Math TEXTbook DIGITAL SAT PREP 1.1 PART I: ALGEBRA BASIC CONCEPTS ORDER OF OPERATIONS PGERMDAS 1. 2. 3. 4. • • • • What is the value of +(-"!)(-/%)"%+ - COMMUTATIVE LAW: When adding or multiplying, order doesn’t matter. Example: 3 + 8 + 12 = 8 + 12 + 3. ASSOCIATIVE LAW: When adding or multiplying, grouping doesn’t matter. Example: 2 + (3 + 4) + 5 = (2 + 3) + (4 + 5) DISTRIBUTIVE LAW: When a grouped sum/difference is being multiplied/divided, you may “distribute” the multiplication/division. Example: 15 (20 + 8) = 15 (20) + 15 (8) SIMPLIFICATION OF EXPRESSIONS • √'#"( PG: Parenthesis and other grouping symbols (inside out) ER: Exponents and roots (inside out) MD: Multiplication and division (left to right) AS: Addition and subtraction (left to right) LAWS OF ARITHMETICS • !"#×% What is the value of !"%×! + #)÷%! ? LAW OF SUBSTITUTION: If two things are equal, you can always substitute one for the other. OPERATIONS: Every operation can be expressed in terms of its inverse. Example: Subtracting -16 is the same as adding 16. FACTORING IDENTITIES: o (π + π)% = (π + π)(π + π) = π% + 2ππ + π% o (π − π)% = (π − π)(π − π) = π% − 2ππ + π% o (π + π)(π − π) = π% − π% ABSOLUTE VALUE ? If x and y are positive numbers such that 3x – 2y = 7, what is %0"1 the value of #- ? 2! /3! ( If m and n are real numbers such that %2/%3 = %, what is the value of m + n? What is the distance between d and -10 on the number line? The absolute value of a number a (|a|) is the distance from a to 0 on the number line. The absolute value of the difference between two numbers (|a-b|) is the distance between a and b on the number line, regardless of which number is greater. This document has been created for educational purposes only and meant to be used exclusively by SCORE – Test Prep and College Counseling. This document may not be distributed or reproduced without the express written consent of its authors. 0.000001 0.00001 Millionths Hundred-Thousandths 0.001 0.0001 Ten-Thousandths Thousandths 0.1 0.01 Hundredths . Tenths 1 Decimal Point 10 Ones 100 DECIMAL PART Tens 1,000 Hundreds 10,000 Thousands 100,000 Ten Thousands 1,000,000 Millions WHOLE PART Hundred Thousands ROUNDING The process of rounding implies adjusting the digits to make rough calculations easier. • If the unit of the number is less than five, the number needs to be rounded down. • If the unit of the number is 5 or above, the number needs to be rounded up. 1 1.2 PART I: ALGEBRA Linear Equations WORD PROBLEMS The easiest way to solve word problems is by breaking them down following these steps: 1. Identify the relevant quantities. 2. Represent those quantities with algebraic expressions. 3. Translate the facts into equations. 4. Solve the equations for the relevant quantities. The Horizon Resort charges $150 per night for a single room, and a one-time valet parking fee of $35. There is a 6.5% state tax on the room charges, but no tax on the valet parking fee. What equation would represent the total charges in dollars, C, for a single room, valet parking, and taxes, for a stay of n nights at the Horizon Resort? CONSTRUCTION AND INTERPRETATION The graph of the line π in the xy-plane passes through the point (2, 5) and has an x-intercept of 7. Which of the following gives the equation of a line that is perpendicular to line π and passes through the point (4, 2)? a. π¦ = −π₯ + 6 b. π¦ = −π₯ + 4 c. π¦ = π₯ − 2 d. π¦ = π₯ + 2 • Linear equations can be written in any of the following forms: Linear form Slope-intercept Standard Point-slope Intercept Equation y = mx + b ax + by = c y - y1 = m(x - x 1) x / a+y / b =1 Slope m -a/b m - Y-intercept b c/b b X-intercept a *The intercept form cannot be used for horizontal lines or lines passing through the origin. a. b. c. The slope of a linear equation shows how much a line goes up or down when you take one step to the right along the line. It can be calculated using the following formula: πππ π π¦# − π¦! π ππππ = = ππ’π π₯# − π₯! Parallel slopes have equal values: π! = π# ! Perpendicular slopes are opposite and reciprocal: π! = − d. e. f. g. h. i. A line with a positive slope goes up as you move to the right. A line with a negative slope goes down as you move to the right. A horizontal line has a 0 slope. A vertical line has an undetermined slope. Distance between two points on a line: 1(π₯! − π₯# )# + (π¦! − π¦# )# " &" ' &' Midpoint: ( " # ! ; " # !) %! ALGEBRAIC MANIPULATION For πΉ = ()% *! , find G in terms of F, M, m and r2. “What is m in terms of p and q” is another way of saying “solve for m” or “use algebra to get m alone”. LAWS OF EQUALITY 1. 2. 3. 4. Whatever you do to change the value of one side of an equation, you must also do to the other side. You may add, subtract, or multiply anything you want on both sides of any equation at any time. You may divide both sides of any equation by any number except 0. If you want to take the square root of both sides of an equation, remember that every positive number has two square roots: one positive and one negative. ! # " $" If + = 4, what is the value of x? This document has been created for educational purposes only and meant to be used exclusively by SCORE – Test Prep and College Counseling. This document may not be distributed or reproduced without the express written consent of its authors. 2 PART I: ALGEBRA 1.3 Inequalities LAWS OF INEQUALITY 1. 2. 3. 4. Whatever you do to change the value of one side of an inequality, you must also do to the other side. You may add or subtract anything you want from both sides of an inequality, or multiply or divide by any positive number without changing the direction of the inequality. You may not perform undefined operations to an inequality (dividing by 0) or operations that have more than one possible result (taking a square root). If you multiply or divide both sides by a negative number, you must switch the direction of the inequality. GRAPHING INEQUALITIES • For inequalities in the form π₯ ≥ π, the expression can be drawn on the number line. Example: π₯ ≥ −3. -5 • -4 -3 -2 -1 0 1 2 3 4 ! ! If − " < −2π₯ + 1 < − # , what is one possible value of x? Graph the following inequalities: a. π¦ > −3 b. π¦ ≥ −π₯ + 1 c. π¦ ≤ 2π₯ + 3 d. π¦ < 12 5 For inequalities on the xy-plane, follow these steps: 1. Change the symbol to “=” and draw the inequality as an equation. 2. Choose a point that does not lie on the line of the inequality. 3. Substitute the coordinates of the chosen point in the inequality. 4. Check if the values comply with the inequality. 5. If the values comply, the inequality contains all points to that side of the line. 6. If the values do not comply, the inequality contains all points to the other side of the line. This document has been created for educational purposes only and meant to be used exclusively by SCORE – Test Prep and College Counseling. This document may not be distributed or reproduced without the express written consent of its authors. 3 1.4 PART I: ALGEBRA Linear Systems TYPES OF SOLUTIONS A system of equations is a set of two or more equations that must be true simultaneously. Systems can have different types of solutions: a. No solution: The equations do not intersect. In the case of linear systems, the lines are parallel. Equal slopes, different y-intercepts. b. Infinite solutions: The equations lie on the same line, so they are coincidental. Equal slopes, same y-intercept. c. One solution: The lines intersect at one point. Different slopes, y-intercept may be the same or different. d. Multiple solutions: The lines intersect at multiple points. This does not apply to linear systems. METHODS FOR SOLVING SYSTEMS OF EQUATIONS Solving a system of equations means finding all the values that make all of the equations true at the same time. There are several methods that can be used to solve these systems: a. Substitution: If one variable is isolated or can be easily isolated in one of the equations, the law of substitution can be used to solve the system. b. Combination: In a system of equations it is possible to add or subtract the corresponding sides of the equation together to get a new equation while trying to eliminate variables. c. Graphing: If a system of equations can be graphed, the solution to the system is the intersection of the graphs. Systems of inequalities are usually solved graphically, as it is easier to visualize the intersection of two or more inequalities on a graph. To graph a system of inequalities, plot each inequality individually and find the common area shaded by all inequalities. ππ₯ + ππ¦ = 12 2π₯ + 8π¦ = 60 In the system of equations above, a and b are constants. If the system has infinitely many solutions, what is the value of a/b? ππ₯ − 3π¦ = 4 4π₯ − 5π¦ = 7 In the system of equations above, k is a constant and x and y are variables. For what value of k will the system of equations have no solution? Solve the following system using substitution: 3π₯ + π¦ = 3π¦ + 4 π₯ + 4π¦ = 6 Solve the following system using combination: 3π₯ + π¦ = 3π¦ + 4 π₯ + 4π¦ = 6 Solve the following system using graphing: 3π₯ + π¦ = 3π¦ + 4 π₯ + 4π¦ = 6 This document has been created for educational purposes only and meant to be used exclusively by SCORE – Test Prep and College Counseling. This document may not be distributed or reproduced without the express written consent of its authors. 