STANDARDS OF LEARNING CONTENT REVIEW NOTES ALGEBRA II 3 rd Nine Weeks, 2016-2017 1
OVERVIEW Algebra II Content Review Notes are designed by the High School Mathematics Steering Committee as a resource for students and parents. Each nine weeks Standards of Learning (SOLs) have been identified and a detailed explanation of the specific SOL is provided. Specific notes have also been included in this document to assist students in understanding the concepts. Sample problems allow the students to see step-by-step models for solving various types of problems. A section has also been developed to provide students with the opportunity to solve similar problems and check their answers. Supplemental online information can be accessed by scanning QR codes throughout the document. These will take students to video tutorials and online resources. In addition, a self-assessment is available at the end of the document to allow students to check their readiness for the nine-weeks test. The document is a compilation of information found in the Virginia Department of Education (VDOE) Curriculum Framework, Enhanced Scope and Sequence, and Released Test items. In addition to VDOE information, Prentice Hall Textbook Series and resources have been used. Finally, information from various websites is included. The websites are listed with the information as it appears in the document. Supplemental online information can be accessed by scanning QR codes throughout the document. These will take students to video tutorials and online resources. In addition, a self-assessment is available at the end of the document to allow students to check their readiness for the nine-weeks test. To access the database of online resources scan this QR code. The Algebra II Blueprint Summary Table is listed below as a snapshot of the reporting categories, the number of questions per reporting category, and the corresponding SOLs. Algebra II Test Blueprint Summary Table Reporting Categories No. of Items SOL Expressions & Operations 13 AII.1a d AII.3 Equations & Inequalities 13 AII.4a d AII.5 Functions & Statistics 24 AII.2 AII.6 AII.7a h AII.8 AII.9 AII.10 AII.11 AII.12 Total Number of Operational Items 50 Field Test Items* 10 Total Number of Items 60 *These field test items will not be used to compute students scores on the tests. It is the Mathematics Instructors desire that students and parents will use this document as a tool toward the students success on the end-of-year assessment. 2
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Radical Functions AII/T.1 The student, given rational, radical, or polynomial expressions, will b) add, subtract, multiply, divide, and simplify radical expressions containing rational numbers and variables, and expressions containing rational exponents; c) write radical expressions as expressions containing rational exponents and vice versa Simplifying Radicals To simplify a radical, you will pull out any perfect square factors (i.e. 4, 9, 16, 25, etc.) The square root of 9 is equal to 3, so you can pull the square root of 9 from underneath the radical sign to find the simplified answer, which means 3 times the square root of 2. You can check this simplification in your calculator by verifying that. Another way to simplify radicals, if you don t know the factors of a number, is to create a factor tree and break the number down to its prime factors. When you have broken the number down to all of its prime factors you can pull out pairs of factors for square roots, which will multiply together to make perfect squares. Example 1: Simplify 2 64 8 8 4 2 2 4 2 2 2 2 Example 2: Simplify 16 2 x x x y 4 4 2 2 2 2 Scan this QR code to go to a video tutorial on simplifying radicals. 4
To simplify a root of a higher index, you pull out factors that occurs the same number of times as the index of the radical. As an example, if you are simplifying would only pull out factors that occurred 5 times, since 5 is the index of the root. you Example 3: Simplify Because this is a 4 th root, pull out things that occur 4 times. Radical Operations In order to multiply or divide radical expressions, they must share the same index. This means that the power of the root must be the same. could not be simplified because one is a 4 th root and one is a 5 th could be simplified because they are both cube roots. If and are real numbers, then If and are real numbers, and, then Example 1: Simplify Because these radicals have the same index, they can be multiplied and simplified. Because this is a cube root, pull out things that occur 3 times. Example 2: Simplify 5
Radical Functions 1. Simplify 2. Simplify 3. Simplify 4. Simplify Rational Exponents Properties of Exponents Property Example Example 1: Simplify 6
