Entry Solving Systems of Equations 5.1.1 The Elimination Method When solving a system of equations, it may be easier to eliminate one of the variables by adding multiples of the two equations. This process is called elimination. The first step is to rewrite the equations so that the x and y variables are lined up vertically. Next, decide what number to multiply each equation by, if necessary, in order to make the coefficients of either the x-terms or the y-terms add up to zero. Be sure that you can justify each step in the solution. 10y − 3x = 14 4y + 2x = −4 For example, consider the system at right. You can eliminate the x-terms by multiplying the top equation by 2 and the bottom equation by 3 and then adding the equations, as shown at right. (10y − 3x = 14) · 2 → 20y − 6x = 28 (4y + 2x = −4) · 3 → 12y + 6x = −12 Add the resulting equations: 32y = 16 Divide: y = 0.5 Finally, substitute 0.5 for y in either original equation: 10(0.5) − 3x = 14 5 − 3x = 14 −3x = 9 x = −3 Thus, the solution to the original system is (−3, 0.5). Check your solution by evaluating for x and y in both of the original equations. 22 5.1.2 Continuous and Discrete Graphs When the points on a graph are connected, and it makes sense to connect them, the graph is said to be continuous. If the graph is not continuous, and is just a sequence of separate points, the graph is called discrete. For example, the graph below left represents the cost of buying x shirts, and it is discrete because you can only buy whole numbers of shirts. The graph below farthest right represents the cost of buying x gallons of gasoline, and it is continuous because you can buy any (non-negative) amount of gasoline. 5.3.2 Types of Sequences An arithmetic sequence is a sequence with an addition (or subtraction) generator. The number added to each term to get the next term is called the common difference. A geometric sequence is a sequence with a multiplication (or division) generator. The number multiplied by each term to get the next term is called the common ratio or the multiplier. A multiplier can also be used to increase or decrease by a given percentage. For example, the multiplier for an increase of 7% is 1.07. The multiplier for a decrease of 7% is 0.93. A recursive sequence is a sequence in which each term depends on the term(s) before it. The equation of a recursive sequence requires at least one term to be specified. A recursive sequence can be arithmetic, geometric, or neither. For example, the sequence –1, 2, 5, 26, 677, … can be defined by the recursive equation: t(1) = –1, t(n + 1) = (t(n))2 + 1 An alternative notation for the equation of the sequence above is: a1 = –1, an + 1 = (an)2 + 1 Prob # 5-5 Date Learning Log Entry Exponential Functions 23 Prob # Date Learning Log Entry 5-94 Multipliers 5-113 Sequences vs. Functions 6.1.2 Interpreting Slope and y-Intercept on Scatterplots The slope of a linear association plays the same role as the slope of a line in algebra. Slope is the amount of change we expect in the dependent variable (Δy) when we change the independent variable (Δx) by one unit. When describing the slope of a line of best fit, always acknowledge that you are making a prediction, as opposed to knowing the truth, by using words like “predict,” “expect,” or “estimate.” The y-intercept of an association is the same as in algebra. It is the predicted value of the dependent variable when the independent variable is zero. Be careful. In statistical scatterplots, the vertical axis is often not drawn at the origin, so the y-intercept can be someplace other than where the line of best fit crosses the vertical axis in a scatterplot. Also be careful about extrapolating the data too far—making predictions that are far to the right or left of the data. The models we create can be valid within the range of 24 the data, but the farther you go outside this range, the less reliable the predictions become. When describing a linear association, you can use the slope, whether it is positive or negative, and its interpretation in context, to describe the direction of the association. 6.1.4 Residuals We measure how far a prediction made by our model is from the actual observed value with a residual: residual = actual – predicted A residual has the same units as the y-axis. A residual can be graphed with a vertical segment that extends from the point to the line or curve made by the best-fit model. The length of this segment (in the units of the y-axis) is the residual. A positive residual means that the actual value is greater than the predicted value; a negative residual means that the actual value is less than the predicted value. 