Core Maths C1. Revision Notes


 Dennis Skinner
 4 years ago
 Views:
Transcription
1 Core Maths C Revision Notes November 0
2 Core Maths C Algebra... Indices... Rules of indices... Surds... 4 Simplifying surds... 4 Rationalising the denominator... 4 Quadratic functions... 4 Completing the square Method... Factorising quadratics... 6 Solving quadratic equations by factorising by completing the square... 6 by using the formula... 6 The discriminant b 4ac... 7 Miscellaneous quadratic equations... 7 Quadratic graphs... 8 Simultaneous equations... 9 Two linear equations... 9 One linear and one quadratic Inequalities... 0 Linear inequalities... 0 Quadratic inequalities... 0 Coordinate geometry... Distance between two points... Gradient... Equation of a line... Parallel and perpendicular lines... 4 Sequences and series... Definition by a formula n = f(n)... Definitions of the form n+ = f( n )... Series and Σ notation... Arithmetic series... Proof of the formula for the sum of an arithmetic series... Curve sketching... 4 Standard graphs... 4 Transformations of graphs... Translations... Stretches... Reflections in the ais... 8 Reflections in the y ais... 8 C 4/04/0 SDB
3 6 Differentiation... 0 General result... 0 Tangents and Normals... 0 Tangents... 0 Normals... 7 Integration... Indefinite integrals... Finding the arbitrary constant... Inde... C 4/04/0 SDB
4 Algebra Indices Rules of indices a m a n = a m+n (a ) n = a mn a m a n = a m n a 0 = a n = n a a n m n = n a n ( ) n m a = a = a m Eamples: (i) 4 = + 4 = = = = 7 4 = 7 6 = (ii) = = + = = = 9 0 = = = + = = (iii) (6 ) 4 = 6 4 = 6 = 6. (iv) = = ( 4) = 6 (v) = since minus means turn upside down =, since on bottom of fraction is cube root, = = Eample: (6 a ) (8 b ) = ( 4 ) a ( ) b = 4a b = 4a b. Eample: Find if 9 = 7 +. Solution: First notice that 9 = and 7 = and so 9 = 7 + ( ) = ( ) + 4 = + 4 = + =. C 4/04/0 SDB
5 Surds A surd is a nasty root i.e. a root which is not rational Thus 64 8 =, 7 =, 4 = are rational and not surds and, 4, 7 are irrational and are surds. Simplifying surds Eample: To simplify 0 we notice that 0 = = 0 = = =. Eample: To simplify 40 we notice that 40 = 8 = 40 = 8 = 8 =. Rationalising the denominator Rationalising means getting rid of surds. We remember that multiplying (a + b) by (a b) gives a b squaring both a and b at the same time!! which has the effect of Eample: Rationalise the denominator of +. Solution: + = = = + 4. Quadratic functions A quadratic function is a function a + b + c, where a, b and c are constants and the highest power of is. Completing the square. Method Rule: ] The coefficient of must be. ] Halve the coefficient of, square it then add it and subtract it. Eample: Complete the square in Solution: ] The coefficient of is already, 4 C 4/04/0 SDB
6 ] the coefficient of is 6, halve it to give then square to give 9 and add and subtract = = ( ). Notice that the minimum value of the epression is when =, since the minimum value of ( ) is 0. Eample: Complete the square in + 4. Solution: ] The coefficient of is not, so we must fiddle it to make it, and then go on to step ]. + 4 = ( + 8) = ( ) = ( + 4) 48 = ( + 4) Notice that the minimum value of the epression is when = 4, since the minimum value of ( + 4) is 0. Thus the verte of the curve is at ( 4, ). Method Eample: Rewrite + 7 in completed square form Solution: We know that we need 6 (half the coefficient of ) and that ( 6) = = ( 6) 7 Notice that the minimum value of the epression is 7 when = 6, since the minimum value of ( 6) is 0. Thus the verte of the curve is at (6, 7). Eample: Epress + in completed square form. Solution: The coefficient of is not so we must take out + = ( + 4) and we know that ( + ) = , so + 4 = ( + ) 4 + = ( + 4) = [( + ) 4] = ( + ) 4 Notice that the minimum value of the epression is 4 when =, since the minimum value of ( ) is 0. Thus the verte of the curve is at (, 4). C 4/04/0 SDB