4 2.1 PART II: Problem Solving and Data Analysis Data Analysis AVERAGE OR ARITHMETIC MEAN The average or arithmetic mean of a set of numbers can be calculated with the following formula: ππ’π ππ ππ’πππππ π΄π£πππππ = # ππ ππ’πππππ The weighted average is the average of two or more groups that do not have the same number of elements. MEDIAN The median of a set of numbers is the value of the number located in the middle when the numbers are ordered increasingly or decreasingly. The average of four numbers is 15. If one of the numbers is 18, what is the average of the remaining three numbers? Ms. Aguilar’s class, which has 20 students, scored an average of 90 points on a test. Mr. Bowle’s class, which has 30 students, scored an average of 80 points on the same test. What was the combined average score, in points, for the two classes? The median of 1, 6, 8 and k is 5. What is the average of these four numbers? When the number or values is odd, the median can be calculated using the following formula: # ππ ππ’πππππ + 1 ππππππ π‘πππ = 2 When the number of values is even, the median will be the average of the two middle values. Divide the number of values by two to find the first middle term. The second middle term will be the next consecutive term. MODE The mode of a set of numbers is the number that appears the most frequently. If all numbers occur equally, then the set does not have a mode. If a set of numbers contains 1, 2, 3, 4, 5, 4, 3, 4, 5, 6, 7, 5, 4. What is the mode of the set? A set of numbers can have more than one mode and be multimodal. For two modes, the set is called bimodal; for three modes, the set is called trimodal. DATA SPREAD Data spread refers to how a set of data is distributed, focusing on how far the values are from the mean or median. Outliers are values that are much larger or smaller than the rest of the values in the set. They generally affect the mean more than the median. The range of a set of data is defined as the absolute difference between the least value and the greatest value in the set. If five positive integers have an average of 10, what is the greatest possible range of this set? Measures of spread: • Range: The difference between the highest and lowest values in a data set. • Standard deviation: The average distance of each element from the mean. The more spread out the data is on a graph, the larger the standard deviation. This document has been created for educational purposes only and meant to be used exclusively by SCORE – Test Prep and College Counseling. This document may not be distributed or reproduced without the express written consent of its authors. 5 2.1 PART II: Problem Solving and Data Analysis Data Analysis CORRELATION POSITIVE AND NEGATIVE CORRELATION • Positive correlation: If one variable increases, the other variable has a tendency to also increase (direct variation). • Negative correlation: If one variable increases, the other variable has a tendency to decrease (inverse variation). • No correlation: If one variable increases, the other variable does not tend to either increase or decrease. STRONG AND WEAK CORRELATION • Strong correlation: If one variable increases or decreases, there is a higher chance of the second variable increasing or decreasing. In a graph, the points tend to form a line at an angle. • Weak correlation: If one variable increases or decreases, there is a lower chance of a relationship between the variables. In a graph, the points tend to spread or form a very flat or vertical line. DATA INFERENCE AND SURVEY INTERPRETATION • • • • Population parameter: A numerical value that describes a characteristic of a population (for example, the percentage of registered voters who would vote for a certain candidate). We often do not know the value of this parameter, and statistics are used to estimate it based on a sample from the population. Selection of a sample: In order for a sample to be representative, subjects must be selected at random, and a significant part of the total population must be considered. Margins of error: The value of the margin of error is affected by the variability of the data (the larger the standard deviation, the larger the margin of error; the smaller the standard deviation, the smaller the margin of error) and the sample size (increasing the size of the random sample provides more information and typically reduces the margin of error). Also, the margin of error applies to the estimated value of the population parameter only, but it does not inform the estimated value for an individual. Generalization: This document has been created for educational purposes only and meant to be used exclusively by SCORE – Test Prep and College Counseling. This document may not be distributed or reproduced without the express written consent of its authors. 6 2.2 PART II: Problem Solving and Data Analysis Rates and ratios RATES Rate expresses a measure, quantity or frequency between two elements. The formula for rate is given by the units in a problem. For example, if the unit used is kilograms per second: π ππ‘π = ππ’ππππ ππ πππππππππ ππ’ππππ ππ π ππππππ A water pump for a dredging project can remove 180 gallons of water per minute, but it can only work for 2.5 consecutive hours, at which time it requires 20 minutes of maintenance before it can be brought back online. While it is offline, a smaller pump is used in its place, which can pump 80 gallons per minute. Using this system, what is the least amount of time it would take to pump 35,800 gallons of water? Rates provide conversion factors that can be used to solve the problem. For example, if the exercise states that a rocket burns fuel at a rate of 15 kilograms per second, two conversion factors can be identified: !" $%&'()*+, ! ,-.'/0 ! ,-.'/0 or !" $%&'()*+, RATIOS A ratio is a mathematical relationship between two quantities expressed as the quotient of those quantities. Ratios can be: • Part-to-whole: They compare a part to the whole. For example, the number of female students in a class compared to the total number of students in the class. These can also be expressed as percentages of the whole. • Part-to-part: They compare one part to another part. For example, the number of female students in a class compared to the number of male students in the class. Probabilities can also be expressed as ratios, where a subset of equally likely events is compared to a larger set of equally likely events. UNIT CONVERSIONS A conversion factor is a fraction in which the numerator and the denominator are equal. For example, 1 mile equals 1.609 kilometers: πΆπππ£πππ πππ ππππ‘ππ (ππ π‘π ππ): 1 ππππ 1.609 πππππππ‘πππ πΆπππ£πππ πππ ππππ‘ππ (ππ π‘π ππ): 1.609 πππππππ‘πππ 1 ππππ When using conversion factors, units must cancel properly to yield the unit needed in each case. Bronze is an alloy consisting of copper and tin. If 50 kg of a bronze alloy of 20% tin and 80% copper is mixed with 70kg of a bronze alloy of 5% tin and 95% copper, what fraction, by weight, of the combined alloy is tin? At the Andromeda Book Store, the ratio of self-help titles to fiction titles is 3:10, and the ratio of biography titles to fiction titles is 2:7. What is the ratio of biography titles to self-help titles? Niko is 27 inches shorter than his father, who is 5 feet 10 inches tall. How tall is Niko? Express your answer in feet (1 foot = 12 inches). If a factory can manufacture b computer screens in n days at a cost c dollars per screen, then which of the following represents the total cost, in dollars, of the computer screens that can be manufactured, at that rate, in m days? 1.+ a. b. c. d. / 1+/ . +. 1/ 1. +/ This document has been created for educational purposes only and meant to be used exclusively by SCORE – Test Prep and College Counseling. This document may not be distributed or reproduced without the express written consent of its authors. 