6.2.1 Least Squares Regression Line There are two reasons for modeling scattered data with a best-fit line. One is so that the trend in the data can easily be described to others without giving them a list of all the data points. The other is so that predictions can be made about points for which we do not have actual data. A unique best-fit line for data can be found by determining the line that makes the residuals, and hence the square of the residuals, as small as possible. We call this line the least squares regression line and abbreviate it LSRL. A calculator can find the LSRL quickly. Statisticians prefer the LSRL to some other best-fit lines because there is one unique LSRL for any set of data. All statisticians, therefore, come up with exactly the same best-fit line and can use it to make similar descriptions of, and predictions from, the scattered data. 6.2.3 Residual Plots A residual plot is created in order to analyze the appropriateness of a best-fit model. A residual plot has an x-axis that is the same as the independent variable for the data. The y-axis of a residual plot is the residual for each point. Recall that residuals have the same units as the dependent variable of the data. If a linear model fits the data well, no interesting pattern will be made by the residuals. That is because a line that fits the data well just goes through the “middle” of all the data. A residual plot can be used as evidence that the description of the form of a linear association has been made appropriately. 6.2.4 Correlation Coefficient The correlation coefficient, r, is a measure of how much or how little data is scattered around the LSRL; it is a measure of the strength of a linear association. The correlation coefficient can take on values between –1 and 1. If r = 1 or r = –1 the association is perfectly linear. There is no scatter about the LSRL at all. A positive correlation coefficient means the trend is increasing (slope is positive), while a negative 25 correlation means the opposite. A correlation coefficient of zero means the slope of the LSRL is horizontal and there is no linear association whatsoever between the variables. The correlation coefficient does not have units, so it is a useful way to compare scatter from situation to situation no matter what the units of the variables are. The correlation coefficient does not have a real-world meaning other than as an arbitrary measure of strength. The value of the correlation coefficient squared, however, does have a real-world meaning. R-squared, the correlation coefficient squared, is written as R² and expressed as a percent. Its meaning is that R²% of the variability in the dependent variable can be explained by a linear relationship with the independent variable. For example, if the association between the amount of fertilizer and plant height has correlation coefficient r = 0.60, we can say that 36% of the variability in plant height can be explained by a linear relationship with the amount of fertilizer used. The rest of the variation in plant height is explained by other variables: amount of water, amount of sunlight, soil type, and so forth. The correlation coefficient, along with the interpretation of R², is used to describe the strength of a linear association. 6.2.5 Non-Linear Models Sometimes a non-linear model best fits the data, and therefore makes better predictions, than a linear model. This is usually made apparent by comparing the residual plots of various models. A good model should also be representative of the physical situation. In problem 6-107 an exponential model made physical sense because we were measuring decay over time. A quadratic model would not have been very satisfactory because of its U-shape. On the other hand, if we were making predictions about the path of a rocket, a quadratic model of the data would make a lot of sense because gravity has a quadratic relationship with height. If we were modeling a relationship between volume and length, a power model would be appropriate, because volume is related to length 1 y x3 2 2 . by a power of 3. A power model has exponents in it, for example y 5 x or The most common models are: Linear y = mx + b Exponential y = abx Power y = axb Quadratic y = ax2 + bx + c 26 Prob # Date Learning Log Entry 6-15 Residuals 6-54 Residual Plots 6-70 Correlation Coefficient, r 27 Prob # 6-98 Date Learning Log Entry Completely Describing Association 7.1.1 Graphs with Asymptotes A mathematically clear and complete definition of an asymptote requires some ideas from calculus, but some examples of graphs with asymptotes might help you recognize them when they occur. In the following examples, the dotted lines are the asymptotes, and their equations are given. In the two lower graphs, the yaxis, x = 0, is also an asymptote. As you can see in the examples above, asymptotes can be diagonal lines or even curves. However, in this course, asymptotes will almost always be horizontal or vertical lines. The graph of a function has a horizontal asymptote if, as you trace along the graph out to the left or right (that is, as you choose x-coordinates farther and farther away from zero, either toward infinity or toward negative infinity), the distance between the graph of the function and the asymptote gets closer to zero. A graph has a vertical asymptote if, as you choose x-coordinates closer and closer to a certain value, from either the left or right (or both), the y-coordinate gets farther away from zero, either toward infinity or toward negative infinity. 