7 Factorising quadratics All the coefficients must be whole numbers (integers) then the factors will also have whole number coefficients. Eample: Factorise Solution: Looking at the 0 and the 6 we see that possible factors are (0 ± ), (0 ± ), (0 ± ), (0 ± 6), ( ± ), ( ± ), ( ± ), ( ± 6), ( ± ), ( ± ), ( ± ), ( ± 6), ( ± ), ( ± ), ( ± ), ( ± 6), Also the 6 tells us that the factors must have opposite signs, and by trial and error or common sense = ( + )( ). Solving quadratic equations. by factorising. Eample: + 6 = 0 ( )( ) = 0 = 0 or = 0 = or =. Eample: + 8 = 0 ( + 8) = 0 = 0 or + 8 = 0 = 0 or = 8 N.B. Do not divide through by first: you will lose the root of = 0. by completing the square Eample: 6 = 0 6 = and ( ) = = + 9 ( ) = ( ) = ± = ± = or by using the formula always try to factorise first. Eample: a + b + c = 0 = b ± b 4ac a = 0 will not factorise, so we use the formula with a =, b =, c = = ± ( ) 4 ( ) =. or C 4/04/0 SDB
8 The discriminant b 4ac In the formula for the quadratic equation a + b + c = 0 = b ± b 4ac a i) there will be two distinct real roots if b 4ac > 0 ii) there will be only one real root if b 4ac = 0 iii) there will be no real roots if b 4ac < 0 Eample: For what values of k does the equation k + = 0 have i) two distinct real roots, ii) eactly one real root iii) no real roots. Solution: The discriminant b 4ac = ( k) 4 = k 60 i) for two distinct real roots k 60 > 0 k > 60 k < 60, or k > + 60 and ii) for only one real root k 60 = 0 k = ± 60 and iii) for no real roots k 60 < 0 60 < k < Miscellaneous quadratic equations a) sin sin = 0. Put y = sin to give y y = 0 (y )(y + ) = 0 y = or y = ½ sin = or ½ = 90, 0 or 0 from 0 to 60. b) = 0 Notice that = ( ) and put y = to give y 0y + 9 = 0 (y 9)(y ) = 0 y = 9 or y = = 9 or = = or = 0. c) y y + = 0. Put y = to give + = 0 and solve to give = or y = = 4 or. C 4/04/0 SDB 7
9 Quadratic graphs a) If a > 0 the parabola will be the right way up i) if b 4ac > 0 the curve will meet the ais in two points. ii) if b 4ac = 0 the curve will meet the ais in only one point (it will touch the ais) iii) if b 4ac < 0 the curve will not meet the ais. b) a) If a < 0 the parabola will be the wrong way up i) if b 4ac > 0 the curve will meet the ais in two points. ii) if b 4ac = 0 the curve will meet the ais in only one point (it will touch the ais) iii) if b 4ac < 0 the curve will not meet the ais. Note: When sketching the curve of a quadratic function you should always show the value on the yais and if you have factorised you should show the values where it meets the ais, if you have completed the square you should give the coordinates of the verte. 8 C 4/04/0 SDB
10 Simultaneous equations Two linear equations Eample: Solve y = 4 and 4 + 7y =. Solution: Make the coefficients of (or y) equal then add or subtract the equations to eliminate (or y). Here 4 times the first equation gives 8y = 6 and times the second equation gives + y = 4 Subtracting gives 9y = 9 y = Put y = in equation one = 4 + y = 4 + = Check in equation two: L.H.S. = = = R.H.S. One linear and one quadratic. Find (or y) from the Linear equation and substitute in the Quadratic equation. Eample: Solve y =, y y = Solution: From first equation = y + Substitute in second (y + ) y y = 4y + y + 9 y y = y + 9y + 4 = 0 (y + )(y + 4) = 0 y = ½ or y = 4 = or = from the linear equation. Check in quadratic for =, y = ½ L.H.S. = ( ½) ( ½) = = R.H.S. and for =, y = 4 L.H.S. = ( ) ( 4) ( 4) = + = = R.H.S. C 4/04/0 SDB 9