7 2.3 PART II: Problem Solving and Data Analysis Percentages and Proportions PERCENTAGES • • • • TRANSLATING PERCENT PROBLEMS: Translation key for word problems: o “is” means equal ( = ) o “of” means multiplication ( * ) o “what” means an unknown ( x ) o “per” means division ( / ) o “percent” means division by 100 ( /100) COMMUTATIVE LAW OF PERCENTAGES: x% of y = y% of x. For example, it is easier to calculate 25% of 80 than 80% of 25. CHANGING BY PERCENTAGES: o To increase a number by a%, multiply by (100 + a)% or by (1 + a%). o To decrease a number by a%, multiply by (100 – a)% or by (1 – a%). PERCENT CHANGE: To find a percent change, use the following formula: πππππ ππππππ − ππππππππ ππππππ %ππππππ = ∗ πππ ππππππππ ππππππ What percent of 150 is 93? On the day it was issued, one share of a stock in Consolidated Energy was priced at $50. If the share price increased by 120% in its first five years and by 150% in its next five years, what was the share price, in dollars, after 10 years? If a population of bacteria increases from 80 cells to 220 cells, what is the percent increase in this population? How many liters of a 40% saline solution must be added to 4 liters of a 10% saline solution to make a 20% saline solution? PROPORTIONS AND SCALING • π PROPORTIONS: o A proportion is a statement that two ratios are equal. π π = π π o Law of cross-multiplication: In any proportion, the crossproducts must be equal. π π If π = π , then ππ = ππ o Law of cross-swapping: Cross-swapping can be done in any proportion. π π π π π If a, b and c are real numbers such that π%π = following must also be true? a. π' + π' − π ' = 0 b. π' − π' − π ' = 0 c. π' − π' + π ' = 0 d. π' + π' + π ' = 0 π&π π , which of the π If π = π , then π = π and π = π • • SCALING: Scaling represents the relationship between a measurement on a model and the corresponding measurement on the actual object. RULE OF THREE: The Rule of Three is a Mathematical Rule that allows you solve problems using proportions. Knowing three values and the relationship between them allows to calculate the value of a fourth unknown value. For direct proportions, direct Rule of Three is used. For inversed proportions, Inverse Rule of Three is used. DIRECT RULE OF THREE On a scale blueprint, the drawing of a rectangular patio has dimensions 5 cm by 7.5 cm. If the longer side of the actual patio measures 21 feet, what is the area, in square feet, of the actual patio? INVERSE RULE OF THREE This document has been created for educational purposes only and meant to be used exclusively by SCORE – Test Prep and College Counseling. This document may not be distributed or reproduced without the express written consent of its authors. 8 2.4 PART II: Problem Solving and Data Analysis Probabilities CONDITIONAL PROBABILITIES A conditional probability is the probability that something is true given that something else is also true. Solving conditional probability problems often require finding population fractions, where the numerator of the fraction is the required population and the denominator of the fraction is the total universe being considered. Test 1 Test 2 Test 3 Test 4 Total A 10 12 7 10 39 B 11 8 12 9 40 C 2 3 6 3 14 C 1 0 0 0 1 Inc. 1 2 0 3 6 Total 25 25 25 25 100 The letter grades on four tests for Ms. Hartman’s 25 students (including incomplete grades marked “Inc.”) are tabulated above. Five students in the class received an A on both test 3 and test 4. If one of the students who received an A on either test 3 or 4 is chosen at random, what is the probability that he or she received an A on test 4? If the incomplete grades are excluded from the statistics for each test in the table above, for which of the tests was the median grade higher than b? a. None of the tests b. Test 2 only c. Tests 1 and 2 only d. It cannot be determined from the given information. ARRANGEMENTS • • ARRANGEMENTS: Arrangements are used to determine how many arrangements of something are possible. For items of different categories, the number of options in each category are multiplied. FACTORIALS: The factorial of n is the number of ways in which the n elements of a group can be ordered. It is expressed as n!, where n! = 1 * 2 * … * (n – 2) * (n – 1) * n. A cafeteria has a lunch special consisting of soup or salad; a sandwich; coffee, tea, or a non-alcoholic beverage; and a dessert. If the menu lists 2 soups, 3 salads, 6 sandwiches, and 10 desserts, how many different lunches can one choose? In how many ways can 4 letters be combined? This document has been created for educational purposes only and meant to be used exclusively by SCORE – Test Prep and College Counseling. This document may not be distributed or reproduced without the express written consent of its authors. 9 2.5 PART II: Problem Solving and Data Analysis Tables and graphs TABLES Tables are useful when analyzing information that falls into nonoverlapping categories. ! ! "! Plot the graph for π(π₯) = #!"!. Tables can be useful for analyzing functions because they help to plot graphs and identify important patterns. For any equation expressing y in terms of x, a table of ordered pairs can be created. VENN DIAGRAMS Venn diagrams are useful when analyzing information that falls into overlapping categories. In a poll of 250 college students, 137 said that they attended at least one athletic event in the past year, and 115 said that they attended at least one career services event in the past year. If 82 of these students attended both an athletic event and a career services event in the past year, how many students attended neither an athletic event nor a career services event in the past year? This document has been created for educational purposes only and meant to be used exclusively by SCORE – Test Prep and College Counseling. This document may not be distributed or reproduced without the express written consent of its authors. 10 2.5 PART II: Problem Solving and Data Analysis Tables and graphs SCATTERPLOTS Scatterplots are graphs of ordered pairs of data. They can show relationships between variables that don’t vary in a highly predictable way. A line of best fit is a line that englobes these points optimally, showing the basic relationship between the variables. To draw a line of best fit, you must roughly estimate that there are the same number of points above and below the line. The scatterplot to the left shows 40 readings for particulate matter (a pollutant) concentration, in micrograms per cubic meter, in a metropolitan area over 9 years. Based on the line of best fit shown as a solid line, which is closest to the average yearly decrease in particulate matter concentration? a. 0.32 mcg/m3 per year b. 0.64 mcg/m3 per year c. 3.2 mcg/m3 per year d. 6.4 mcg/m3 per year According to the line of best fit to the data to the left, which of the following is closest to the percent decrease in average particulate matter concentration for 2007 to 2012? a. 9% b. 18% c. 36% d. 60% PIE GRAPHS A pie chart is a circular statistical graph that is divided into slices to illustrate numerical proportion. When analyzing graphs, the following formula is essential: ππππ‘ πππππππ ππ π‘βπ π πππ‘ππ = π€βπππ 360 Maria is constructing a pie graph to represent the expenses for her project. Here, expenses fall into three categories: marketing, design, and development. She knows that the marketing expenses are $12,000 and the design expenses are $30,000, but the development expenses could range anywhere from $30,000 to $48,000. Based on this information, which of the following could be the measure of the central angle of the sector representing marketing expenses. a. 45° b. 54° c. 62° d. 65° This document has been created for educational purposes only and meant to be used exclusively by SCORE – Test Prep and College Counseling. This document may not be distributed or reproduced without the express written consent of its authors. 