Example 2: Write in radical form. Example 3: Write in simplest form Write each radical in exponential form, then use the properties of exponents to simplify. Scan this QR code to go to a video tutorial on rational exponents. Radical Functions 5. Write in radical form. 6. Write in exponential form. 7. Simplify 7
Radical Equations and Graphs AII/T.4 The student will solve, algebraically and graphically, d) equations containing radical expressions. AII/T.6 The student will recognize the general shape of function (absolute value, square root, cube root, rational, polynomial, exponential, and logarithmic) families and will convert between graphic and symbolic forms of functions. A transformational approach to graphing will be employed. Graphing calculators will be used as a tool to investigate the shapes and behaviors of these functions. Radical Equations may require you to square both sides of the equation. This has the potential to introduce extraneous solutions. For this reason, it is very important to always check your solutions to radical equations. Example 1: Solve for y Cube both sides of the equation to cancel out the cube root. Don t forget to check! Scan this QR code to go to a video tutorial on radical equations. To solve equations with a fractional exponent, raise each side of the equation to the reciprocal power. This is because when you raise a power to a power you multiply the exponents, and multiplying by the reciprocal will make the exponent 1. 8
Example 2: Solve for x: Raise both sides to the reciprocal power. Don t forget to check! Sometimes it might be helpful to rewrite a radical using a rational exponent to solve. Example 3: What is the solution of Isolate the radical expression. Rewrite in exponential form. Raise both sides to the reciprocal power. Graphs of Radical Functions Don t forget to check! Only in the 1 st quadrant, because square roots are only defined for positive values of x. 9 In the 1 st and 3 rd quadrant because when x is positive y will be positive, and when x is negative y will also be negative.
Example 4: The maximum walking speed of a large animal can be modeled by the equation:, where S is the speed (ft/sec), g is a constant, and L is the length (ft) of the animal s leg. Use this model to predict how long an animal s leg would need to be in order to travel 15 ft/sec. This equation could be solved algebraically or graphically. We will show both examples here. Algebraically: Plug in the given values for S and g Square both sides of the equation to cancel out the square root. The animal s leg would need to be just over 7 feet long in order to travel 15 ft/sec. Graphically: Graph the equation in Y1 as. The y value that we want to consider is 15 ft/sec, so graph in Y2. You will need to adjust your zoom in order to see the intersection. You can do this by pressing ZOOM then 0 for ZoomFit. Your graph should look like this ---> Press 2 nd then TRACE to take you to the CALC menu. Scroll down to 5: intersect. The calculator will ask for first curve?, second curve? and guess?. Just scroll near the intersection and press ENTER until your intersection is displayed. 1. Radical Equations This again shows your answer as X = 7.03 feet. 2. 3. 4. 5. You can model the population,, of St. Louis, Missouri between 1900 and 1940 by the function, where is the year. Using this model, in what year was the population of St. Louis 175,000? 10
Rational Expressions AII/T.1 The student, given rational, radical, or polynomial expressions, will a) add, subtract, multiply, divide, and simplify rational algebraic expressions; A rational expression is the quotient of two polynomials (or a polynomial fraction). To simplify rational expressions you need to factor both the numerator and denominator, and then divide out common factors. Example 1: Simplify The GCF is. If you remove this from both the numerator and the denominator, you can divide it out. This is because where. If x were 0, then the original problem would be undefined (the denominator would equal 0). Example 2: Simplify Scan this QR code to go to a video tutorial on simplifying rational expressions. Factor the numerator! So, we are looking for factors of 12 that add up to! Find the greatest common factor in each row and each column. These will give you your two binomials! Put the now factored trinomial back into the original problem and divide out factors that are in both the numerator and denominator. Don t forget: anytime the term is negative and the term is positive, your answer will have two minus signs! To multiply rational expressions, you multiply the numerators and multiply the denominators. You can use factoring to simplify. Example 3: Multiply and Simplify Divide out common factors that occur in both the numerator and denominator. Your answer is the product of what remains after common factors are divided out. 11