11 Inequalities Linear inequalities Solving algebraic inequalities is just like solving equations, add, subtract, multiply or divide the same number to, from, etc. BOTH SIDES EXCEPT  if you multiply or divide both sides by a NEGATIVE number then you must TURN THE INEQUALITY SIGN ROUND. Eample: Solve + < Solution: sub from B.S. < + 4 sub 4 from B.S.  < divide B.S. by  and turn the inequality sign round > .. Eample: Solve > 6 Solution: We must be careful here since the square of a negative number is positive giving the full range of solutions as < 4 or > +4. Quadratic inequalities Always sketch a graph and find where the curve meets the ais Eample: Find the values of which satisfy 0. Solution: = 0 y ( + )( ) = 0 = / or We want the part of the curve which is above or on the ais / or  0 C 4/04/0 SDB
12 Coordinate geometry Distance between two points Distance between P (a, b ) and Q (a, b ) is ( a a ) + ( b b ) Gradient Gradient of PQ is m b = b a a Equation of a line Equation of the line PQ, above, is y = m + c and use a point to find c or the equation of the line with gradient m through the point (, y ) is y y = m( ) or the equation of the line through the points (, y ) and (, y ) is y y = y y (you do not need to know this one!!) Parallel and perpendicular lines Two lines are parallel if they have the same gradient and they are perpendicular if the product of their gradients is. Eample: Find the equation of the line through (4½, ) and perpendicular to the line joining the points A (, 7) and B (6, ). Solution: 7 Gradient of AB is = 4 6 gradient of line perpendicular to AB is ¼, (product of perpendicular gradients is ) so we want the line through (4½, ) with gradient ¼. Using y y = m( ) y = ¼ ( 4½) 4y =  ½ or + 8y + = 0. C 4/04/0 SDB
13 4 Sequences and series A sequence is any list of numbers. Definition by a formula n = f(n) Eample: The definition n = n gives = =, = = 7, = =,. Definitions of the form n+ = f( n ) These have two parts: (i) a starting value (or values) (ii) a method of obtaining each term from the one(s) before. Eamples: (i) The definition = and n = n + defines the sequence,,, 07,... (ii) The definition =, =, n = n + n defines the sequence,,,,, 8,,, 4, 6,.... This is the Fibonacci sequence. Series and Σ notation A series is the sum of the first so many terms of a sequence. For a sequence whose nth term is n = n + the sum of the first n terms is a series S n = n = (n + ) n i i= This is written in Σ notation as S n = n = ( i + ) and is a finite series of n terms. i= An infinite series has an infinite number of terms S = i= Arithmetic series An arithmetic series is a series in which each term is a constant amount bigger (or smaller) than the previous term: this constant amount is called the common difference. Eamples:, 7,,, 9,,.... with common difference 4 8,,, 9, 6,,... with common difference. i. Generally an arithmetic series can be written as S n = a + (a + d) + (a + d) + (a + d) + (a + 4d) +... upto n terms, where the first term is a and the common difference is d. The nth term n = a + (n )d The sum of the first n terms of the above arithmetic series is C 4/04/0 SDB
14 n S n = ( a + ( n ) d), or S n = ( L ) n a + where L is the last term. Eample: Find the nth term and the sum of the first 00 terms of the arithmetic series with rd term and 7 th term 7. Solution: 7 = + 4 d 7 = + 4 d d = = d = 6 = nth term n = a + (n )d = + (n ) and S 00 = 00 / ( + (00 ) ) = 470. Proof of the formula for the sum of an arithmetic series You must know this proof. First write down the general series and then write it down in reverse order S n = a + (a + d) + (a + d) (a + (n )d) + (a + (n )d S n = (a + (n )d + (a + (n )d) + (a + (n )d) + + (a + d) + a ADD S n = (a + (n )d) + (a + (n )d) + (a + (n )d) + (a + (n )d) + (a + (n )d) S n = n(a + (n )d) S n = n / (a + (n )d), Which can be written as S n = n / (a + a + (n )d) = n / (a + L), where L is the last term. C 4/04/0 SDB