11 2.5 PART II: Problem Solving and Data Analysis Tables and graphs OTHER GRAPHS Bar graphs: They compare numeric values of any type and can be either horizontal or vertical. One axis represents the categories being compared and the other axis represents the value of each category. Histograms: They are a specific type of bar graph that illustrated the distribution of numeric data across categories. Line graphs: They show how related data changes over a period of time. Maps: They usually show a geographical area. BOXPLOTS A boxplot is a standardized way of displaying the distribution of data based on a five-number summary: • Minimum • First quartile (Q1): The middle number between the smallest number and the median. • Median: The middle value of the data set. Note that the median does not need to be in the middle of the box. • Third quartile (Q3): The middle number between the largest number and the median. • Maximum A sample of 10 boxes of raisins has these weights in grams: 25, 28, 29, 29, 30, 34, 35, 35, 37, 38 Make a box plot of the data. To draw a boxplot, follow these steps: • Order the data from smallest to largest. • Find the median. • Find the quartiles. • Complete the five-number summary by finding the min and max. This document has been created for educational purposes only and meant to be used exclusively by SCORE – Test Prep and College Counseling. This document may not be distributed or reproduced without the express written consent of its authors. 12 3.1 PART IiI: Advanced Math Functions BASIC CONCEPTS • Find the extraneous solution: √−π₯ = π₯ + 2 Function: A function is a recipe for turning any input number (usually x) into an output number (usually y or f(x)). Extraneous or spurious solutions: Solutions that emerge from the process of solving a problem but are not valid solutions to the problem. Undefined functions: Functions are said to be undefined at points outside of their domain. Pay special attention to expressions with polynomials in the denominator. • • For what value is REPRESENTATION OF FUNCTIONS Functions can be represented as: • Ordered pairs in a table: • • x 0 1 2 3 4 &! undefined? x 1 2 3 4 5 6 7 8 9 y 3 5 7 9 11 Equations in functional notation: π(π₯) = 2π₯ + 3 Graphs in the xy-plane: QII ! ! $% g(x) 2 4 6 8 10 12 14 16 18 h(x) -9 -6 -3 0 3 6 9 12 15 What would the equation of g(x) be? QI π¦ = 2π₯ + 3 What would the equation of h(x) be? QIII QIV COMPOSITION OF FUNCTIONS The notation π(π(π₯)) indicated the composition of two functions (g and f). Composition means that an initial value x is put into the function g, and the resulting value is put into the function f. If π(π₯) = π₯ + 2 and π(π(1)) = 6 , which of the following could be g(x)? a. π(π₯) = 3π₯ b. π(π₯) = π₯ + 3 c. π(π₯) = π₯ − 3 d. π(π₯) = 2π₯ + 1 If π(π₯ − 1) = 2π₯ + 3 for all values of x, what is the value of π(−3)? If π(2π₯) = π₯ + 2 for all values of x, which of the following is equivalent to f(x)? !"# a. # b. c. d. ! +2 # !$# # 2π₯ − 2 This document has been created for educational purposes only and meant to be used exclusively by SCORE – Test Prep and College Counseling. This document may not be distributed or reproduced without the express written consent of its authors. 13 3.1 PART IiI: Advanced Math Functions TRANSFORMATION OF FUNCTIONS For π(π₯) = π₯ # + 2π₯ + 1 • • • • Movement in the x-axis: o To the right: π(π₯ − 1) o To the left: π(π₯ + 1) Movement in the y-axis: o Upwards: π(π₯) + 1 o Downwards: π(π₯) − 1 Reflection over an axis: o Over the x-axis: -π(π₯) o Over the y-axis: π(−π₯) Stretching and shrinking: o Vertical stretch: ππ(π₯), when k > 1 o Vertical shrink: ππ(π₯), when k < 1 For π(π₯) = π₯ # − 1, what functions will yield… a. π(π₯) displaced one unit upwards and then reflected over the y-axis b. π(π₯) reflected over the x-axis and then displaced five units to the right c. π(π₯) stretched by a factor of 3 and then displaced 2 units to the left This document has been created for educational purposes only and meant to be used exclusively by SCORE – Test Prep and College Counseling. This document may not be distributed or reproduced without the express written consent of its authors. 14 3.2 PART IiI: Advanced Math Quadratic Expressions QUADRATIC FORMS A quadratic expression (or second-degree polynomial) is any expression of the form If ππ₯ ! + ππ₯ + π. Quadratic equations can be expressed in three forms: a. Standard form: π¦ = ππ₯ ! + ππ₯ + π, where: • a determines the orientation and width of the parabola: o If a positive, the parabola opens upwards. o If a is negative, the parabola opens downwards. o An increasing a narrows the parabola. o A decreasing a widens the parabola. • b indicates the slope of the parabola at the y-intercept. • c indicates the y-intercept (0;c) and indicates and upwards or downwards movement of the parabola. " • The vertical axis of symmetry is found at π₯ = − !#. " b. c. "! • The vertex is found at (− !# ; π − $#). Factored form: π¦ = π(π₯ − π)(π₯ − π) • a is a real number. • p and q are the roots or x-intercepts. Vertex form: π¦ = π(π₯ − β)! + π • a is a real number. • h is the x-coordinate of the vertex of the parabola. • k is the y-coordinate of the vertex of the parabola. SOLVING QUADRATIC EQUATIONS FACTORING 1. Factor out any common factors using the distributive law. For example, 3π₯ ! − 12π₯ + 12 = 3(π₯ ! − 4π₯ + 4). 2. Apply factoring formulas. For example, 3(π₯ ! − 4π₯ + 4) = 3(π₯ − 2)(π₯ − 2). 3. If no formulas can be applied, use the product-sum method for 0 = ππ₯ ! + ππ₯ + π to find two factors of the form (dx+e) and (mx+n) where the cross product of d*n + e*m = b, and e*n = c. 4. If no factors can be found with the product-sum method, use the quadratic formula to find the roots and apply the Factor Theorem (if a polynomial has a zero factor at x = b, then it must have a factor of (x – b). QUADRATIC FORMULA Any quadratic equation can be solved using the quadratic formula: −π ± √π! − 4ππ π₯= 2π The discriminant of a quadratic equation is π· = π! − 4ππ, and it may be: • D < 0, if the equation has no real roots (in this case it has two imaginary roots). • D = 0, if the equation has one real root. • D > 0, if the equation has two distinct real roots. Additionally, the sum of the roots of an equation is & product of the roots of an equation is #. %" # Write 3π₯ ! + 11π₯ + 10 in factored form. Write (π₯ + 3)(2π₯ − 5) in standard form. Write 2(π₯ − 3)! − 5 in standard and factored forms. Which of the following is a factor of π₯ ! + 8π₯ + 16? a. π₯ − 4 b. π₯ − 8 c. π₯ + 4 d. π₯ + 8 Factorize 2π₯ ! − 18 using common factors and applying factoring formulas. Solve for x: 5π₯ ! + 10π₯ + 3 using the quadratic formula: How many roots does 0.5π₯ ! + 4π₯ + 8 have? What is the sum of its roots? How many roots does 2π₯ ! + 8π₯ + 3 have? What is the product of its roots? , and the This document has been created for educational purposes only and meant to be used exclusively by SCORE – Test Prep and College Counseling. This document may not be distributed or reproduced without the express written consent of its authors. 15 3.2 PART IiI: Advanced Math Quadratic Expressions QUADRATIC GRAPHS The graph of a quadratic expression will always be a parabola. Depending on the form of the expression given, it is possible to use known values to graph the expression. Graph 2π₯ ! + 8π₯ + 6. This document has been created for educational purposes only and meant to be used exclusively by SCORE – Test Prep and College Counseling. This document may not be distributed or reproduced without the express written consent of its authors. 16 3.3 PART IiI: Advanced Math Higher order equations and operations HIGHER ORDER POLYNOMIALS Higher order polynomials are similar to quadratic expressions, but their highest power is three or more. What is the degree of (π₯ + 3)(π₯ − 5)(π₯ + 1)? What is the degree of π₯ ! 3π₯ + 3π₯ + 10? OPERATIONS WITH POLYNOMIALS ADDITION AND SUBTRACTION To add polynomials, identify the like terms (terms with the same exact variables raised to the same exact power) and combine them. (8π" π# + 6π# π − 4π# + 5) + (10π# π − 4π" π# + 6π# − 7) (8π" π# + 6π# π − 4π# + 5) − (10π# π − 4π" π# + 6π# − 7) MULTIPLICATION To multiply polynomials, use distribution and multiply the coefficients of terms and use the rules of exponents to find the exponents for each variable. (9π − ππ)(5π# π + 7ππ − π) This document has been created for educational purposes only and meant to be used exclusively by SCORE – Test Prep and College Counseling. This document may not be distributed or reproduced without the express written consent of its authors. 17 3.3 PART IiI: Advanced Math Higher order equations and operations DIVISION • To divide monomials, divide the coefficients and then use the rules of exponents to divide the variables. • To divide a polynomial by a monomial, distribute the monomial as denominator for each term of the polynomial and divide as monomials. • To divide two polynomials: o Write each of them in descending order, and complete missing terms with a coefficient of 0. o Divide the terms with the highest powers and multiply the answer by the divisor (the polynomial you are dividing by). o Subtract and bring down the next term. o Repeat the process until there are no more terms left. o If there is a remainder, it must be written as a fraction in the final answer. 