7.1.3 Compound Interest A bank can pay simple interest in which case the amount in the bank grows linearly. For example, 3% simple interest compounded annually on an initial investment of $2500 would grow in a sequence with a common difference: 0.03(2500) = $75.The equation and table follow: t(n) = 2500 + 75n 28 If the bank compounds interest, the relationship is exponential. For example, 3% annual interest, compounded annually, would have a multiplier of 1.03 every year. The equation and table using the example above are: t(n) = 2500 · 1.03n If the bank compounds monthly, the 3% annual interest becomes 3% 0.25% 12 months / year per month, and the multiplier becomes 1.0025. The equation and table for the first ten years follows: t(m) = 2500 · 1.0025m 7.1.5 Step Functions A step function is a special kind of piecewise function ( a function composed of parts of two or more functions). A step function has a graph that is a series of line segments that often looks like a set of steps. Step functions are used to model realworld situations where there are abrupt changes in the output of the function. The endpoints of the segments on step functions are either open circles (indicating this point is not part of the segment) or filled-in circles (indicating this point is part of the segment). The graph below models a situation in which a tour bus company has busses that can each hold up to 20 passengers with 5 available busses. 7.2.2 Negative and Fractional Exponents For all x not equal to zero: x0 = 1 Examples: For positive values of x : 7.2.3 Equations for Sequences Arithmetic Sequences The equation for an arithmetic sequence is: t(n) = mn + b or an = mn + a0 where n is the term number, m is the sequence generator (the common difference), and b or a0 is the zeroth term. Compare these equations to a continuous linear function f(x) = mx + b where m is the growth (slope) and b is the starting value (y-intercept). 29 For example, the arithmetic sequence 4, 7, 10, 13, … could be represented by t(n) = 3n + 1 or by an = 3n + 1. (Note that “4” is the first term of this sequence, so “1” is the zeroth term.) Another way to write the equation of an arithmetic sequence is by using the first term in the equation, as in an = m(n – 1) + a1, where a1 is the first term. The sequence in the example could be represented by an = 3(n – 1) + 4. You could even write an equation using any other term in the sequence. The equation using the fourth term in the example would be an = 3(n – 4) + 13. Geometric Sequences The equation for a geometric sequence is: t(n) = abn or an = a0 · bn where n is the term number, b is the sequence generator (the multiplier or common ratio), and a or a0 is the zeroth term. Compare these equations to a continuous exponential function f(x) = abx where b is the growth (multiplier) and a is the starting value (y-intercept). For example, the geometric sequence 6, 18, 54, … could be represented by t(n) = 2 · 3n or by an = 2 · 3n. You can write a first term form of the equation for a geometric sequence as well: an = a1 · bn–1. For the example, first term form would be an = 6 · 3n–1. Prob # Date Learning Log Entry 7-6 Investigating y=bx 7-22 Multiple Representations Web for Exponential Functions TABLE EQUATION GRAPH SITUATION 30 Prob # Date Learning Log Entry 7-60 Graph Equation for Exponential Functions 7-72 Important Ideas about Exponential Functions 7-86 Zero, Negative, and Fractional Exponents 31 8.1.1 More Vocabulary for Expressions Since algebraic expressions come in several different forms, there are special words used to help describe these expressions. For example, if the expression can be written in the form ax2 +bx + c and if a is not 0, it is called a quadratic expression. Review the examples of quadratic expressions below. Examples of quadratic expressions: The way an expression is written can also be named. When an expression is written in product form, it is said to be factored. When factored, each of the expressions being multiplied is called a factor. For example, the factored form of x2 − 15x + 26 is (x − 13)(x − 2), so x − 13 and x − 2 are each factors of the original expression. Finally, if the expression is a polynomial (see Math Notes box in Lesson 3.2.3) the number of terms can help you name the polynomial. If the polynomial has one term, it is called amonomial, while a polynomial with two terms is called a binomial. If the polynomial has three terms, it is called a trinomial. Review the examples below. Examples of monomials: Examples of binomials: Examples of trinomials: 15y 2 and 2 16m 25 and 7h9 1 2 12 3k 5k and x 15x 26 3 2 8.1.2 Diagonals of Generic Rectangles Why does Casey’s pattern from problem 8-4 work? That is, why does the product of the terms in one diagonal of a 2-by-2 generic rectangle always equal the product of the terms in the other diagonal? Examine the generic rectangle at right for (a + b)(c + d). Notice that each of the resulting diagonals have a product of abcd. Thus, the product of the terms in the diagonals are equal. 