Revision notes for Pure 1(9709/12)
|
|
- Franklin Pearson
- 5 years ago
- Views:
Transcription
1 Revision notes for Pure 1(9709/12) By WaqasSuleman A-Level Teacher Beaconhouse School System
2 Contents 1. Sequence and Series 2. Functions & Quadratics 3. Binomial theorem 4. Coordinate Geometry 5. Trigonometry 6. Circular Measure 7. Vectors 8. Differentiation 9. Integration
3 Arithmetic Progressions If you have the sequence 2, 8, 14, 20, 26, then each term is 6 more than the previous term. This is an example of an arithmetic progression (AP) and the constant value that defines the difference between any two consecutive terms is called the common difference. If an arithmetic difference has a first term a and a common difference of d, then we can write a, (a + d), (a + 2d),... {a + (n-1) d} where the n th term = a + (n 1)d Sum of Arithmetic series The sum of an arithmetic series of n terms is found by making n/2 pairs each with the value of the sum of the first and last term. (Try this with the sum of the first 10 integers, by making 5 pairs of 11.) This gives us the formula: where a = first term and l = last term. As the last term is the n th term = a + (n 1)d we can rewrite this as: (Use the first formula if you know the first and last terms; use the second if you know the first term and the common difference.) Geometric Progressions If you have a sequence such as: 81, 27, 9, 3, 1, 1/3, 1/9,... then each term is one third of the term before. This can be written as 81, 81(1/3), 81(1/3) 2, 81(1/3) 3, 81(1/3) 4,... It is an example of a Geometric Progression (GP) where the each term is a multiple of the previous one. The multiplying factor is called the common ratio. So a GP with a first term a and a common ratio r with n terms, can be stated as a, ar, ar 2, ar 3, ar 4...ar n-1, where the n th term = ar n-1 Example:
4 In the sequence, 400, 200, 100, 50,... find the 8 th term. a = 400, r = 0.5 and so the 8 th term = = Note: To find which term has a certain value you will need to use logarithms. Example: In the sequence, 2, 6, 18, which is the first term to exceed 1,000,000? a = 2, r = n-1 > 1,000,000 3 n-1 > (n 1) log 3 > log n > Therefore: n = 13 Example: In the earlier sequence, 400, 200, 100, which is the first term that is less than 1? (n-1] < (n-1) < (n-1) log 0.5 < log Therefore: n > 9, or n = 10 Note: The inequality sign changed because we divided by a negative (log 0.5 < 0) Sum of Geometric series The sum of the terms can be written in two ways.
5 where a = first term, r = common ratio and r 1. (use this formula when r < 1). Example: Evaluate, (Note: there are 9 terms.) The first term is when n = 2 (i.e = ) Using the formula for the sum of a geometric progression gives: which is approximately 9300 (to 3 s.f.). Convergence The sum of an infinite series exists if: -1 < r < 1 or r < 1 This is because each successive term is getting smaller and so the series will tend towards a certain limit. This limit is found using the second of our two formulae: If r < 1 then as n, r n 0 and so:
6 Example: the series 1/3 + (1/3) 2 + (1/3) 3 + (1/3) converges and its sum is 1 as n approaches. (A sequence such as n 3 has the first 6 terms as As n approaches infinity, the sum also increases. Therefore, it is not convergent. This series is divergent. Every AP has a sum that approaches infinity as n increases, so every AP is divergent.) Example Find 1-1/2 + 1/4-1/ /2 + 1/4-1/ = 1 + (-1/2) + (-1/2) 2 + (-1/2) This is a geometric progression where r = -½, so r < 1. Therefore this series converges to: Two final pieces of information that may be useful: Example: The 7 th term of a GP is 6, the 9 th is 1.5. The 8 th term is: (6 1.5) = 9 = 3 Here r = 0.5 and a = 384.
7 Homework Questions 1 Terms of a Sequences 1. Find the next 3 terms of the following sequences and state the rule to find the next term in each case a) 5, 9, 13, 17 b) 1, 3, 5, 7 c) 9, 13, 17, 21 d) -2, 6, 14, 22 e) 15, 22, 31, 42, 55 f) 4, 13, 26, 43, 64
8 g) 3, 7, 13, 21, 31 h) 5, 12, 21, 32, 45 i) 1, 2, 2, 4, 8 j) 1, 3, 6, 10, 15
9 Homework Questions 2 Using the Nth Term of A Sequences 1. Find the value of U 1, U 2, U 3 and U 20 a) U n = 3n b) U n = 7n 2 c) U n = 2n 2 d) U n = n A sequence is generate according to the formula U n =an-b. Given that U 3 =7 and U 5 =13.find the value of a and b 3. Find the value of n for which U n =(3n-2) 2 has the given value of U n = A sequence is generated from the formula U n =pn 2 -q where p and q are constants. Given that U 1 =- 1 and U 3 =7, find the value of the constants p and q.
10 5. Find the value of n for which U n has the given value a) U n =4n-1 and U n =23 b) U n = 2n3 1 3 and U n = 5 c) U n = 5n + 6 and U n = 31
11 Homework Questions 3 Recursive Formula 1. Find the next 3 terms of the following sequences given both the first term and the recursive formula. a) U 1 = 5 U n + 1 = 3U n b) U 1 = 3 U n + 1 = 2U n c) U 1 = 2 U n + 1 = 3U n 4 d) U 1 = 16 U n + 1 = U n 4 2. By writing down the first 4 terms or otherwise, find the recursive formula that defines the following sequence. a) U n =2n-1
12 b) U n =3n-2 3. Find the next 4 terms of these recursively defined sequences a) U n+1 =U n -U n-1 when U 1 =6 and U 2 =2 b) U n+1 =3U n +2U n-1 when U 1 =1 and U 2 =-3 c) U n+1 =5U n -11 when U 1 =3 4. Write down the first 3 terms of the sequence defined by U n+1 =12-U n when U 1 =10
13 Homework Questions 4 General Term of an Arithmetic Sequence 1. Which of the following sequences are arithmetic? a) 7, 17, 27, 37 b) 12, 5, 0, -9, -17 c) 24, 15, 6, -3, a) Find the 10 th term and b) Find the formula for the nth term a) 4, 7, 10, 13 b) -3, -1, 1, 3 c) 1, -4, -9, -13
14 3. Find the 20 th term, if the sequence begins a) 2, 6, 10, 14, 18 b) 5, -3, -11, -19 c) 21, 27.5, 34, 40.5, Find the number of terms in the arithmetic sequence 4, 9, 14,
15 Homework Questions 5 Arithmetic Sequences 1. Find the number of terms in the following sequences if you are given the first few and the last term. a) 12, 25, 38,..155 b) 198, 192, 186, 180, Find the first term of the sequence and the common difference if a) U 2 = 2 U 5 = 17 b) U 4 = 10 U 8 = 6 3. Find the 22 nd term and the nth term of the following sequences a) 5, 11, 17, 23.. b) 25, 21, 17, 13.