11 (14π₯ " π¦)/(28π₯ % π¦ $ ) (4π₯ $ π¦ & − 2π₯ ' π¦ " + 6π₯ " π¦ # )/(2π₯ # π¦) Use long division to divide (π₯ " − 4π₯ # + 2π₯ − 3) ππ¦ (π₯ + 2). Ruffini’s rule may be used when the divisor has the form (x – a). For example, to divide (π₯ $ − 16) by (π₯ + 2): Use Ruffini’s rule to divide (π₯ " − 4π₯ # + 2π₯ − 3) ππ¦ (π₯ + 2). The Remainder Theorem calculates the remainder of a division when the divisor has the form (x – a) by making (x – a) = 0, finding a, and replacing the value in the polynomial. If y = 0, then there is no remainder. Use the Remainder Theorem to calculate the remainder of (π₯ " − 4π₯ # + 2π₯ − 3) /(π₯ + 2). For an expression “x” to be a factor of another expression “y”, the remainder of their division must be zero. This document has been created for educational purposes only and meant to be used exclusively by SCORE – Test Prep and College Counseling. This document may not be distributed or reproduced without the express written consent of its authors. 18 PART IiI: Advanced Math 3.4 Exponents and Radicals THE LAWS OF EXPONENTIALS 1. 2. 3. 4. 5. 6. 7. ! If n is a positive integer, π₯ means x multiplied by itself n times. If n is a negative integer, π₯ ! means (1/x) multiplied by itself n times. π₯ " = 1. π₯ # *π₯ ! = π₯ #$! π₯ # /π₯ ! = π₯ #%! (π₯ # )! = π₯ #! ! √π₯ # = π₯ #/! '" If π − 3π = 6, which of the following is equivalent to ()# ? a. 3* + ' b. c. d. -,. (3%( )( 3%* $ Which of the following is equal to 8%% ? a. −2+ b. − (% + c. +" + d. THE LAWS OF RADICALS ! 1. 2. √π₯ # = π₯ #/! ! ! ! ( 1π₯ )( 1π¦ ) = 1π₯π¦ 3. ! ! ./ 01 Which of the following is equivalent to a. b. c. d. ! / = 31 Radicals can be simplified by: • Factoring out perfect squares from the radicand. • Multiplying top and bottom by the radical (in fractions). • Multiplying top and bottom by the conjugate of the denominator (if the denominator includes a sum or a difference with radicals). *Remember that expressions must only have radicals in the numerator (never in the denominator). Therefore, you must rationalize your final expressions using these methods. SOLVING EXPONENTIAL AND RADICAL EQUATIONS ($ (√($3√+4 √( ? 6√10 7 14 19 If π₯ ( = 4 and π¦ ( = 9, and if (π₯ − 2)(π¦ + 3) ≠ 0, what is the value of x + y? a. -5 b. -1 c. 1 d. 5 If + /$( = √2 , solve for x. To solve exponential and radical equations, apply the laws of exponentials and radicals to isolate the desired expression. If + (& = 4√2 , solve for k. + %! If - . √/ ! = π¦ $ , and π ≠ 0 , find x in terms of y. This document has been created for educational purposes only and meant to be used exclusively by SCORE – Test Prep and College Counseling. This document may not be distributed or reproduced without the express written consent of its authors. 19 3.4 PART IiI: Advanced Math Exponents and Radicals EXPONENTIAL FUNCTIONS An exponential function with base b is defined by π(π₯) = ππ / + π, where π ≠ 0, π > 0, π ≠ 1, and x is any real number. The effect of q: • The line y = q is the horizontal asymptote (the number that the function approaches but never reaches). • For q > 0, f(x) moves vertically upwards by q units. • For q < 0, f(x) moves vertically downwards by q units. The effect of a and b: • If b > 1: • If 0 < b < 1: • If b ≤ 0, f(x) is not defined. EXPONENTIAL GROWTH AND DECAY • In exponential growth and decay functions, the growth factor in the exponential function (“b”) is replaced by (1 + r) or (1 – r): o Exponential growth: π¦ = π(1 + π)/ o Exponential decay: π¦ = π(1 − π)/ • In these functions: o a represents the initial value before measuring growth • If b ≤ 0, f(x) is not defined. or decay. o r represents the growth or decay rate, and it is often represented as a percentage and expressed as a decimal. o 1+r or 1-r represents the growth or decay factor. o x represents the number of time intervals that have passed. Graph the function π¦ = 2(3)/ + 1 and identify: • Value of q: • Effect of q: • Value of a: • Effect of a: • Value of b: • Effect of b: • Asymptote: • Domain: • Range: In 2016, the population of Town X was estimated to be 35,000 people, with an annual rate of increase of 2.4%. a. What is the growth factor for Town X? b. Write an equation to model future growth. c. Use the equation to estimate the population in 2020 to the nearest hundred people. This document has been created for educational purposes only and meant to be used exclusively by SCORE – Test Prep and College Counseling. This document may not be distributed or reproduced without the express written consent of its authors. 20 3.5 PART IiI: Advanced Math Rational Expressions OPERATIONS WITH RATIONAL EXPRESSIONS A rational expression is a fraction that has polynomials in both the numerator and the denominator. To add or subtract rational expressions, find a common denominator and add or subtract like terms in the numerator. To multiply rational expressions, multiply the numerators and the denominators. ! # For !"# − ! , which of the following is equivalent to the expression for all positive values of x? a. b. c. d. ! ! $!$# ! ! "! ! ! $!"# ! ! "! ! ! $# ! ! "! % π₯ −1 To divide rational expressions, invert the second fraction and operate as a multiplication. SIMPLIFYING RATIONAL EXPRESSIONS Rational expressions, such as complex fractions, can be simplified by cancelling common factors or by multiplying top and bottom by a convenient factor. If π₯ = 3π and π ≠ 2 , which of the following is equivalent to ! ! $&' a. b. c. d. SOLVING RATIONAL EQUATIONS When solving equations that include fractions or rational expressions, it may be useful to simplify the equation by multiplying both sides by the common denominator. (!$')! *"% ? *$% &*"% &*$% &*"% &* +*! $&' +"&' # # If x > 0 and !$# − !"# = 2, what is the value of x? The function f is defined by the equation π(π₯) = π₯ % − 3π₯ − 18. If the function h is defined by the equation β(π₯) = ,(!) , for what value of x does h(x) = 6? %!$#% This document has been created for educational purposes only and meant to be used exclusively by SCORE – Test Prep and College Counseling. This document may not be distributed or reproduced without the express written consent of its authors. 21 3.6 PART IiI: Advanced Math Non-linear Expressions PARENT FUNCTIONS In order to understand how to manipulate non-linear expressions, it is essential to know the types of functions and equations that exist. A parent function is defined as the simplest function that satisfies all requirements for a specific type of function. This document has been created for educational purposes only and meant to be used exclusively by SCORE – Test Prep and College Counseling. This document may not be distributed or reproduced without the express written consent of its authors. 22 3.6 PART IiI: Advanced Math Non-linear Expressions MANIPULATION OF NON-LINEAR EXPRESSIONS To manipulate non-linear expressions, respect the order of operations and use the properties you have learned so far. ! Leaf’s Law states that πΏ = √2π₯ "#!, where p represents price per SAT book, q represents quantity of books purchased in bulk, and x represents the number of years after 2016. What is p in terms of L, q, and x? For πΌ = SOLVING NON-LINEAR SYSTEMS When solving non-linear systems, you can use the same methods used for linear systems: substitution, combination, and graphing. However, if the system has equations of different degrees or forms, it is better to use substitution instead of combination. $% √'() Find x and y if: , find R in terms of P, T and I. π¦ = π₯* − π₯ π¦ =π₯−1 11 This document has been created for educational purposes only and meant to be used exclusively by SCORE – Test Prep and College Counseling. This document may not be distributed or reproduced without the express written consent of its authors. 23 4.1 PART Iv: Geometry and Trigonometry Angles and PArallel Lines THE INTERSECTING LINES THEOREM When two lines cross, the vertical angles that are formed are congruent (they are the same), while the adjacent angles that are formed are supplementary (their sum is 180 degrees). If two lines cross forming angle A with a measure of 35°, what is the measure of angles B, C, and D? C A = 35° B D THE PARALLEL LINES THEOREM When two parallel lines are crosses by a third line, they form either eight right 90° angles, or four acute angles and four obtuse angles, where: a. All acute angles are congruent, b. All obtuse angles are congruent, c. And any acute angle is supplementary to any obtuse angle. In the figure above, ABCD is a parallelogram, and point B lies #### . If x = 40, what is the value of y? on π΄πΈ THE “ZCUF” ANGLES When two parallel lines are crosses by a third line, and the diagram gets too complicated, the “ZCUF” angles can be used. If lines l and m are parallel in the figure above, what is the value of x? This document has been created for educational purposes only and meant to be used exclusively by SCORE – Test Prep and College Counseling. This document may not be distributed or reproduced without the express written consent of its authors. 