15 Curve sketching Standard graphs y y y y = y =  y = y y y y = y = / y = / y y y = The eponential curve y = y = is like y = but steeper: similarly for y = and y = 7 /, etc. a b c > a b > y = ( a)( b)( c) y = ( a) ( b) a b c > a b > y = (a )( b)( c) y = (a )( b) 4 C 4/04/0 SDB
16 Transformations of graphs Translations (i) If the graph of y = + is translated through + in the ydirection the equation of the new graph is y = + + ; and in general if we know that y is given by some formula involving, which we write as y = f (), then the new curve after a translation through + units in the y direction is y = f () +. (ii) If the graph of y = + is translated through + in the direction the equation of the new graph is y = (  ) + (  ); and if y = f (), then the new curve after a translation through + units in the direction is y = f (  ): i.e. we replace by (  ) everywhere in the formula for y: note the minus sign, , which seems wrong but is correct! Eample: 4 Y The graph of y = (  ) + is the graph of y = + after a translation of + y = y = (  ) X In general y = or y = f () becomes y = (  a) + b a or y = f (  a) + b after a translation through b Stretches (i) If the graph of y = + is stretched by a factor of + in the ydirection.then the equation of the new graph is y = ( + ); and in general if y is given by some formula involving, which we write as y = f (), then the new curve after a stretch by a factor of + in the ydirection is y = f (). C 4/04/0 SDB
17 Eample: y =  4 Y The graph of y =  becomes y = (  ) after a stretch of factor in the ydirection X y = (  ) 6 In general y = or y = f () becomes y = a or y = af () after a stretch in the ydirection of factor a. (ii) If the graph of y = + is stretched by a factor of + in the direction then the equation of the new graph is y = + and in general if y is given by some formula involving, which we write as y = f (), then the new curve after a a stretch by a factor of + in the direction is y = f i.e. we replace by everywhere in the formula for y. 6 C 4/04/0 SDB
18 Eample: In this eample the graph of y = 4 has been stretched by a factor of in the direction to form a new graph with equation y = 4 y = y = 4 8 Y X (iii) Note that to stretch by a factor of ¼ in the direction we replace by that y = f () becomes y = f (4) = so 4 4 Eample: In this eample the graph of y =  9 has been stretched by a factor of in the direction to form a new graph with equation ( ) 9( ) y = = 7 7, The new equation is formed by replacing by = in the original equation. Y y = X ( ) 9( ) y = C 4/04/0 SDB 7
19 Reflections in the ais When reflecting in the ais all the positive ycoordinates become negative and vice versa and so the image of y = f () after reflection in the ais is y = f (). Eample: The image of y = + after reflection in the ais is y = f () = ( + ) = + y = +. y y = +. Reflections in the y ais When reflecting in the yais the ycoordinate for = + becomes the ycoordinate for = and the ycoordinate for = becomes the ycoordinate for = +. Thus the equation of the new graph is found by replacing by and the image of y = f () after reflection in the y ais is y = f ( ). Eample: The image of y = + after reflection in the yais is y = ( ) + ( ) = y y = + y = C 4/04/0 SDB
20 Old equation Transformation New equation y = f () y = f () Translation through Stretch with factor a in the ydirection. a b y = f (  a) + b y = a f () y = f () y = f () Stretch with factor a in the direction. Stretch with factor in the direction. a y = f a y = f( a ) y = f () Reflection in the ais y =  f () y = f () Reflection in the yais y = f () C 4/04/0 SDB 9