8.1.3 Standard Form of a Quadratic Expression A quadratic expression in the form ax2 + bx + c is said to be in standard form. Notice that the terms are in order from greatest exponent to least. Examples of quadratic expressions in standard form: 3m2 + m − 1, x2 − 9, and 3x2 + 5x. Notice that in the second example, b = 0, while in the third example, c = 0. 8.1.4 Factoring Quadratic Expressions Review the process of factoring quadratics developed in problem 8-13 and outlined below. This example demonstrates how to factor 3x2 + 10x + 8. 1. Place the x2 and constant terms of the quadratic expression in opposite corners of a generic rectangle. Determine the sum and product of the two remaining 32 2. 3. 4. corners: The sum is simply the x-term of the quadratic expression, while the product is equal to the product of the x2 and constant terms. Place this sum and product into a Diamond Problem and solve it. Place the solutions from the Diamond Problem into the generic rectangle and find the dimensions of the generic rectangle. Write your answer as a product: (3x + 4)(x + 2). 8.2.2 Zero Product Property When the product of two or more numbers is zero, one of those numbers must be zero. This is known as the Zero Product Property. If the two numbers are represented by a and b, this property can be written as follows: If a and b are two numbers where a · b = 0, then a = 0 or b = 0. For example, if (2x − 3)(x + 5) = 0, then 2x − 3 = 0 or x + 5 = 0. Solving yields the solutions x 3 or x = −5. This property helps you solve quadratic equations when the 2 equation can be written as a product of factors. 8.2.4 Forms of a Quadratic Function There are three main forms of a quadratic function: standard form, factored form, and graphing form. Study the examples below. Assume that a ≠ 0 and that the meaning of a, b, and c are different for each form below. Standard form: f(x) = ax2+ bx + c. The y-intercept is (0, c). Factored form: f(x) = a(x + b)(x + c). The x-intercepts are (–b, 0) and(–c, 0). Graphing form (vertex form): f(x) = a(x – h)2+ k. The vertex is (h, k). Prob # 8-5 Date Learning Log Entry Diagonals of a Generic Rectangle 33 Prob # Date Learning Log Entry 8-28 Factoring Quadratics 8-48 Factoring Shortcuts 8-57 Quadratic Web TABLE EQUATION GRAPH SITUATION 34 Prob # Date Learning Log Entry 8-66 Zero Product Property 8-82 Strategies for Finding x-intercepts 9.1.1 Zeros and Roots of Quadratics A root or zero of a quadratic expression is a value of x that makes the expression equal to zero. For example, the roots or zeros of the quadratic expression x2 − 2x − 8 are the solutions to the equation x2 − 2x − 8= 0 The x-intercepts of any quadratic function are roots. You find the x-intercepts by setting the function equal to zero and solving for x. For example, the quadratic function f(x) = x2 − 2x − 8= (x + 2)(x − 4) is graphed at right. The x-intercepts are at (0, –2) and (0, 4). The roots or zeros are –2 and 4 because the solutions to the equation x2 − 2x − 8 = 0 are –2 and 4. 35 9.1.2 The Quadratic Formula 2 Why is x b b 4ac 2a formula is shown below. 1. a solution of ax2 + bx + c = 0? One way to derive this Begin with the quadratic equation in standard form. ax2 + bx + c = 0 4a(ax2 + bx + c) = 4a(0) 2. Multiply each side by 4a. 4a2x2 + 4abx + 4ac = 0 3. 2 Add b − 4ac to each side in order to get a factorable quadratic on the left. 4. The left side can be factored as (2ax + b)2, which is demonstrated in the generic rectangle shown at right. 5. Take the square root of each side. Since a square root refers to thepositive root, the absolute value of 2ax + b is used. Then by “looking inside” there are two possible values for 2ax + b: b2 4ac and 4a2x2 + 4abx + b2 = b2 − 4ac (2ax + b)2 = b2 − 4ac b2 4ac . 6. Now continue to solve for x by subtracting b from both sides and dividing by 2a. Notice that a cannot equal zero or else you will get an error! However, if a = 0, then this equation would not be quadratic and you would not use this formula. are solutions of the equation ax2 + bx + c = 0. Thus, 9.1.3 Solving a Quadratic Equation So far in this course, you have learned two algebraic methods to solve a quadratic equation of the form ax2 + bx + c = 0. Example 1: Solve 3x2 + x − 14 = 0 for x using the Zero Product Property. Solution: First, factor the quadratic so it is written as a product: (3x + 7)(x − 2) = 0. (If factoring is not possible, one of the other methods of solving must be used.) The Zero Product Property states that if the product of two terms is 0, then at least one of the factors must be 0. Thus, 3x + 7 = 0 or x − 2 = 0. Solving these equations for x reveals that x = 7 or that x = 2. 