16 4. If the first term of an arithmetic sequence is 8 and the common difference is -5. what is the 22 nd term? 5. An arithmetic sequence has a first term of 15 and the 8 th term is 43. What are the first four terms of the sequence? 6. The first two terms of an arithmetic sequence are a+2b and 7b. Find the 3 rd term. 7. What is the common difference of the arithmetic sequence with a 6 th term of -56 and an 11 th term of 11?
17 Homework Questions 6 Partial sums of Arithmetic Sequences 1. Find the sum of the following series a) 17, 25, 33, 41 (25 terms) b) 15, 26, 37, 42.(15 terms) c) 143, 130, 117, 104.(22 terms) d) 96, 90.5, 85, (21 terms) 2. Find the sum of the following arithmetic sequences if you are given the first and the last term a) 5, 19, 33, 47,..243
18 b) 271, 263, 255, 247,..95 c) 78, 65, 52, 39, After how many terms does the sum of the sequence equal the following a) 6, 13, 20, 27 equal 1596 b) 18, 44, 70, 96 equal Find the 3 rd term of the arithmetic sequence if the 6 th term is 24 and the 15 th term is 21
19 Homework Questions 7 Sigma Notation 1. Rewrite the following sums using the sigma notation a) b) c) Multiples of 4 less than 50 d) , Calculate the following a) r = 7 r = 1 r 2 b) r = 5 r = 1 2r + 1
20 c) r = 10 r = 1 r 2 3 h) r = 4 r = 1 (r 6) 2 3. For what values of n does r = 1 n (n 2 + 5) first exceed 500? 4. For what value of n would r (25 6r) = 7 = 1 n
21 Answers Questions 1 Terms of a Sequences 1. Find the next 3 terms of the following sequences and state the rule to find the next term in each case a) 5, 9, 13, 17 21, 25, 29 b) 1, 3, 5, 7 9, 11, 13 c) 9, 13, 17, 21 25, 29, 33 d) -2, 6, 14, 22 30, 38, 46 e) 15, 22, 31, 42, 55 78, 87, 106 f) 4, 13, 26, 43, 64 89, 118, 151
22 g) 3, 7, 13, 21, 31 43, 57, 73 h) 5, 12, 21, 32, 45 60, 77, 96 i) 1, 2, 2, 4, 8 32, 256, 8192 j) 1, 3, 6, 10, 15 21, 28, 36
23 Questions 2 Using the Nth Term of A Sequences 1. Find the value of U 1, U 2, U 3 and U 20 a) U n = 3n 3, 6, 9 b) U n = 7n 2 5, 12, 19 c) U n = 2n 2 2, 8, 18 d) U n = n 2 4-3, 0, 5 2. A sequence is generate according to the formula U n =an-b. Given that U 3 =7 and U 5 =13.find the value of a and b a=3, b=2 3. Find the value of n for which U n =(3n-2) 2 has the given value of U n =100 n=4 4. A sequence is generated from the formula U n =pn 2 -q where p and q are constants. Given that U 1 =- 1 and U 3 =7, find the value of the constants p and q.
24 p=1 q=2 5. Find the value of n for which U n has the given value a) U n =4n-1 and U n =23 n=6 b) U n = 2n3 1 3 and U n = 5 n=2 c) U n = 5n + 6 and U n = 31 n=5
25 Questions 3 Recursive Formula 1. Find the next 3 terms of the following sequences given both the first term and the recursive formula. a) U 1 = 5 U n + 1 = 3U n 15, 45, 135 b) U 1 = 3 U n + 1 = 2U n -6, -12, -24 c) U 1 = 2 U n + 1 = 3U n 4-24, -76, -232 d) U 1 = 16 U n + 1 = U n 4 4, 1, By writing down the first 4 terms or otherwise, find the recursive formula that defines the following sequence. a) U n =2n-1
26 U n+1 =U n +2 b) U n =3n-2 U n+1 =U n Find the next 4 terms of these recursively defined sequences a) U n+1 =U n -U n-1 when U 1 =6 and U 2 =2 6, 2, 8, 10, 18, 28 b) U n+1 =3U n +2U n-1 when U 1 =1 and U 2 =-3 1, -3, -7, -27, -95, -339 c) U n+1 =5U n -11 when U 1 =3 3, 4, 9, 34, Write down the first 3 terms of the sequence defined by U n+1 =12-U n when U 1 =10 10, 2, 10
27 Questions 4 General Term of an Arithmetic Sequence 1. Which of the following sequences are arithmetic? a) 7, 17, 27, 37 d=10 so yes b) 12, 5, 0, -9, -17 No c) 24, 15, 6, -3, -12 d=-9 so yes 2. a) Find the 10 th term and b) Find the formula for the nth term a) 4, 7, 10, U n =3n+1 b) -3, -1, 1, 3 15 U n =2n-5 c) 1, -4, -9, -13
28 -44 U n =-5n+6 3. Find the 20 th term, if the sequence begins a) 2, 6, 10, 14, b) 5, -3, -11, c) 21, 27.5, 34, 40.5, Find the number of terms in the arithmetic sequence 4, 9, 14,
29 Questions 5 Arithmetic Sequences 1. Find the number of terms in the following sequences if you are given the first few and the last term. a) 12, 25, 38, b) 198, 192, 186, 180, Find the first term of the sequence and the common difference if a) U 2 = 2 U 5 = 17 d=5 a=-3 b) U 4 = 10 U 8 = 6 d=1 a= Find the 22 nd term and the nth term of the following sequences a) 5, 11, 17, U n =6n-1
30 b) 25, 21, 17, U n =-4n If the first term of an arithmetic sequence is 8 and the common difference is -5. what is the 22 nd term? An arithmetic sequence has a first term of 15 and the 8 th term is 43. What are the first four terms of the sequence? 15, 19, 23, The first two terms of an arithmetic sequence are a+2b and 7b. Find the 3 rd term. 12b-a 7. What is the common difference of the arithmetic sequence with a 6 th term of -56 and an 11 th term of 11? d=9
31 Questions 6 Partial sums of Arithmetic Sequences 1. Find the sum of the following series a) 17, 25, 33, 41 (25 terms) 732 b) 15, 26, 37, 42.(15 terms) 906 c) 143, 130, 117, 104.(22 terms) 858 d) 96, 90.5, 85, (21 terms) Find the sum of the following arithmetic sequences if you are given the first and the last term
32 a) 5, 19, 33, 47,..243 N= b) 271, 263, 255, 247,..95 N= c) 78, 65, 52, 39, N= After how many terms does the sum of the sequence equal the following a) 6, 13, 20, 27 equal b) 18, 44, 70, 96 equal Find the 3 rd term of the arithmetic sequence if the 6 th term is 24 and the 15 th term is 21 a = d = 1 3 3rd term = 25