24 4.2 PART Iv: Geometry and Trigonometry Triangles INTERIOR AND EXTERIOR ANGLES • In any triangle, the sum of the interior angles is 180°. Triangle ABC has three angles (A, B and C). If A measures 60° and B measures 90°, what is the measure of C? A B a + b + c = 180° • In any triangle, if any side is extended, it makes an exterior angle with the adjacent side. The measure of any exterior angle equals the sum of the two remote interior angles. C If side BC was extended, what would the measure of exterior angle C be? SIDES AND ANGLES • Side-Angle Theorem: The biggest interior angle in a triangle will always be opposite the biggest side, and the smallest interior angle will always be opposite the smallest side. In the figure above, point D is on side AC of triangle ABC. If AD = DB = DC, what is the value of x + y? • Types of triangles based on the length of the sides: o Equilateral: All three sides have the same measure, and consequently all three internal angles have the same measure. o Isosceles: Two sides have the same measure and two angles have the same measure. According to the Isosceles Triangle Theorem, the two congruent angles will always be opposite the two congruent sides. o Scalene: All sides and all angles have different measures. THE TRIANGLE INEQUALITY The sum of any two sides of a triangle must always be greater than the third side. Therefore, the length of any side must be between the sum and the difference of the other two sides. This document has been created for educational purposes only and meant to be used exclusively by SCORE – Test Prep and College Counseling. This document may not be distributed or reproduced without the express written consent of its authors. 25 4.2 PART Iv: Geometry and Trigonometry Triangles CONGRUENCE AND SIMILARITY • • • • • • • • Congruence means that two figures have the same shape and size. Similarity means that two figures have the same shape but not the same size. If two figures are similar, all corresponding angles are congruent, and all corresponding sides are proportional. Dilation is a technique used to create similar triangles, where each point is stretched outwards from the center point by multiplying distances by the scale factor. The Angle-Angle (AA) Theorem: If two triangles have two congruent pairs of corresponding angles, then the triangles are similar, and all corresponding sides are proportional. Perimeters: If two similar polygons have corresponding sides in a ratio of a:b, then their perimeters have a ratio of a:b. Areas: If two similar polygons have corresponding sides in a ratio of a:b, then their areas have a ratio of a2:b2. Volumes: If two similar solids have corresponding sides in a ratio of a:b, then their volumes have a ratio of a3:b3. #### and #### In the figure above, π΄π΅ πΆπ· are line segments that intersect at point P. What is the value of m? If two triangles have corresponding sides in a ratio of 1:2, what is the ratio of their perimeters, areas and volumes? THE PITHAGOREAN THEOREM • • The Pythagorean Theorem: If a, b and c are the lengths of the sides of a right triangle, where c is the longest side, then π! + π! = π ! . Special Right Triangles: What is the perimeter of quadrilateral ABCD in the figure above? • The Distance Formula: The Pythagorean Theorem can be used to calculate distance: o In two dimensions: π = ,(ππ − ππ )π + (ππ − ππ )π o In three dimensions: π = ,(ππ − ππ )π + (ππ − ππ )π + (ππ − ππ )π AREA • The area of any triangle can be calculated by multiplying its base by its height and dividing the product by two. π∗π π¨= π • The area of an equilateral triangle can be calculated using the length of its side: √π π π¨= π π The figure above shows a rectangular box with a length of 24, a width of 10, and a height of x. If AB = √712 , what is the volume of the box in cubic units? If the sides of an equilateral triangle measure 2 cm, use both formulas to calculate the area. This document has been created for educational purposes only and meant to be used exclusively by SCORE – Test Prep and College Counseling. This document may not be distributed or reproduced without the express written consent of its authors. 26 4.3 PART Iv: Geometry and Trigonometry Circles EQUATIONS Since a circle is defined as the set of all points in a plane that are at a fixed distance r from the center (h, k), the Pythagorean Theorem can be used to find the equation of the circle: (π₯ − β)! + (π¦ − π)! = π ! Which of the following equations represents a circle in the xyplane that passes through the point (1, 5) and has a center of (3, 2)? a. (π₯ − 3)! + (π¦ − 2)! = √13 b. (π₯ − 3)! + (π¦ − 2)! = 13 c. (π₯ − 1)! + (π¦ − 5)! = 13 d. (π₯ − 3)! + (π¦ − 2)! = 25 CIRCUMFERENCE AND AREA • Circumference: Since π is defined as the ratio of the circumference of any circle to its diameter, πΆππππ’ππππππππ πΆππππ’ππππππππ π= = π·πππππ‘ππ 2π What is the area, in square centimeters, of a circle with a circumference of 16π centimeters? πΆππππ’ππππππππ = 2ππ • Area: If a circle is cut into tiny sectors and rearranged as shown, a parallelogram-like shape with a height of r and a length of half the circumference is created. Since the area of a parallelogram equals its base time its height, π΄ = (ππ)(π) = ππ ! This document has been created for educational purposes only and meant to be used exclusively by SCORE – Test Prep and College Counseling. This document may not be distributed or reproduced without the express written consent of its authors. 27 4.3 PART Iv: Geometry and Trigonometry Circles TANGENTS TO A CIRCLE A line is tangent to a curve if it touches the curve at only one point and is perpendicular to the radius at the point of tangency. In the figure below, AAAAA ππ is tangent to the circle at point P, ππ = √269, and ππ = √244. If the circle has an area of 100π, what is the area of triangle MOQ? RADIANS A radian is a unit for measuring angles, and it represents the ratio of an arc to a radius. "#$ For 360°, since πππππ’ππππππππ = 2π, πππππππ = #"%&'( = !)# # What is the degree measure of an angle that measures 4.5 radians? = 2π. CHORDS A chord is a line segment that connects two points on a circle. The perpendicular segment from the center of the circle to a chord will always bisect the chord. The circle above has an area of 100π square centimeters. If AAAA from the center of the circle? AB = 8, how far is π΄π΅ This document has been created for educational purposes only and meant to be used exclusively by SCORE – Test Prep and College Counseling. This document may not be distributed or reproduced without the express written consent of its authors. 28 4.3 PART Iv: Geometry and Trigonometry Circles ARCS An arc is a portion of a circumference that has a corresponding central angle. The ratio of an arc length to the circumference equals the ratio of its central angle to 360°. ππ΄π΅ π₯ = 2ππ 360 *Not to scale. In the figure below, AC is a diameter of the circle with center O, OB = 12, and the length of arc AB is 7π. What is the value of x? A minor arc is less than 180° and is equal to the central angle. A major arc is greater than 180°. Minor arcs are named with two letters (its endpoints), and major arcs are named with three letters (its endpoint and any other point in between). SECTORS A sector is a “pie slice” of a circle. The ratio of a sector area to the area of the circle is equal to the radio of its central angle to 360°. π΄πππ ππ π πππ‘ππ π₯ = ! ππ 360 INSCRIBED AND CIRCUMSCRIBED ANGLES AND POLYGONS • Inscribed vs. circumscribed: Inscribed • *Not to scale. In the figure below, AC is a diameter of the circle with center O and OB = 7. If the measure of ACB is 20°, what is the area of the shaded sector? The measure of the inscribed angle ACD is x, and the measure of the central angle CAB is y. Find CAB in terms of x. Circumscribed Inscribed Angle Theorems: This document has been created for educational purposes only and meant to be used exclusively by SCORE – Test Prep and College Counseling. This document may not be distributed or reproduced without the express written consent of its authors. 