3 Example 2: Solve 3x2 + x − 14 = 0 for x using the Quadratic Formula. Solution: This method works for any quadratic. First, identify a, b, and c. a equals the number of x2 terms, b equals the number of x terms, and c equals the constant. For 3x2 + x − 14 = 0, a = 3, b = 1, and c = −14. Substitute the values of a, b, and c into the 36 Quadratic Formula and evaluate the expression twice: once with addition and once with subtraction. Examine this method below: Example 3: Solve x2 + 5x + 4 = 0 by completing the square. Solution: This method works most efficiently when the coefficient of x2 is 1. Rewrite the equation as x2 + 5x = −4. Rewrite the left side as an incomplete square: Take the square root of both sides, x + 2.5 = ±1.5. Solving for x reveals that x = −1 or x = −4. 9.1.4 Simplifying Square Roots Before calculators were universally available, people who wanted to use approximate decimal values for numbers like 45 had a few options: 1. Carry around copies of long square-root tables. 2. Use Guess and Check repeatedly to get desired accuracy. 3. “Simplify” the square roots. A square root is simplified when there are no more perfect square factors (square numbers such as 4, 25, and 81) under the radical sign. Simplifying square roots was by far the fastest method. People factored the number as the product of integers hoping to find at least one perfect square number. They memorized approximations of the square roots of the integers from one to ten. Then they could figure out the decimal value by multiplying these memorized facts with the roots of the square numbers. Here are some examples of this method. Example 1: Simplify 45 . First rewrite 45 in an equivalent factored form so that one of the factors is a perfect square. Simplify the square root of the perfect square. Verify with your calculator that both 3 5 and 45 ≈ 6.71. Examine Example 2 and Example 3 at right. Note that in Example 3, 72 was rewritten as 36 2 , rather than as 9 8 or 4 18 , because 36 is the largest perfect square factor of 72. However, since 4 18 2 9 2 2 9 2 2 3 2 6 2 and 9 8 3 4 2 3 4 2 3 2 2 6 2 , you can still get the same answer if you simplify it using different methods. When you take the square root of an integer that is not a perfect square, the result is a decimal that never repeats itself and never ends. It is a number that cannot be written as a fraction using integers. This result is called an irrational number. The irrational 37 numbers and the rational numbers together form the real numbers. Generally, since it is now the age of technology, when a decimal approximation of an irrational square root is desired, a calculator is used. However for an exact answer, called exact form or radical form, the number must be written using the symbol. 9.2.1 Inequality Symbols Just as the symbol “=” is used to represent that two quantities are equal in mathematics, the inequality symbols at right are used when describing the relationships between quantities that are not necessarily equal. < less than < less than or equal to > greater than > greater than or equal to When graphing an inequality on a number line, such as x ≥ −1, a filled circle (point) indicates that the value is a solution of the inequality, as shown at right. An open circle indicates that the value is not part of the solution, as in x < –3, as shown at right. 9.3.1 Curve Fitting an Exponential Function You can find an exponential function that goes through two points (if both points are above the x-axis). Recall that an exponential function with an asymptote of the x-axis has an equation of the form y = abx. To find an exponential function that goes through two given points, create a system of equations by substituting one (x, y) point into y = abx, then substituting the other point. Rewrite both equations in “a =” form. Solve the system with the equal values method to find a and b and now you can write the equation. For example, find an exponential function that passes through (2, 14) and (5, 112). Create a system of equations by substituting (x, y) = (2, 14) into y = abx, and then substituting again with (x, y) = (5, 112): 14 112 14 = ab2 and 112 = ab5 Rewrite as a 2 and a 5 . b b Use the equal values method to find b: 38 The equation of the exponential function that passes through the two given points is y = 3.5 · 2x. 9.3.2 Solving One-Variable Inequalities To solve a one-variable inequality, first treat the problem as if it were an equality. The solution to the equality is called the boundary point. For example, x = 12 is the boundary point for the inequality 10 − 2(x − 3) > −8, as shown below. Problem: 10 − 2(x − 3) > −8 10 − 2(x − 3) = −8 10 − 2x + 6 = −8 −2x + 16 = −8 −2x = −24 x = 12 First change the problem to an equality and solve for x: Since the original inequality is true when x = 12, place your boundary point on the number line as a solid point. Then test one value on either side in the original inequality to determine which set of numbers makes the inequality true. Therefore, the solution is x < 12. Test: x=8 10 − 2(8 − 3) > −8 10 − 2(5) > −8 0 > −8 TRUE! Test: x = 15 10 − 2(15 − 3) > −8 10 − 2(12) > −8 −14 > 17 FALSE! When the inequality is < or >, the boundary point is not included in the answer. On a number line, this would be indicated with an open circle at the boundary point. 