21 6 Differentiation General result Differentiating is finding the gradient of the curve. y = n dy d = n n, or f () = n f () = n n Eamples: a) y = dy = 6 7 d b) 7 f () = 7 = 7 f ( ) 7 = = c) 8 dy 4 4 y = = 8 = 8 = 4 d d) f () = ( + )( ) multiply out first = f ( ) = 4 e) 7 4 y = split up first 7 4 dy 4 = = 4 = 6 = d Tangents and Normals Tangents Eample: Find the equation of the tangent to y = 7 + at the point where =. Solution: We first find the gradient when =. y = 7 + dy d = 6 7 and when =, dy d = 6 7 =. so the gradient when = is. The equation is of the form y = m + c where m is the gradient so we have y = + c. To find c we must find a point on the line, namely the point on the curve when =. When = the y value on the curve is 7 + =, i.e. when =, y =. Substituting these values in y = + c we get = + c c = 7 the equation of the tangent is y = 7. 0 C 4/04/0 SDB
22 Normals (The normal to a curve is the line at 90 o to the tangent at that point). We first remember that if two lines with gradients m and m are perpendicular then m m =. Eample: Find the equation of the normal to the curve y = + at the point where =. Solution: We first find the gradient of the tangent when =. y = + dy d = = when = gradient of the tangent is m = 4 = ½. If gradient of the normal is m then m m = ½ m = m = Thus the equation of the normal must be of the form y = + c. From the equation y = + the value of y when = is y = + = Substituting = and y = in the equation of the normal y = + c we have = + c c = 7 equation of the normal is y = + 7. C 4/04/0 SDB
23 7 Integration Indefinite integrals n+ n d = + C provided that n n + N.B. NEVER FORGET THE ARBITRARY CONSTANT + C. Eamples: = d 4 4 = + C = + C ( )( + ) d = + d a) d 6 b) = + ½ + C. + 4 d = + d = + + C = + C 9 c) multiply out first split up first Finding the arbitrary constant If you know the derivative (gradient) function and a point on the curve you can find C. dy Eample: Solve =, given that y = 4 when =. d Solution: y = d = + C but y = 4 when = 4 = + C C = 6 y = + 6. C 4/04/0 SDB
24 Inde Differentiation, 0 Distance between two points, Equation of a line, Gradient, Graphs Reflections, 8 Standard graphs, 4 Stretches, Translations, Indices, Inequalities linear inequalities, 0 quadratic inequalities, 0 Integrals finding arbitrary constant, indefinite, Normals, Parallel and perpendicular lines, Quadratic equations b 4ac, 7 completing the square, 6 factorising, 6 formula, 6 miscellaneous, 7 Quadratic functions, 4 completing the square, 4 factorising, 6 verte of curve, Quadratic graphs, 8 Series Arithmetic, Arithmetic, proof of sum, Sigma notation, Simultaneous equations one linear equations and one quadratic, 9 two linear equations, 9 Surds, 4 rationalising the denominator, 4 Tangents, 0 C 4/04/0 SDB
y intercept Gradient Facts Lines that have the same gradient are PARALLEL
CORE Summar Notes Linear Graphs and Equations = m + c gradient = increase in increase in intercept Gradient Facts Lines that have the same gradient are PARALLEL If lines are PERPENDICULAR then m m = or
More informationCore Maths C2. Revision Notes
Core Maths C Revision Notes November 0 Core Maths C Algebra... Polnomials: +,,,.... Factorising... Long division... Remainder theorem... Factor theorem... 4 Choosing a suitable factor... 5 Cubic equations...
More informationCore Maths C3. Revision Notes
Core Maths C Revision Notes October 0 Core Maths C Algebraic fractions... Cancelling common factors... Multipling and dividing fractions... Adding and subtracting fractions... Equations... 4 Functions...