33 Homework Questions 7 Sigma Notation 1. Rewrite the following sums using the sigma notation a) r = 13 r = 1 6r 4 b) r = 12 r = 1 7r c) Multiples of 4 less than 50 r = 12 r = 1 4r d) r = 5 r = 1 4r + 4 2, Calculate the following a) r = 7 r = 1 r 2 140
34 b) r = 5 r = 1 2r c) r = 10 r = 1 r h) r = 4 r = 1 (r 6) For what values of n does r = 1 n (n 2 + 5) first exceed 500? n=11 4. For what value of n would r = 1 n (25 6r) = 7 n=7
35 Functions Mapping A function is similar to a number machine or formula, in that you put values in to the function and out come new values. The input is x, the output is f(x). Mapping is the way of showing the results from a function. It is basically saying if you have a value of x, what would be the value of the function f(x). Take a function f:x 6x + 2, (which can also be written as, f(x) = (6x + 2). If we input x = 2, we get an output of 14. This means that the function f maps 2 to 14, or This diagram shows how each input value of x maps to only one output value of f(x). This type of mapping is called 'one-to-one mapping'. There are certain functions where many values of x give the same value of f(x). This is called 'many-to-one mapping'. An example that shows this is: f:x x 2 + 3x 2. This function maps 0 to -2 and -3 to -2. So, some values of f(x) come from more than one input value of x.
36 Domain and Range We need to be able to state which values of x produce values for the function f(x) and the set of these values is called the domain. Using the example function shown above f:x x 2 + 3x 2. This function produces solutions for any value of x. This means that x can be any real number - (in Further Maths there are such things as imaginary numbers!). This means we write that for, f:x x 2 + 3x 2, x R (where R is the set of real numbers). The set of output solutions produced by the function is called the range of the function. Taking the example above, we have already noted that its minimum is at x = -17/4 Therefore all the output values of the function are greater than or equal to -17/4, which is written as: Using a table At times you will need to sketch a function to see what it looks like. An easy way of doing this is: 1.
37 Select values of x and then calculate the corresponding values of the function. 2. Put these values in a table. 3. Use this table to sketch the graph. Using the above example, where f:x x 2 + 3x 2. Select values of x and put the corresponding values of f(x) and into an organized table: x f(x) Now we can plot the values of f(x) on a graph, we can see a pattern in the values of f(x): There are several important pieces of information about the function that need to be found. In particular where the graph crosses the x- and y-axes, and where the graph turns. The graph shows us that: a) The curve has a line of symmetry at the line (because values of x that are symmetrical about the line x = -3/2, give the same value for f(x)). b) The lowest value of y = -17/4 and this happens when
38 c) Using the quadratic formula,...we can calculate the roots of this equation (where f(x) = 0). So, And, Quadratics All quadratics have this same symmetrical shape and for a general quadratic function in the form, f(x) = ax 2 + bx + c Where a, b, and c are constants. The main features we need to sketch a quadratic are: 1. Where the graph crosses the y-axis. (At (0, c) as when x = 0, y = c). 2. Where the graph crosses the x-axis. (Factorise or use the quadratic formula to solve f(x) = 0.) 3. Where the graph turns. You can use differentiation, or completing the square (the quadratic formula), to find that:
39 Graphically, we see that this means: Once you know this information you can sketch any quadratic function. For example: Sketch the curve that represents f(x) -x 2 + 2x When x = 0, y = 0. Therefore it crosses the y-axis at (0,0) f(x) = 0 when -x 2 + 2x = 0, or x(2 - x) = 0. For instance, when x = 0, or when x = 2. It is a - x 2 therefore it is a symmetrical shape, with its maximum value when x = 1 (a = -1, b = 2, therefore -b/2a = 1) and y = 1. So, the graph can be sketched as:
40 More Complex Graphs If we don't already know what a graph will look like we need to find its main features. These are: 1. Where the graph crosses the y-axis, which is when x = 0. (i.e. at the constant). 2. Where the graph crosses the x-axis. To find the roots (where the graph crosses the x-axis), we solve the equation y = 0 3. Where the stationary points are. The stationary points occur when the gradient is 0. (i.e. differentiate.) Whether there are any discontinuities. 4. Are there any discontinuities? A discontinuity occurs when the graph is undefined for a certain value of x. This occurs when x appears in the denominator of a fraction (you can't divide by zero). 5. What happens as x approaches ±? When x becomes a large positive or a large negative number the graph will tend towards a certain value or pattern. Now put all this information onto the graph and join up the points. Example 1: Sketch the graph
41 If x = -3 then the denominator is zero. As we cannot divide by zero the graph is undefined, and there is a discontinuity at x = 3. As x +, y 2 (The -1 and +3 become insignificant.) As x -, y 2 as well. This means there is a horizontal asymptote (value that the graph tends towards) at x = 2. So the final graph looks like this: Example 2: The graph of the function f(x) = 2/x looks like this: The two asymptotes are the x-axis and y-axis. This curve has a special discontinuity at x = 0 where f(0) is undefined.