29 4.4 PART Iv: Geometry and Trigonometry PERIMETERS, Areas, and Volumes BASIC FORMULAS Right rectangular prism Right circular cylinder Sphere Right circular cone Rectangular pyramid What is the area, in square units, of the triangle? THE STRANGE AREA RULE If you need to find the area of a figure that does not have a fixed formula, think of the area as the sum of simpler shapes. What is the area, in square units, of the quadrilateral? This document has been created for educational purposes only and meant to be used exclusively by SCORE – Test Prep and College Counseling. This document may not be distributed or reproduced without the express written consent of its authors. 30 PART Iv: Geometry and Trigonometry 4.5 Trigonometry TRIGONOMETRIC FUNCTIONS (SOH CAH TOA) In the triangle below, x represents the measure in radians of the smallest angle. What is the tangent of 2x? QII Students Sin and cosec positive Take Tan and cot positive a. QI All b. All positive c. d. Calculus √( ( √' ' √' ( √3 Cos and sec positive QIII QIV When working with trigonometric functions, the lengths of the sides are not necessarily the same as expressed by the trigonometric function, as this could be a simplified ratio. Side lengths are always proportional to the values of the functions. If sin x = 4/5 and BC = 9, what is the length of AC? A x B UNIT CIRCLE C ! ! Which of the following is equivalent to πππ " − π ππ # ? a. b. c. d. $%√' ( √(%√' ( √(%$ ( √'%√( ( For any ordered pair (x;y), πππ Θ = x and π ππΘ = y. This document has been created for educational purposes only and meant to be used exclusively by SCORE – Test Prep and College Counseling. This document may not be distributed or reproduced without the express written consent of its authors. 31 4.5 PART Iv: Geometry and Trigonometry Trigonometry THE PYTHAGOREAN IDENTITY For all values of x, π ππ( π₯ + πππ ( π₯ = 1. ! If ( < π₯ < π and sin x = 0.8, what is the value of cos x? According to the Pythagorean Theorem, (πππππ ππ‘π)( + (ππππππππ‘)( = (βπ¦πππ‘πππ’π π)( If both sides are divided by (βπ¦πππ‘πππ’π π)( , πππππ ππ‘π ( ππππππππ‘ ( βπ¦πππ‘πππ’π π ( ( ) +( ) =( ) βπ¦πππ‘πππ’π π βπ¦πππ‘πππ’π π βπ¦πππ‘πππ’π π Which is equivalent to: π ππ( π₯ + πππ ( π₯ = 1. COMPLEMENTARY ANGLES The two acute angles in a right triangle are always complements of one another (their sum is 90° or π/2 radians). If one angle measures x, the other measures π/2 – x. If sin x = a and x + y = π/2, what is the value of sin y? a. π b. 1 − π( c. √1 − π d. √1 − π( As a result, the sine of an angle equals the cosine of its complement, and the cosine of an angle equals the sine of its complement. π ππ = π − π₯@ = cos π₯ 2 πππ = π − π₯@ = sin π₯ 2 This document has been created for educational purposes only and meant to be used exclusively by SCORE – Test Prep and College Counseling. This document may not be distributed or reproduced without the express written consent of its authors. 32 5.1 PART v: TIPS AND STRATEGY Graphing AND SCIENTIFIC Calculator RECOMMENDED CALCULATORS As in most standardized examinations, you will only be allowed to use one calculator. Therefore, you must be familiar with the calculator you choose to bring on test day. The best option is to use a graphing calculator. However, if you don’t have one, you can use a scientific calculator with equation-solving functions. If your calculator does not have this option, you will be at a disadvantage during the exam. If you decide to borrow a calculator from a friend, make sure to get it with anticipation so that you can learn how to use it. SCORE recommends the Casio fx-CG50 graphing calculator (USD 90 on Amazon) or the Casio fx-991LAX scientific calculator (S/. 100 in Tailoy). If you have another calculator, make sure to check the list of acceptable calculators: https://collegereadiness.collegeboard.org/sat/taking-thetest/calculator-policy. WHEN TO USE THE CALCULATOR The SAT has two math sections: • Section 3: Math – No Calculator • Section 4: Math – Calculator Even though a calculator is allowed for all questions in section 4, not all questions can be answered using it. Many questions can only be solved without a calculator. Make sure to read each question carefully before deciding to use your calculator. Read the following questions and decide whether you would solve it manually (M) or with your calculator (C). This document has been created for educational purposes only and meant to be used exclusively by SCORE – Test Prep and College Counseling. This document may not be distributed or reproduced without the express written consent of its authors. 33 PART v: TIPS AND STRATEGY 5.1 Graphing AND SCIENTIFIC Calculator MAIN MENU The graphing calculator has 20 different menu options. To perform well in the SAT, you must know how to use 4 functions: Run-Matrix: Arithmetic calculations. Statistics: Regression (to find equations from a table of values). Graph: Graphs of functions and equations. Equation: Systems of equations and polynomial equations. RUN-MATRIX Use this option to perform arithmetic calculations (addition, subtraction, multiplication, division). Battery status SHIFT à MENU (SET UP). Choose “Deg” for Angle. Activates options in yellow Activates options in red Fractions Exponents Fraction/Decimal This document has been created for educational purposes only and meant to be used exclusively by SCORE – Test Prep and College Counseling. This document may not be distributed or reproduced without the express written consent of its authors. 34 PART v: TIPS AND STRATEGY 5.1 Graphing AND SCIENTIFIC Calculator STATISTICS Use this function to find equations from a table of values. GRAPH Use this function to graph functions and equations, and to solve non-linear systems of equations or inequalities. Press EXE Press SHIFT Vz This document has been created for educational purposes only and meant to be used exclusively by SCORE – Test Prep and College Counseling. This document may not be distributed or reproduced without the express written consent of its authors. 35 5.1 PART v: TIPS AND STRATEGY Graphing AND SCIENTIFIC Calculator Press SHIFT Press EXE ZOOM Press EXE Use arrows to move around Press SHIFT TRACE Press SHIFT G-SOLVE TYPE *Choose type before inserting the function. This document has been created for educational purposes only and meant to be used exclusively by SCORE – Test Prep and College Counseling. This document may not be distributed or reproduced without the express written consent of its authors. 36 5.1 PART v: TIPS AND STRATEGY Graphing AND SCIENTIFIC Calculator EQUATION Use this function solve systems of equations, polynomial equations, and other equations. SIMULTANEOUS POLYNOMIAL SOLVER This document has been created for educational purposes only and meant to be used exclusively by SCORE – Test Prep and College Counseling. This document may not be distributed or reproduced without the express written consent of its authors. 37 PART v: TIPS AND STRATEGY 5.2 CHEAT SHEET PART I: HEART OF ALGEBRA FACTORING IDENTITIES: (π + π)! = (π + π)(π + π) = π ! + 2ππ + π ! • (π − π)! = (π − π)(π − π) = π ! − 2ππ + π ! • (π + π)(π − π) = π ! − π ! • LINEAR EQUATIONS: Linear form Slope-intercept Standard Point-slope Intercept ROUNDING: b. c. 0.00001 Y-intercept b c/b b X-intercept a (! )(" Slope formula: π ππππ = • Slope interpretation: o Positive slope: The line goes up as you move to the right. o Negative slope: The line goes down as you move to the right. o Zero slope: The line is horizontal. o Undefined slope: The line is vertical. o Parallel slopes have equal values: π+ = π! + o Perpendicular slopes are opposite and reciprocal: π+ = − • • Distance between two points on a line: /(π₯+ − π₯! )! + (π¦+ − π¦! )! * -* ( -( Midpoint: ( " ! ; " !) "&' = *! )*" ,! LINEAR SYSTEMS: a. "#$% Slope m -a/b m - • 0.000001 Millionths Hundred-Thousandths 0.001 0.0001 Thousandths Ten-Thousandths 0.1 0.01 Hundredths . Tenths 1 Decimal Point 10 Ones 100 DECIMAL PART Tens 1,000 Hundreds 10,000 Thousands 100,000 Ten Thousands 1,000,000 Hundred Thousands Millions WHOLE PART Equation y = mx + b ax + by = c y - y1 = m(x - x 1) x / a+y / b =1 No solution: The equations do not intersect. In the case of linear systems, the lines are parallel. Equal slopes, different yintercepts. Infinite solutions: The equations lie on the same line, so they are coincidental. Equal slopes, same y-intercept. One solution: The lines intersect at one point. Different slopes, y-intercept may be the same or different. ! ! INEQUALITIES: • If you multiply or divide both sides by a negative number, you must switch the direction of the inequality. • Systems of inequalities must be solved by graphing. PART II: Problem Solving and Data Analysis PERCENTAGES: DATA ANALYSIS: • π΄π£πππππ = .