To divide rational expressions, you can simply multiply the first expression by the reciprocal of the second. Example 4: Divide and Simplify Multiply by the reciprocal. Scan this QR code to go to a video tutorial on multiplying and dividing rational expressions. Factor all polynomials and divide out common factors. If you want to add or subtract rational expressions, you must first find a common denominator. Once both of your fractions have a common denominator, you can add or subtract the numerators and simplify. You can find the least common denominator (LCD) by finding the least common multiple (LCM) of each denominator. Do this by writing the prime factorization of each expression, and then write the product of the prime factors raised to the highest power in either expression. Example 5: Find the LCM of Start by writing the prime factorization of each expression Simplify Then, look at each factor individually. If the factor occurs in both expressions, choose the one that has the highest power. The LCM would be Scan this QR code to go to a video tutorial on adding and subtracting rational expressions. 12
Example 6: Subtract and simplify Start by factoring the denominators and finding the least common denominator (LCD) In this case, the LCD is first fraction by denominator.. This means that you will need to multiply the, and the second fraction by. This will give each fraction the same Multiply both fractions to create LCD. Rewrite as one fraction over LCD Simplify numerator and check for common factors between numerator and denominator. A complex fraction is a rational expression that has a fraction in either the numerator, denominator or both (basically this is a fraction in a fraction). There are two methods to simplify a complex fraction. Only one will be shown in these notes. The second method will be explained in a video tutorial. Example 7: Simplify The LCD of all of the rational expressions is. If we multiply both the numerator and denominator by this LCD, it will help us to simplify. We can now check both the numerator and denominator for common factors. In this case, we can factor an x out of both. 13
Rational Expressions Scan this QR code to go to a video tutorial on complex fractions. 1. 2. 3. 4. 5. 6. Graphing Rational Functions AII/T.6 The student will recognize the general shape of function (absolute value, square root, cube root, rational, polynomial, exponential, and logarithmic) families and will convert between graphic and symbolic forms of functions. A transformational approach to graphing will be employed. Graphing calculators will be used as a tool to investigate the shapes and behaviors of these functions. AII/T.7 The student will investigate and analyze functions algebraically and graphically. a) domain and range, including limited and discontinuous domains and ranges; b) zeros; e) asymptotes; f) end behavior; AII/T.10 The student will identify, create, and solve real-world problems involving inverse variation, joint variation, and a combination of direct and inverse variations. To graph a rational function, first check for horizontal and vertical asymptotes. Find x and y-intercepts, look for holes (where the graph is undefined), and then use a few points to determine the general shape of the graph. 14
Vertical Asymptote Horizontal Asymptote A rational function, has a vertical asymptote at each real zero of if and have no common zeros. If they do have a common zero [i.e. ], and the degree of the term in the denominator is greater, then also has a vertical asymptote at The horizontal asymptote of a rational function is found by comparing the degree of the numerator (m) and denominator (n). If m < n, the horizontal asymptote is. If m > n, there is no horizontal asymptote. If m = n, the horizontal asymptote is, where a is the coefficient of the highest degree term in the numerator and b is the coefficient of the highest degree term in the denominator. This graph has a vertical asymptote at x = 1 This graph has a horizontal asymptote at The numerator and denominator have the same degree (1), so the asymptote is found by dividing the coefficients ½. The x-intercepts are found by determining where the numerator is equal to zero. The y-intercepts are found by setting x = 0. Example 1: Graph Start by factoring the numerator and denominator: Vertical Asymptote: Because the numerator and the denominator do not share any common zeros, the function will have vertical asymptotes at any value that makes the denominator zero Horizontal Asymptote: The degree of the denominator (2) is higher than the degree of the numerator (1) therefore the horizontal asymptote is X-Intercepts: The x-intercepts are where the numerator is equal to zero. Y-Intercepts: Set x = 0. Therefore, Now you can sketch your asymptotes, and plot your intercepts. 15