PLC Papers. Created For:
PLC Papers Created For: Algebra and proof 2 Grade 8 Objective: Use algebra to construct proofs Question 1 a) If n is a positive integer explain why the expression 2n + 1 is always an odd number. b) Use
More informationMCPS Algebra 2 and Precalculus Standards, Categories, and Indicators*
Content Standard 1.0 (HS) Patterns, Algebra and Functions Students will algebraically represent, model, analyze, and solve mathematical and real-world problems involving functional patterns and relationships.
More informationSemester Review Packet
MATH 110: College Algebra Instructor: Reyes Semester Review Packet Remarks: This semester we have made a very detailed study of four classes of functions: Polynomial functions Linear Quadratic Higher degree
More informationevaluate functions, expressed in function notation, given one or more elements in their domains
Describing Linear Functions A.3 Linear functions, equations, and inequalities. The student writes and represents linear functions in multiple ways, with and without technology. The student demonstrates
More informationPre AP Algebra. Mathematics Standards of Learning Curriculum Framework 2009: Pre AP Algebra
Pre AP Algebra Mathematics Standards of Learning Curriculum Framework 2009: Pre AP Algebra 1 The content of the mathematics standards is intended to support the following five goals for students: becoming
More informationAlgebra 2 Khan Academy Video Correlations By SpringBoard Activity
SB Activity Activity 1 Creating Equations 1-1 Learning Targets: Create an equation in one variable from a real-world context. Solve an equation in one variable. 1-2 Learning Targets: Create equations in
More informationAlgebra 2 Khan Academy Video Correlations By SpringBoard Activity
SB Activity Activity 1 Creating Equations 1-1 Learning Targets: Create an equation in one variable from a real-world context. Solve an equation in one variable. 1-2 Learning Targets: Create equations in
More informationCenterville High School Curriculum Mapping Algebra II 1 st Nine Weeks
Centerville High School Curriculum Mapping Algebra II 1 st Nine Weeks Chapter/ Lesson Common Core Standard(s) 1-1 SMP1 1. How do you use a number line to graph and order real numbers? 2. How do you identify
More informationRoots and Coefficients of a Quadratic Equation Summary
Roots and Coefficients of a Quadratic Equation Summary For a quadratic equation with roots α and β: Sum of roots = α + β = and Product of roots = αβ = Symmetrical functions of α and β include: x = and
More informationS4 (4.3) Quadratic Functions.notebook February 06, 2018
Daily Practice 2.11.2017 Q1. Multiply out and simplify 3g - 5(2g + 4) Q2. Simplify Q3. Write with a rational denominator Today we will be learning about quadratic functions and their graphs. Q4. State
More informationPre-Calculus and Trigonometry Capacity Matrix
Review Polynomials A1.1.4 A1.2.5 Add, subtract, multiply and simplify polynomials and rational expressions Solve polynomial equations and equations involving rational expressions Review Chapter 1 and their
More informationMiller Objectives Alignment Math
Miller Objectives Alignment Math 1050 1 College Algebra Course Objectives Spring Semester 2016 1. Use algebraic methods to solve a variety of problems involving exponential, logarithmic, polynomial, and
More informationAlgebra I Assessment. Eligible Texas Essential Knowledge and Skills
Algebra I Assessment Eligible Texas Essential Knowledge and Skills STAAR Algebra I Assessment Mathematical Process Standards These student expectations will not be listed under a separate reporting category.
More informationTopic Outline for Algebra 2 & and Trigonometry One Year Program
Topic Outline for Algebra 2 & and Trigonometry One Year Program Algebra 2 & and Trigonometry - N - Semester 1 1. Rational Expressions 17 Days A. Factoring A2.A.7 B. Rationals A2.N.3 A2.A.17 A2.A.16 A2.A.23
More informationAlgebra III. Mathematics Curriculum Framework. Revised 2004
Algebra III Mathematics Curriculum Framework Revised 2004 Title: Algebra III (Fourth-year Course) Course/Unit Credit: 1 Course Number: Teacher Licensure: Secondary Mathematics Pre-requisite: Algebra II
More informationAlgebra 1. Correlated to the Texas Essential Knowledge and Skills. TEKS Units Lessons
Algebra 1 Correlated to the Texas Essential Knowledge and Skills TEKS Units Lessons A1.1 Mathematical Process Standards The student uses mathematical processes to acquire and demonstrate mathematical understanding.
More informationRevision Questions. Sequences, Series, Binomial and Basic Differentiation
Revision Questions Sequences, Series, Binomial and Basic Differentiation 1 ARITHMETIC SEQUENCES BASIC QUESTIONS 1) An arithmetic sequence is defined a=5 and d=3. Write down the first 6 terms. ) An arithmetic
More information1 Solving Algebraic Equations
Arkansas Tech University MATH 1203: Trigonometry Dr. Marcel B. Finan 1 Solving Algebraic Equations This section illustrates the processes of solving linear and quadratic equations. The Geometry of Real
More informationGUIDED NOTES 5.6 RATIONAL FUNCTIONS
GUIDED NOTES 5.6 RATIONAL FUNCTIONS LEARNING OBJECTIVES In this section, you will: Use arrow notation. Solve applied problems involving rational functions. Find the domains of rational functions. Identify
More informationSequence. A list of numbers written in a definite order.