&, 01 '&,2%"$ # 01 '&,2%"$ • # 01 '&,2%"$-+ • ππππππ π‘πππ (πππ ππ’ππππ ππ π‘ππππ ) = • Standard deviation: The average distance of each element from the mean. The more spread out the data is on a graph, the larger the standard deviation. ! • CHANGING BY PERCENTAGES: o To increase a number by a%, multiply by (100 + a)% or by (1 + a%). o To decrease a number by a%, multiply by (100 – a)% or by (1 – a%). PERCENT CHANGE: To find a percent change, use the following formula: πππππ ππππππ − ππππππππ ππππππ %ππππππ = ∗ πππ ππππππππ ππππππ ARRANGEMENTS: • • ARRANGEMENTS: Arrangements are used to determine how many arrangements of something are possible. For items of different categories, the number of options in each category are multiplied. FACTORIALS: The factorial of n is the number of ways in which the n elements of a group can be ordered. It is expressed as n!, where n! = 1 * 2 * … * (n – 2) * (n – 1) * n. PART III: Passport to Advanced Math TRANSFORMATION OF FUNCTIONS: • • • • Movement in the x-axis: o To the right: π(π₯ − 1) o To the left: π(π₯ + 1) Movement in the y-axis: o Upwards: π(π₯) + 1 o Downwards: π(π₯) − 1 Reflection over an axis: o Over the x-axis: -π(π₯) o Over the y-axis: π(−π₯) Stretching and shrinking: o Vertical stretch: ππ(π₯), when k > 1 o Vertical shrink: ππ(π₯), when k < 1 RUFFINI’S RULE AND REMAINDER THEOREM: • Ruffini’s rule may be used when the divisor has the form (x – a) (for example, to divide (π₯ 5 − 16) by (π₯ + 2)). • The Remainder Theorem calculates the remainder of a division when the divisor has the form (x – a) by making (x – a) = 0, finding a, and replacing the value in the polynomial. If y = 0, then there is no remainder. QUADRATIC FUNCTIONS: • Standard form: π¦ = ππ₯ ! + ππ₯ + π, where: o a determines the orientation and width of the parabola: § If a positive, the parabola opens upwards. § If a is negative, the parabola opens downwards. § An increasing a narrows the parabola. § A decreasing a widens the parabola. o b indicates the slope of the parabola at the y-intercept. o c indicates the y-intercept (0;c) and indicates and upwards or downwards movement of the parabola. 2 o The vertical axis of symmetry is found at π₯ = − . o • • The vertex is found at (− 2 !4 ;π − 2! 54 !4 ). Factored form: π¦ = π(π₯ − π)(π₯ − π) o a is a real number. o p and q are the roots or x-intercepts. Vertex form: π¦ = π(π₯ − β)! + π o a is a real number. o h is the x-coordinate of the vertex of the parabola. o k is the y-coordinate of the vertex of the parabola. This document has been created for educational purposes only and meant to be used exclusively by SCORE – Test Prep and College Counseling. This document may not be distributed or reproduced without the express written consent of its authors. 38 5.2 PART v: TIPS AND STRATEGY CHEAT SHEET QUADRATIC FORMULA: • Any quadratic equation can be solved using the quadratic formula: −π ± √π ! − 4ππ π₯= 2π • The discriminant of a quadratic equation is π· = π ! − 4ππ, and it may be: o D < 0, if the equation has no real roots (in this case it has two imaginary roots). o D = 0, if the equation has one real root. o D > 0, if the equation has two distinct real roots. )2 Sum of the roots of an equation: • Product of the roots of an equation: • 4 6 4 LAWS OF EXPONENTS: • If n is a positive integer, π₯ ' means x multiplied by itself n times. • If n is a negative integer, π₯ ' means (1/x) multiplied by itself n times. • π₯ 7 = 1. • π₯ , *π₯ ' = π₯ ,-' • π₯ , /π₯ ' = π₯ ,)' • (π₯ , )' = π₯ ,' # • √π₯ , = π₯ ,/' LAWS OF RADICALS: # • √π₯ , = π₯ ,/' # # # • ( /π₯ )( /π¦ ) = /π₯π¦ • EXPONENTIAL FUNCTIONS: • An exponential function with base b is defined by π(π₯) = ππ * + π, where π ≠ 0, π > 0, π ≠ 1, and x is any real number. • The effect of q: The line y = q is the horizontal asymptote (the number that the function approaches but never reaches). • For q > 0, f(x) moves vertically upwards by q units. • For q < 0, f(x) moves vertically downwards by q units. • The effect of a and b: o If b > 1: # 9* # :( # * = a ( SIMPLIFICATION OF RADICALS: • Radicals can be simplified by: o Factoring out perfect squares from the radicand. o Multiplying top and bottom by the radical (in fractions). o Multiplying top and bottom by the conjugate of the denominator (if the denominator includes a sum or a difference with radicals). • Remember that expressions must only have radicals in the numerator (never in the denominator). Therefore, you must rationalize your final expressions using these methods. GRAPHING FUNCTIONS: o If 0 < b < 1: o If b ≤ 0, f(x) is not defined. EXPONENTIAL GROWTH AND DECAY: • In exponential growth and decay functions, the growth factor in the exponential function (“b”) is replaced by (1 + r) or (1 – r): o Exponential growth: π¦ = π(1 + π)* o Exponential decay: π¦ = π(1 − π) * • In these functions: o a represents the initial value before measuring growth or decay. o r represents the growth or decay rate, and it is often 1. If b ≤ represented 0, f(x) is not defined. as a percentage and expressed as a decimal. o 1+r or 1-r represents the growth or decay factor. o x represents the number of time intervals that have passed. PART IV: ADDITIONAL TOPICS INTERSECTING LINES THEOREM: PARALLEL LINES THEOREM: “ZCUF” ANGLES: C This document has been created for educational purposes only and meant to be used exclusively by SCORE – Test Prep and College Counseling. This document may not be distributed or reproduced without the express written consent of its authors. 39 5.2 PART v: TIPS AND STRATEGY CHEAT SHEET INTERIOR ANGLES: EXTERIOR ANGLES: CONGRUENCE AND SIMILARITY: CONGRUENCE: • Congruence means that two figures have the same shape and size. SIMILARITY: • Similarity means that two figures have the same shape but not the same size. • If two figures are similar, all corresponding angles are congruent, and all corresponding sides are proportional. The Angle-Angle (AA) Theorem: If two triangles have two congruent • pairs of corresponding angles, then the triangles are similar, and all corresponding sides are proportional. • Ratios: If two similar polygons have corresponding sides in a ratio of a:b, then: Perimeters: a:b o Areas: a2:b2 o Volumes: a3:b3 o AREA OF A TRIANGLE: • • The area of any triangle can be calculated by multiplying its base by its height and dividing the product by two. π∗π π¨= π The area of an equilateral triangle can be calculated using the length of its side: √π π π¨= π π CIRCLES: Equation: (π₯ − β)! + (π¦ − π)! = π ! , where: • o h is the x-coordinate of the center of the circle. o k is the y-coordinate of the center of the circle. o r is the radius of the circle. Circumference: πΆππππ’ππππππππ = 2ππ • Area: π΄πππ = ππ ! • Units: 360° = 2π πππππππ • TRIANGLE INEQUALITY: SIDE-ANGLE THEOREM: THE PITHAGOREAN THEOREM: The Pythagorean Theorem: If a, b and c are the lengths of the sides of a right • triangle, where c is the longest side, then π ! + π ! = π ! . Special Right Triangles: • • The Distance Formula: The Pythagorean Theorem can be used to calculate distance: In two dimensions: o π = /(ππ − ππ )π + (ππ − ππ )π In three dimensions: o π = /(ππ − ππ )π + (ππ − ππ )π + (ππ − ππ )π CHORD: TANGENT TO A CIRCLE: SECTOR: ARC: INSCRIBED AND CIRCUMSCRIBED ANGLE THEOREM: π΄πππ ππ π πππ‘ππ π₯ = ππ ! 360 ππ΄π΅ π₯ = 2ππ 360 INSCRIBED AND CIRCUMSCRIBED ANGLES AND POLYGONS: Inscribed Circumscribed This document has been created for educational purposes only and meant to be used exclusively by SCORE – Test Prep and College Counseling. This document may not be distributed or reproduced without the express written consent of its authors. 40 5.2 PART v: TIPS AND STRATEGY CHEAT SHEET Right rectangular prism Sphere Right circular cylinder Right circular cone Rectangular pyramid TRIGONOMETRIC FUNCTIONS (SOH CAH TOA): VALUE OF SINE AND COSINE: For any ordered pair (x;y), πππ Θ = x and π ππΘ = y. QII Students Sin and cosec positive Take Tan and cot positive All positive Calculus Cos and sec positive QIII COMPLEMENTARY ANGLES: QI All QIV THE PYTHAGOREAN IDENTITY: π ππ s π − π₯t = cos π₯ 2 πππ s π − π₯t = sin π₯ 2 This document has been created for educational purposes only and meant to be used exclusively by SCORE – Test Prep and College Counseling. This document may not be distributed or reproduced without the express written consent of its authors. π ππ! π₯ + πππ ! π₯ = 1. 41