More informationSTRAND: ALGEBRA Unit 3 Solving Equations
CMM Subject Support Strand: ALGEBRA Unit Solving Equations: Tet STRAND: ALGEBRA Unit Solving Equations TEXT Contents Section. Algebraic Fractions. Algebraic Fractions and Quadratic Equations. Algebraic
More informationSection 5.0A Factoring Part 1
Section 5.0A Factoring Part 1 I. Work Together A. Multiply the following binomials into trinomials. (Write the final result in descending order, i.e., a + b + c ). ( 7)( + 5) ( + 7)( + ) ( + 7)( + 5) (
More informationBiggar High School Mathematics Department. National 5 Learning Intentions & Success Criteria: Assessing My Progress
Biggar High School Mathematics Department National 5 Learning Intentions & Success Criteria: Assessing My Progress Expressions & Formulae Topic Learning Intention Success Criteria I understand this Approximation
More informationLINEAR INEQUALITIES. less than, < 2x + 5 x 3 less than or equal to, greater than, > 3x 2 x 6 greater than or equal to,
LINEAR INEQUALITIES When we use the equal sign in an equation we are stating that both sides of the equation are equal to each other. In an inequality, we are stating that both sides of the equation are
More informationPOLYNOMIAL FUNCTIONS
POLYNOMIAL FUNCTIONS Polynomial Division.. 314 The Rational Zero Test.....317 Descarte s Rule of Signs... 319 The Remainder Theorem.....31 Finding all Zeros of a Polynomial Function.......33 Writing a
More informationReview of Intermediate Algebra Content
Review of Intermediate Algebra Content Table of Contents Page Factoring GCF and Trinomials of the Form + b + c... Factoring Trinomials of the Form a + b + c... Factoring Perfect Square Trinomials... 6
More informationMathematics 31 Precalculus and Limits
Mathematics 31 Precalculus and Limits Overview After completing this section, students will be epected to have acquired reliability and fluency in the algebraic skills of factoring, operations with radicals
More informationAnswers to Basic Algebra Review
Answers to Basic Algebra Review 1. 1.1 Follow the sign rules when adding and subtracting: If the numbers have the same sign, add them together and keep the sign. If the numbers have different signs, subtract
More informationYear 9 set 1 Mathematics notes, to accompany the 9H book.
Part 1: Year 9 set 1 Mathematics notes, to accompany the 9H book. equations 1. (p.1), 1.6 (p. 44), 4.6 (p.196) sequences 3. (p.115) Pupils use the Elmwood Press Essential Maths book by David Raymer (9H
More information10.1. Solving Quadratic Equations. Investigation: Rocket Science CONDENSED
CONDENSED L E S S O N 10.1 Solving Quadratic Equations In this lesson you will look at quadratic functions that model projectile motion use tables and graphs to approimate solutions to quadratic equations
More informationHigher Education Math Placement
Higher Education Math Placement Placement Assessment Problem Types 1. Whole Numbers, Fractions, and Decimals 1.1 Operations with Whole Numbers Addition with carry Subtraction with borrowing Multiplication
More informationSUNY ECC. ACCUPLACER Preparation Workshop. Algebra Skills
SUNY ECC ACCUPLACER Preparation Workshop Algebra Skills Gail A. Butler Ph.D. Evaluating Algebraic Epressions Substitute the value (#) in place of the letter (variable). Follow order of operations!!! E)
More informationMath 0980 Chapter Objectives. Chapter 1: Introduction to Algebra: The Integers.
Math 0980 Chapter Objectives Chapter 1: Introduction to Algebra: The Integers. 1. Identify the place value of a digit. 2. Write a number in words or digits. 3. Write positive and negative numbers used
More informationwww.mathsbox.org.uk ab = 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 informationMethod To Solve Linear, Polynomial, or Absolute Value Inequalities:
Solving Inequalities An inequality is the result of replacing the = sign in an equation with ,, or. For example, 3x 2 < 7 is a linear inequality. We call it linear because if the < were replaced with
More informationRational Exponents. Squaring both sides of the equation yields. and to be consistent, we must have
8.6 Rational Exponents 8.6 OBJECTIVES 1. Define rational exponents 2. Simplify expressions containing rational exponents 3. Use a calculator to estimate the value of an expression containing rational exponents
More informationThis is a square root. The number under the radical is 9. (An asterisk * means multiply.)