Sequence A list of numbers written in a definite order. Terms of a Sequence a n = 2 n 2 1, 2 2, 2 3, 2 4, 2 n, 2, 4, 8, 16, 2 n We are going to be mainly concerned with infinite sequences. This means we
More informationab = c a If the coefficients a,b and c are real then either α and β are real or α and β are complex conjugates
Further Pure Summary Notes. Roots of Quadratic Equations For a quadratic equation ax + bx + c = 0 with roots α and β Sum of the roots Product of roots a + b = b a ab = c a If the coefficients a,b and c
More informationSCORE BOOSTER JAMB PREPARATION SERIES II
BOOST YOUR JAMB SCORE WITH PAST Polynomials QUESTIONS Part II ALGEBRA by H. O. Aliu J. K. Adewole, PhD (Editor) 1) If 9x 2 + 6xy + 4y 2 is a factor of 27x 3 8y 3, find the other factor. (UTME 2014) 3x
More informationHoles in a function. Even though the function does not exist at that point, the limit can still obtain that value.
Holes in a function For rational functions, factor both the numerator and the denominator. If they have a common factor, you can cancel the factor and a zero will exist at that x value. Even though the
More informationCollege Algebra & Trig w Apps
WTCS Repository 10-804-197 College Algebra & Trig w Apps Course Outcome Summary Course Information Description Total Credits 5.00 This course covers those skills needed for success in Calculus and many
More informationPacing Guide Algebra 1
Pacing Guide Algebra Chapter : Equations and Inequalities (one variable) Section Section Title Learning Target(s) I can. Evaluate and Simplify Algebraic Expressions. Evaluate and simplify numeric and algebraic
More informationFinding the Equation of a Graph. I can give the equation of a curve given just the roots.
National 5 W 7th August Finding the Equation of a Parabola Starter Sketch the graph of y = x - 8x + 15. On your sketch clearly identify the roots, axis of symmetry, turning point and y intercept. Today
More informationMathematics Precalculus: Academic Unit 1: Polynomial and Transcendental Functions
Understandings Questions Knowledge Functions can be used as models for real-life problems. Functions can be graphed, evaluated, transformed, analyzed, manipulated and combined using algebraic & graphical
More informationThe Quadratic Formula, the Discriminant, and Solving Quadratic Equations and Inequalities
CHAPTER The Quadratic Formula, the Discriminant, and Solving Quadratic Equations and Inequalities 009 Carnegie Learning, Inc. The Chinese invented rockets over 700 years ago. Since then rockets have been
More informationAlgebra II Vocabulary Word Wall Cards
Algebra II Vocabulary Word Wall Cards Mathematics vocabulary word wall cards provide a display of mathematics content words and associated visual cues to assist in vocabulary development. The cards should
More informationLIMITS AT INFINITY MR. VELAZQUEZ AP CALCULUS
LIMITS AT INFINITY MR. VELAZQUEZ AP CALCULUS RECALL: VERTICAL ASYMPTOTES Remember that for a rational function, vertical asymptotes occur at values of x = a which have infinite its (either positive or
More informationSection 2.6 Limits at infinity and infinite limits 2 Lectures. Dr. Abdulla Eid. College of Science. MATHS 101: Calculus I
Section 2.6 Limits at infinity and infinite its 2 Lectures College of Science MATHS 0: Calculus I (University of Bahrain) Infinite Limits / 29 Finite its as ±. 2 Horizontal Asympotes. 3 Infinite its. 4
More informationPre-Calculus and Trigonometry Capacity Matrix
Pre-Calculus and Capacity Matri Review Polynomials A1.1.4 A1.2.5 Add, subtract, multiply and simplify polynomials and rational epressions Solve polynomial equations and equations involving rational epressions
More informationPolynomials and Rational Functions. Quadratic Equations and Inequalities. Remainder and Factor Theorems. Rational Root Theorem
Pre-Calculus Pre-AP Scope and Sequence - Year at a Glance Pre-Calculus Pre-AP - First Semester Pre-calculus with Limits; Larson/Hostetler Three Weeks 1 st 3 weeks 2 nd 3 weeks 3 rd 3 weeks 4 th 3 weeks
More informationChapter 2 Polynomial and Rational Functions
Chapter 2 Polynomial and Rational Functions Overview: 2.2 Polynomial Functions of Higher Degree 2.3 Real Zeros of Polynomial Functions 2.4 Complex Numbers 2.5 The Fundamental Theorem of Algebra 2.6 Rational
More informationMODULE 1: FOUNDATIONS OF MATHEMATICS
MODULE 1: FOUNDATIONS OF MATHEMATICS GENERAL OBJECTIVES On completion of this Module, students should: 1. acquire competency in the application of algebraic techniques; 2. appreciate the role of exponential
More informationAlgebra II Learning Targets
Chapter 0 Preparing for Advanced Algebra LT 0.1 Representing Functions Identify the domain and range of functions LT 0.2 FOIL Use the FOIL method to multiply binomials LT 0.3 Factoring Polynomials Use
More informationVolusia County Mathematics Curriculum Map. Pre-Calculus. Course Number /IOD
Volusia County Mathematics Curriculum Map Pre-Calculus Course Number 1202340/IOD Mathematics Department Volusia County Schools Revised June 9, 2012 Pre- Calculus Curriculum Map 120234/IOD COMPONENTS OF
More informationMath ~ Exam #1 Review Guide* *This is only a guide, for your benefit, and it in no way replaces class notes, homework, or studying
Math 1050 2 ~ Exam #1 Review Guide* *This is only a guide, for your benefit, and it in no way replaces class notes, homework, or studying General Tips for Studying: 1. Review this guide, class notes, the
More informationUNIT 3 MATHEMATICAL METHODS ALGEBRA
UNIT 3 MATHEMATICAL METHODS ALGEBRA Substitution of Values Rearrangement and Substitution Polynomial Expressions Expanding Expressions Expanding Expressions by Rule Perfect Squares The Difference of Two
More informationHow do we analyze, evaluate, solve, and graph quadratic functions?