If you chose to plot a few more points, such as your graph would start to take shape. Scan this QR code to go to a video tutorial on graphing rational functions. You can solve a rational equation algebraically or graphically. To solve a rational equation graphically, graph both sides of the equation in your calculator and check for intersection points. This procedure will be shown in the video tutorial. Example 2 will show you how to solve algebraically. Example 2: Solve. Be sure to check for extraneous solutions! Rewrite with factored denominators so that we can find the LCD. (x+2)(x-2) Multiply both sides of the equation by the LCD to clear the denominators. Solve the equation! Scan this QR code to go to a video tutorial on solving rational equations. Check for extraneous solutions! This is not an extraneous solution. Therefore, the solution to the equation is: 16 x = 1
Direct and Inverse Variation Direct Variation ( Inverse Variation Direct variations are always linear and pass through the origin. Inverse variations are rational functions and will never pass through the origin. If the ratio between two variables is a constant, then a direct variation exists. A direct variation can be written in the form, where is the constant of variation. If the product of two variables is a constant, then an inverse variation exists. An inverse variation can be written in the form. Example 3: Determine if each relation is a direct variation, inverse variation, or neither. x y 1 3 2 4 3 5 x y -1 2 0 0 1-2 x y 3 4 1 12-2 -6 First check the ratios: Does the ratio? NO! Therefore this is NOT a direct variation! Next check the products: Does? NO! Therefore this is NOT an inverse variation! First check the ratios: Does the ratio First check the ratios: Does the ratio? YES! Therefore this IS a direct variation! Notice that we did not use the ordered pair to check the ratios. It is impossible to divide by zero. Therefore, we used the other point.? NO! Therefore this is NOT a direct variation! Next check the products: Does? Does this also equal? YES! Therefore, this IS an inverse variation! 17
To write an equation of a direct variation, use the given point plugged into solve for. For an inverse variation, plug into., to Example 4: Suppose varies inversely with, and when. What inverse variation equation relates and? Start with. We are given a value for and, so plug those in and solve for. This is the constant of variation! This is the inverse variation equation! Once you have a direct or inverse variation equation, you can use that equation to determine other values. Example 5: The distance that you jog,, varies directly with the amount of time you jog,. If you can jog 4 miles in 0.9 hours, how long will it take you to jog 6.5 miles? Jogging varies directly with time Now we need to solve for in order to write a direct variation equation. This is the constant of variation! This is the direct variation equation! Now, we can use this equation to solve for the time it takes to jog 6.5 miles. We are given that you jog 6.5 miles. This will be plugged in for. Then, solve for. Therefore, it would take you almost 1.5 hours to jog 6.5 miles. Scan this QR code to go to a video tutorial on direct, inverse, and joint variations. 18
Sometimes, a quantity varies with respect to more than one quantity. This creates a combined variation. If the quantities vary directly or jointly, those quantities will be in the numerator (with k). If the quantities vary inversely, those quantities will be in the denominator. Example 6: The amount of cake,, that it takes to feed a birthday party varies directly with the number of children,, and inversely with the age of the children,. If a party of sixteen 8 year-old children eat 112 in² of birthday cake, how much cake would a party of ten 6 year-olds require? Start by setting up the formula. varies directly with and inversely with. Substitute for. = 112 in² of cake, = 16 children and = 8 years old. Write your new combined variation equation with substituted in. Substitute in your new values for n and a to solve for c. Graphing Rational Functions 1. State all asymptotes, x and y-intercepts, and restrictions of 2. Sketch the graph of 3. Solve for x. 4. y varies inversely with x. If y is 240 when x is 3, what is the constant of variation? 5. The amount of hats knitted by a group is directly proportional to the number of people in the group. If 5 people can knit 12 hats, how many hats can 12 people knit? 6. varies jointly with and and inversely with both and. Write a formula relating these variables. 19