9.4.1 Graphing Inequalities with Two Variables To graph solve an inequality with two variables, first graph the boundary line or curve. If the inequality does not include an equality (that is, if it is > or < rather than > or <), then the graph of the boundary is dashed to indicate that it is not included in the solution. Otherwise, the boundary is a solid line or curve. Once the boundary is graphed, choose a point that does not lie on the boundary to test in the inequality. If that point makes the inequality true, then the entire region in which that point lies is a solution. Examine the two examples below. There are infinite solutions to each of the inequalities. The shaded portion of the graph is a diagram of all of the solutions. 39 Prob # Date Learning Log Entry 9-16 Quadratic Formula 9-35 Choosing a Strategy to Solve Quadratics 9-69 Graphing Linear Inequalities 40 Prob # Date Learning Log Entry 9-104 Graphing Systems of Inequalities 10.2.1 Equivalent Equations Two equations are equivalent if all of their solutions are the same. There are several ways to change one equation into a different, equivalent equation. Common ways include: adding the same number to both sides, subtracting the same number from both sides, multiplying both sides by the same number, dividing both sides by the same (non-zero) number, and rewriting one or both sides of the equation. For example, the equations below are all equivalent to 2x + 1 = 3: 20x + 10 = 30 2(x + 0.5) = 3 0.002x + 0.001 = 0.003 10.2.2 Solving Equations with Fractions (also known as Fraction Busters) Example: Solve x 3 x 5 2 for x. This equation would be much easier to solve if it had no fractions. Therefore, the first goal is to find an equivalent equation that has no fractions. To eliminate the denominators, multiply both sides of the equation by the common denominator. In this example, the lowest common denominator is 15, so multiplying both sides of the equation by 15 eliminates the fractions. Another approach is to multiply both sides of the equation by one denominator and then by the other. Either way, the result is an equivalent equation without fractions: 41 5x + 3x = 30 8x = 30 The number used to eliminate the denominators is called a fraction buster. Now the equation looks like many you have seen before, and it can be solved in the usual way. Once you have found the solution, remember to check your answer. 10.2.3 Methods to Solve Single-Variable Equations Here are three different approaches you can take to solve a one-variable equation: Rewriting: Use algebraic techniques to rewrite the equation. This will often involve using the Distributive Property to get rid of parentheses. Then solve the equation using solution methods you know. Looking inside: Choose a part of the equation that includes the variable and is grouped together by parentheses or another symbol. (Make sure it includes all occurrences of the variable!) Ask yourself, “What must this part of the equation equal to make the equation true?” Use that information to write and solve a new, simpler equation. Undoing: Start by undoing the last operation that was done to the variable. This will give you a simpler equation, which you can solve either by undoing again or with some other approach. 10.2.4 Forms of a Quadratic Equation There are three main forms of a quadratic function: standard form, factored form, and graphing form. (See the Math Notes box in Lesson 8.2.4) Similarly, there are three forms of a single-variable quadratic equation. Standard form: Any quadratic equation written in the form ax2 + bx + c = 0. Factored form: Any quadratic function written in the form a(x + b)(x + c) = 0. Perfect Square form: Any quadratic function written in the form (ax − b)2 = c2. Notice that when the expression on the left side of the equation below is built with tiles, it forms a perfect square, as shown at right. (2x + 3)2 = 5 Solutions to a quadratic equation can be written in exact form (radical form) as in: or solutions can be estimated in approximate decimal form: x = −0.38 or x = −2.62 42 10.2.5 The Number System The collection (set) of all numbers is organized into categories. 