Topic: 4. Quadratic Functions and Factoring Days: 18 Key Learning: Students will be able to analyze, evaluate, solve and graph quadratic functions. Unit Essential Question(s): How do we analyze, evaluate,
More informationSummer Packet A Math Refresher For Students Entering IB Mathematics SL
Summer Packet A Math Refresher For Students Entering IB Mathematics SL Name: PRECALCULUS SUMMER PACKET Directions: This packet is required if you are registered for Precalculus for the upcoming school
More informationMath 115 Spring 11 Written Homework 10 Solutions
Math 5 Spring Written Homework 0 Solutions. For following its, state what indeterminate form the its are in and evaluate the its. (a) 3x 4x 4 x x 8 Solution: This is in indeterminate form 0. Algebraically,
More informationAmarillo ISD Math Curriculum
Amarillo Independent School District follows the Texas Essential Knowledge and Skills (TEKS). All of AISD curriculum and documents and resources are aligned to the TEKS. The State of Texas State Board
More informationAlgebra I Learning Targets Chapter 1: Equations and Inequalities (one variable) Section Section Title Learning Target(s)
Chapter 1: Equations and Inequalities (one variable) Section Learning Target(s) I can 1.2 Evaluate and Simplify Algebraic Expressions 1. Evaluate and simplify numeric and algebraic expressions (order of
More informationAmarillo ISD Algebra I Standards
Amarillo Independent School District follows the Texas Essential Knowledge and Skills (TEKS). All of AISD curriculum and documents and resources are aligned to the TEKS. The State of Texas State Board
More informationTwitter: @Owen134866 www.mathsfreeresourcelibrary.com Prior Knowledge Check 1) Simplify: a) 3x 2 5x 5 b) 5x3 y 2 15x 7 2) Factorise: a) x 2 2x 24 b) 3x 2 17x + 20 15x 2 y 3 3) Use long division to calculate:
More informationAnalysis of Functions
Lecture for Week 11 (Secs. 5.1 3) Analysis of Functions (We used to call this topic curve sketching, before students could sketch curves by typing formulas into their calculators. It is still important
More informationModule 4: Equations and Inequalities in One Variable
Module 1: Relationships between quantities Precision- The level of detail of a measurement, determined by the unit of measure. Dimensional Analysis- A process that uses rates to convert measurements from
More informationAlgebra 2 and Trigonometry
Algebra 2 and Trigonometry Number Sense and Operations Strand Students will understand meanings of operations and procedures, and how they relate to one another. Operations A2.N.1 Evaluate numerical expressions
More informationHIGH SCHOOL MATH CURRICULUM GRADE ELEVEN ALGEBRA 2 & TRIGONOMETRY N
VALLEY CENTRAL SCHOOL DISTRICT 944 STATE ROUTE 17K MONTGOMERY, NY 12549 Telephone Number: (845) 457-2400 ext. 8121 FAX NUMBER: (845) 457-4254 HIGH SCHOOL MATH CURRICULUM GRADE ELEVEN ALGEBRA 2 & TRIGONOMETRY
More informationWAYNESBORO AREA SCHOOL DISTRICT CURRICULUM ALGEBRA II
UNIT: Review of Basic Algebra Skills as Needed SR1 and any Supplemental Materials UNIT : What skills from Algebra I are used in Algebra II? Review Algebra I Skills as Needed SR1 and any additional resources
More informationPractice Test - Chapter 2
Graph and analyze each function. Describe the domain, range, intercepts, end behavior, continuity, and where the function is increasing or decreasing. 1. f (x) = 0.25x 3 Evaluate the function for several
More informationCourse Number 432/433 Title Algebra II (A & B) H Grade # of Days 120
Whitman-Hanson Regional High School provides all students with a high- quality education in order to develop reflective, concerned citizens and contributing members of the global community. Course Number
More informationAgile Mind Algebra I Scope and Sequence, Texas Essential Knowledge and Skills for Mathematics
In the three years prior to Algebra I, students have already begun their study of algebraic concepts. They have investigated variables and expressions, solved equations, constructed and analyzed tables,
More informationInstructional Unit Conic Sections Pre Calculus #312 Unit Content Objective Performance Performance Task State Standards
Instructional Unit Conic Sections Conic Sections The student will be -Define conic sections -Homework 2.8.11E -Ellipses able to create conic as conic slices and -Classwork -Hyperbolas sections based on
More informationAS PURE MATHS REVISION NOTES
AS PURE MATHS REVISION NOTES 1 SURDS A root such as 3 that cannot be written exactly as a fraction is IRRATIONAL An expression that involves irrational roots is in SURD FORM e.g. 2 3 3 + 2 and 3-2 are
More informationAlgebra 2 Honors Curriculum Pacing Guide
SOUTH CAROLINA ACADEMIC STANDARDS FOR MATHEMATICS The mathematical processes provide the framework for teaching, learning, and assessing in all high school mathematics core courses. Instructional programs
More informationTopic Outline for Integrated Algebra 2 and Trigonometry-R One Year Program with Regents in June
Topic Outline for Integrated Algebra 2 and Trigonometry-R One Year Program with Regents in June Integrated Algebra 2 & Trigonometry - R Semester 1 1. Rational Expressions 7 Days A. Factoring A2.A.7 Factor
More informationDefinition (The carefully thought-out calculus version based on limits).