Exponential and Logarithmic Functions AII/T.6 AII/T.7 The student will recognize the general shape of function (absolute value, square root, cube root, rational, polynomial, exponential, and logarithmic) families and will convert between graphic and symbolic forms of functions. The student will investigate and analyze functions algebraically and graphically. a) domain and range, including limited and discontinuous domains and ranges; e) asymptotes; f) end behavior; Rules of Exponents Multiplying Dividing Raising a power to a power Negative Exponents When multiplying two exponential terms with the same base, you add the exponents. When dividing two exponential terms with the same base, you subtract the exponents. When raising an exponential term to a power, you multiply the exponents. To simplify negative exponents, write the term as its reciprocal with a positive exponent. Example 1: Simplify Scan this QR code to go to a video tutorial on rules of exponents. Example 2: Remember that anything to the zero power equals 1! Example 3: 20
Exponential Growth The graph crosses the y-axis at. The asymptote is Exponential Decay The graph crosses the y-axis at. The asymptote is Logarithmic Functions A logarithmic function is the inverse of an exponential function. You can model exponential growth and decay with the function where is the final amount, is the initial amount, is the rate of growth or decay, and is the number of time periods. Example 4: If you invested $5,000 in a savings account today that pays 3.5% interest annually, how much will you have in 10 years? Plug in your given values and solve for 21
Continuously compounded interest can be calculated using the formula where A(t) is the final amount, P is the initial amount, r is the annual interest rate, and t is the number of years. Example 5: If you invested $2,000 in a savings account today that pays 2.75% interest compounded continuously, in how many years would you have $5,000? To find the number of years, let s graph the function and look at the table of values in our calculator. We will graph Once you ve graphed the function, press 2 nd and GRAPH to show the table of values. We want to know where the final value (Y) equals 5000, so scroll through the table until you find it. It would take 34 years for your investment of $2000 to grow to $5000. 34 years Exponential and Logarithmic Functions 1. 2. Scan this QR code to go to a video tutorial on exponential growth and decay. 3. A population of 800,000 decreases at a rate of 2.2% annually. After how many years will the population go below 600,000? (remember that your growth is negative now!) 4. If $1000 is invested in a savings account with 5.1% interest compounded continuously, how much money will you have after 3 years? 22
Properties of Logarithms Product Property Quotient Property Power Property Change of Base Formula Example 6: Write as a single logarithm. Using the properties of logarithms, this can be condensed to Example 7: Write in expanded form. Scan this QR code to go to a video tutorial on properties of logarithms. Example 8: Evaluate. (Hint: Use Change of Base) Use a calculator to approximate the answer. Exponential and Logarithmic Functions 5. Write as a single logarithm: 6. Write as a single logarithm: 7. Expand: 8. Expand: 9. Evaluate: 10. Evaluate: 23
Exponential and Logarithmic Equations AII/T.4 The student will solve, algebraically and graphically, c) equations containing rational algebraic expressions; You can use the properties of exponents and logarithms to help you solve exponential and logarithmic equations. One way to solve an exponential equation is to rewrite terms on both sides of the equation with a common base. Example 1: Solve Rewrite each term with a common base Set exponents equal to each other (Power Property of Exponents) Solve If you cannot find like bases, you can solve by taking the logarithm of both sides. This is only true as long as both bases are positive. Example 2: Solve Take the logarithm of both sides Power Property of Logarithms Use your calculator to estimate the answer You can also solve an exponential or logarithmic equation by graphing both sides of the equation in your calculator and finding where the two graphs intersect. Exponential and Logarithmic Equations 1. 2. 3. 4. (Remember that logarithms are only defined for positive numbers) Scan this QR code to go to a video tutorial on logarithmic and exponential equations. 24
Answers to the Radical Functions 1. 2. 3. 4. 5. 6. 7. Radical Equations 1. 2. 3. 4. 5. Rational Expressions 1. 2. 3. 4. 5. 6. problems: Graphing Rational Functions 3. 4. 5. 6. Exponential and Logarithmic Functions 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. Exponential and Logarithmic Equations 1. 2. 3. 4. Graphing Rational Functions 1. vertical asymptote: horizontal asymptote: x-intercept: y-intercept: 2. 25