10.2.6 Solving Absolute-Value Equations To solve an equation with an absolute value algebraically, first “isolate” the absolute value on one side of the equation. Determine the possible values of the quantity inside the absolute value. For example, if 2 x 3 7 , then the quantity (2x + 3) must equal 7 or –7. With these two values, set up new equations and solve as shown below. Note that distributing over an absolute value is not allowed. For example, Prob # 10-66 Date Learning Log Entry Number of Solutions 43 Prob # Date Learning Log Entry 10-80 The Number System 10-102 Intercepts and Intersections 10-136 Solving Inequalities with Absolute Value 44 11.1.2 Number of Solutions to a Quadratic Equation The solutions to a quadratic equation in standard form, ax2 + bx + c = 0, 2 are x b b 4ac . 2a Therefore you can tell how many solutions a quadratic equation in standard form has by looking at the value of the expression b2– 4ac. The expression b2– 4ac is called the discriminant. If b2– 4ac is negative, there are no real solutions, because the square roots of negative numbers are not real numbers. However, there are two solutions that contain imaginary numbers whenb2– 4ac is negative. If b2– 4ac is zero, there is only one solution, because 0 simplifies only to 0. If b2– 4ac is positive, there are two solutions involving b2 4ac . If a quadratic is in perfect square form, (ax – b)2 = c, you may solve by taking the square root of both sides: ax b c . The number of solutions is determined by c: If c is negative, there are no real solutions. If c is zero, there is one solution. If c is positive, there are two solutions. If a quadratic is in factored form, a(x + b)(x + c) = 0, the Zero Product Property usually gives two solutions: x + b = 0 and x + c = 0. However, if b = c then there is only one solution. 11.2.1 Interquartile Range and Boxplots Quartiles are points that divide a data set into four equal parts (and thus, the use of the prefix “quar” as in “quarter”). One of these points is the median. The first quartile (Q1) is the median of the lower half, and the third quartile (Q3) is the median of the upper half. To find quartiles, the data set must be placed in order from smallest to largest. Note that if there are an odd number of data values, the median is not included in either half of the data set. Suppose you have the data set: 22, 43, 14, 7, 2, 32, 9, 36, and 12. The interquartile range (IQR) is the difference between the third and first quartiles. It is used to measure the spread (the variability) of the middle fifty percent of the data. The interquartile range is 34 – 8 = 26. A boxplot (also known as a box-and-whisker plot) displays a five number summary of data: minimum, first quartile, median, third quartile, and maximum. The box contains “the middle half” of the data and visually displays how large the IQR is. The right segment represents the top 25% of the data and the left segment represents the bottom 25% of the data. A boxplot makes it easy to see where 45 the data are spread out and where they are concentrated. The wider the box, the more the data are spread out. 11.2.2 Describing Shape (of a Data Distribution) Statisticians use the words below to describe the shape of a data distribution. Outliers are any data values that are far away from the bulk of the data distribution. In the example at right, data values in the right-most bin are outliers. Outliers are marked on a modified boxplot with a dot. 11.2.3 Describing Spread (of a Data Distribution) A distribution of data can be summarized by describing its center, shape, spread, and outliers. You have learned three ways to describe the spread. Interquartile Range (IQR) The variability, or spread, in the distribution can be numerically summarized with the interquartile range (IQR). The IQR is found by subtracting the first quartile from the third quartile. The IQR is the range of the middle half of the data. IQR can represent the spread of any data distribution, even if the distribution is not symmetric or has outliers. Standard Deviation Either the interquartile range or standard deviation can be used to represent the spread if the data is symmetric and has no outliers. The standard deviation is the square root of the average of the distances to the mean, after the distances have been made positive by squaring. For example, for the data 10 12 14 16 18 kilograms: The mean is 14 kg. The distances of each data value to the mean are –4, –2, 0, 2, 4 kg. The distances squared are 16, 4, 0, 4, 16 kg2. The mean distance-squared is 8 kg2. The square root is 2.83. The standard deviation is 2.83 kg. Range The range (maximum minus minimum) is usually not a very good way to describe the spread because it considers only the extreme values in the data, rather than how the bulk of the data is spread. 46 Prob # Date Learning Log Entry 11-3 Transforming Functions 11-19 Finding and Checking Inverses 11-27 Representing Data Graphically 47 Prob # Date Learning Log Entry 11-45 Describing Single Variable Data 11-66 Standard Deviation 48 49