4.1. Continuity and Graphs Definition 4.1.1 (Intuitive idea used in algebra based on graphing). A function, f, is continuous on the interval (a, b) if the graph of y = f(x) can be drawn over the interval
More informationA video College Algebra course & 6 Enrichment videos
A video College Algebra course & 6 Enrichment videos Recorded at the University of Missouri Kansas City in 1998. All times are approximate. About 43 hours total. Available on YouTube at http://www.youtube.com/user/umkc
More informationThe degree of a function is the highest exponent in the expression
L1 1.1 Power Functions Lesson MHF4U Jensen Things to Remember About Functions A relation is a function if for every x-value there is only 1 corresponding y-value. The graph of a relation represents a function
More informationAlgebra 1 Khan Academy Video Correlations By SpringBoard Activity and Learning Target
Algebra 1 Khan Academy Video Correlations By SpringBoard Activity and Learning Target SB Activity Activity 1 Investigating Patterns 1-1 Learning Targets: Identify patterns in data. Use tables, graphs,
More informationMATHEMATICAL ANALYSIS CURRICULUM GUIDE
MATHEMATICAL ANALYSIS CURRICULUM GUIDE Loudoun County Public Schools 2010-2011 Complete scope, sequence, pacing and resources are available on the CD and will be available on the LCPS Intranet. INTRODUCTION
More informationPractice Test - Chapter 2
Graph and analyze each function. Describe the domain, range, intercepts, end behavior, continuity, and where the function is increasing or decreasing. 1. f (x) = 0.25x 3 Evaluate the function for several
More information30 Wyner Math Academy I Fall 2015
30 Wyner Math Academy I Fall 2015 CHAPTER FOUR: QUADRATICS AND FACTORING Review November 9 Test November 16 The most common functions in math at this level are quadratic functions, whose graphs are parabolas.
More informationThings to remember: x n a 1. x + a 0. x n + a n-1. P(x) = a n. Therefore, lim g(x) = 1. EXERCISE 3-2
lim f() = lim (0.8-0.08) = 0, " "!10!10 lim f() = lim 0 = 0.!10!10 Therefore, lim f() = 0.!10 lim g() = lim (0.8 - "!10!10 0.042-3) = 1, " lim g() = lim 1 = 1.!10!0 Therefore, lim g() = 1.!10 EXERCISE
More informationFactors of Polynomials Factoring For Experts
Factors of Polynomials SUGGESTED LEARNING STRATEGIES: Shared Reading, Activating Prior Knowledge, Discussion Group, Note-taking When you factor a polynomial, you rewrite the original polynomial as a product
More informationarb where a A, b B and we say a is related to b. Howdowewritea is not related to b? 2Rw 1Ro A B = {(a, b) a A, b B}
Functions Functions play an important role in mathematics as well as computer science. A function is a special type of relation. So what s a relation? A relation, R, from set A to set B is defined as arb
More information) = nlog b ( m) ( m) log b ( ) ( ) = log a b ( ) Algebra 2 (1) Semester 2. Exponents and Logarithmic Functions
Exponents and Logarithmic Functions Algebra 2 (1) Semester 2! a. Graph exponential growth functions!!!!!! [7.1]!! - y = ab x for b > 0!! - y = ab x h + k for b > 0!! - exponential growth models:! y = a(
More informationAlgebra II Standards of Learning Curriculum Guide
Expressions Operations AII.1 identify field properties, axioms of equality inequality, properties of order that are valid for the set of real numbers its subsets, complex numbers, matrices. be able to:
More informationCore A-level mathematics reproduced from the QCA s Subject criteria for Mathematics document
Core A-level mathematics reproduced from the QCA s Subject criteria for Mathematics document Background knowledge: (a) The arithmetic of integers (including HCFs and LCMs), of fractions, and of real numbers.
More informationCh. 7.6 Squares, Squaring & Parabolas
Ch. 7.6 Squares, Squaring & Parabolas Learning Intentions: Learn about the squaring & square root function. Graph parabolas. Compare the squaring function with other functions. Relate the squaring function
More informationPure Mathematics P1
1 Pure Mathematics P1 Rules of Indices x m * x n = x m+n eg. 2 3 * 2 2 = 2*2*2*2*2 = 2 5 x m / x n = x m-n eg. 2 3 / 2 2 = 2*2*2 = 2 1 = 2 2*2 (x m ) n =x mn eg. (2 3 ) 2 = (2*2*2)*(2*2*2) = 2 6 x 0 =
More informationPre-Calculus and Trigonometry Capacity Matrix
Information Pre-Calculus and Capacity Matri Review Polynomials A1.1.4 A1.2.5 Add, subtract, multiply and simplify polynomials and rational epressions Solve polynomial equations and equations involving
More informationNew Jersey Quality Single Accountability Continuum (NJQSAC) A-SSE 1-2; A-CED 1,4; A-REI 1-3, F-IF 1-5, 7a
ALGEBRA 2 HONORS Date: Unit 1, September 4-30 How do we use functions to solve real world problems? What is the meaning of the domain and range of a function? What is the difference between dependent variable
More informationAlgebra 2 CP Curriculum Pacing Guide First Half of Semester
Algebra 2 CP Curriculum Pacing Guide 207-208 Unit Functions A2.AAPR.* A2.ACE.2* A2.FBF.3* A2.FIF.6* Add, subtract, and multiply polyomials and understand that polynomials are closed under these operations.
More informationAlgebra 2 (2006) Correlation of the ALEKS Course Algebra 2 to the California Content Standards for Algebra 2
Algebra 2 (2006) Correlation of the ALEKS Course Algebra 2 to the California Content Standards for Algebra 2 Algebra II - This discipline complements and expands the mathematical content and concepts of
More informationUNIVERSITY OF CAMBRIDGE
UNIVERSITY OF CAMBRIDGE DOWNING COLLEGE MATHEMATICS FOR ECONOMISTS WORKBOOK This workbook is intended for students coming to Downing College Cambridge to study Economics 2018/ 19 1 Introduction Mathematics
More informationModule 2: Reflecting on One s Problems
MATH55 Module : Reflecting on One s Problems Main Math concepts: Translations, Reflections, Graphs of Equations, Symmetry Auxiliary ideas: Working with quadratics, Mobius maps, Calculus, Inverses I. Transformations
More informationYEAR 12 - Mathematics Pure (C1) Term 1 plan
Week YEAR 12 - Mathematics Pure (C1) Term 1 plan 2016-2017 1-2 Algebra Laws of indices for all rational exponents. Use and manipulation of surds. Quadratic functions and their graphs. The discriminant
More informationMEI Core 2. Sequences and series. Section 1: Definitions and Notation
Notes and Eamples MEI Core Sequences and series Section : Definitions and Notation In this section you will learn definitions and notation involving sequences and series, and some different ways in which
More information1. Find the domain of the following functions. Write your answer using interval notation. (9 pts.)
MATH- Sample Eam Spring 7. Find the domain of the following functions. Write your answer using interval notation. (9 pts.) a. 9 f ( ) b. g ( ) 9 8 8. Write the equation of the circle in standard form given
More information6.1 Polynomial Functions
6.1 Polynomial Functions Definition. A polynomial function is any function p(x) of the form p(x) = p n x n + p n 1 x n 1 + + p 2 x 2 + p 1 x + p 0 where all of the exponents are non-negative integers and
More informationTropical Polynomials
1 Tropical Arithmetic Tropical Polynomials Los Angeles Math Circle, May 15, 2016 Bryant Mathews, Azusa Pacific University In tropical arithmetic, we define new addition and multiplication operations on
More informationPreCalculus. Curriculum (447 topics additional topics)
PreCalculus This course covers the topics shown below. Students navigate learning paths based on their level of readiness. Institutional users may customize the scope and sequence to meet curricular needs.
More informationCandidates are expected to have available a calculator. Only division by (x + a) or (x a) will be required.
Revision Checklist Unit C2: Core Mathematics 2 Unit description Algebra and functions; coordinate geometry in the (x, y) plane; sequences and series; trigonometry; exponentials and logarithms; differentiation;
More informationAlgebra I, Adopted 2012 (One Credit).
111.39. Algebra I, Adopted 2012 (One Credit). (a) General requirements. Students shall be awarded one credit for successful completion of this course. This course is recommended for students in Grade 8
More informationMath Curriculum Map: Integrated Algebra II Unit: 1 Quarter: Time Frame: Review of Algebra 13 days Essential Questions: Key Concepts: Key Vocabulary:
Math Curriculum Map: Integrated Algebra II Unit: 1 Quarter: Time Frame: Review of Algebra 1 13 days Essential Questions: How does the order of operations help solve one- and two- step equations? How is
More informationAlgebra 2 Math Curriculum Pacing Guide (Revised 2017) Amherst County Public Schools. Suggested Sequence of Instruction and Pacing
Algebra 2 Math Curriculum Pacing Guide (Revised 2017) Amherst County Public Schools Suggested Sequence of Instruction and Pacing 1st 9-weeks Unit 1 - Solving Equations and Inequalities (including absolute
More informationTrigonometry Self-study: Reading: Red Bostock and Chandler p , p , p
Trigonometry Self-study: Reading: Red Bostock Chler p137-151, p157-234, p244-254 Trigonometric functions be familiar with the six trigonometric functions, i.e. sine, cosine, tangent, cosecant, secant,
More informationCoach Stones Expanded Standard Pre-Calculus Algorithm Packet Page 1 Section: P.1 Algebraic Expressions, Mathematical Models and Real Numbers
Coach Stones Expanded Standard Pre-Calculus Algorithm Packet Page 1 Section: P.1 Algebraic Expressions, Mathematical Models and Real Numbers CLASSIFICATIONS OF NUMBERS NATURAL NUMBERS = N = {1,2,3,4,...}
More informationQuestionnaire for CSET Mathematics subset 1
Questionnaire for CSET Mathematics subset 1 Below is a preliminary questionnaire aimed at finding out your current readiness for the CSET Math subset 1 exam. This will serve as a baseline indicator for
More informationThe final is cumulative, but with more emphasis on chapters 3 and 4. There will be two parts.
Math 141 Review for Final The final is cumulative, but with more emphasis on chapters 3 and 4. There will be two parts. Part 1 (no calculator) graphing (polynomial, rational, linear, exponential, and logarithmic
More informationINDEX UNIT 3 TSFX REFERENCE MATERIALS 2014 ALGEBRA AND ARITHMETIC
INDEX UNIT 3 TSFX REFERENCE MATERIALS 2014 ALGEBRA AND ARITHMETIC Surds Page 1 Algebra of Polynomial Functions Page 2 Polynomial Expressions Page 2 Expanding Expressions Page 3 Factorising Expressions
More informationAlgebra Vocabulary. abscissa
abscissa The x-value of an ordered pair that describes the horizontal distance from the x-axis. It is always written as the first element in the ordered pair. 3 is the abscissa of the ordered pair (3,
More informationNYS Algebra II and Trigonometry Suggested Sequence of Units (P.I's within each unit are NOT in any suggested order)
1 of 6 UNIT P.I. 1 - INTEGERS 1 A2.A.1 Solve absolute value equations and inequalities involving linear expressions in one variable 1 A2.A.4 * Solve quadratic inequalities in one and two variables, algebraically
More informationLimits at Infinity. Horizontal Asymptotes. Definition (Limits at Infinity) Horizontal Asymptotes
Limits at Infinity If a function f has a domain that is unbounded, that is, one of the endpoints of its domain is ±, we can determine the long term behavior of the function using a it at infinity. Definition
More information2.4 The Precise Definition of a Limit
2.4 The Precise Definition of a Limit Reminders/Remarks: x 4 < 3 means that the distance between x and 4 is less than 3. In other words, x lies strictly between 1 and 7. So, x a < δ means that the distance
More informationFoundations of Math II Unit 5: Solving Equations
Foundations of Math II Unit 5: Solving Equations Academics High School Mathematics 5.1 Warm Up Solving Linear Equations Using Graphing, Tables, and Algebraic Properties On the graph below, graph the following
More informationMathematics AKS
Integrated Algebra I A - Process Skills use appropriate technology to solve mathematical problems (GPS) (MAM1_A2009-1) build new mathematical knowledge through problem-solving (GPS) (MAM1_A2009-2) solve
More information