Page of Review of Radical Expressions and Equations Skills involving radicals can be divided into the following groups: Evaluate square roots or higher order roots. Simplify radical expressions. Rationalize
More informationEQUATIONS and INEQUALITIES
EQUATIONS and INEQUALITIES Linear Equations and Slope 1. Slope a. Calculate the slope of a line given two points b. Calculate the slope of a line parallel to a given line. c. Calculate the slope of a line
More informationSolving Rational Equations and Inequalities
85 Solving Rational Equations and Inequalities TEKS 2A.10.D Rational functions: determine the solutions of rational equations using graphs, tables, and algebraic methods. Objective Solve rational equations
More informationExponents and Radicals
Exponents and Radicals (a + b) 10 Exponents are a very important part of algebra. An exponent is just a convenient way of writing repeated multiplications of the same number. Radicals involve the use of
More informationNational 5 Mathematics Course Assessment Specification (C747 75)
National 5 Mathematics Course Assessment Specification (C747 75) Valid from August 013 First edition: April 01 Revised: June 013, version 1.1 This specification may be reproduced in whole or in part for
More information1.3 Algebraic Expressions
1.3 Algebraic Expressions A polynomial is an expression of the form: a n x n + a n 1 x n 1 +... + a 2 x 2 + a 1 x + a 0 The numbers a 1, a 2,..., a n are called coefficients. Each of the separate parts,
More informationWhat are the place values to the left of the decimal point and their associated powers of ten?
The verbal answers to all of the following questions should be memorized before completion of algebra. Answers that are not memorized will hinder your ability to succeed in geometry and algebra. (Everything
More informationAlgebra 1 Course Title
Algebra 1 Course Title Course wide 1. What patterns and methods are being used? Course wide 1. Students will be adept at solving and graphing linear and quadratic equations 2. Students will be adept
More informationZero: If P is a polynomial and if c is a number such that P (c) = 0 then c is a zero of P.
MATH 11011 FINDING REAL ZEROS KSU OF A POLYNOMIAL Definitions: Polynomial: is a function of the form P (x) = a n x n + a n 1 x n 1 + + a x + a 1 x + a 0. The numbers a n, a n 1,..., a 1, a 0 are called
More informationVocabulary Words and Definitions for Algebra
Name: Period: Vocabulary Words and s for Algebra Absolute Value Additive Inverse Algebraic Expression Ascending Order Associative Property Axis of Symmetry Base Binomial Coefficient Combine Like Terms
More information6 EXTENDING ALGEBRA. 6.0 Introduction. 6.1 The cubic equation. Objectives
6 EXTENDING ALGEBRA Chapter 6 Extending Algebra Objectives After studying this chapter you should understand techniques whereby equations of cubic degree and higher can be solved; be able to factorise
More informationLesson 9.1 Solving Quadratic Equations
Lesson 9.1 Solving Quadratic Equations 1. Sketch the graph of a quadratic equation with a. One intercept and all nonnegative yvalues. b. The verte in the third quadrant and no intercepts. c. The verte
More information7.7 Solving Rational Equations
Section 7.7 Solving Rational Equations 7 7.7 Solving Rational Equations When simplifying comple fractions in the previous section, we saw that multiplying both numerator and denominator by the appropriate
More informationSubstitute 4 for x in the function, Simplify.
Page 1 of 19 Review of Eponential and Logarithmic Functions An eponential function is a function in the form of f ( ) = for a fied ase, where > 0 and 1. is called the ase of the eponential function. The
More informationIB Maths SL Sequence and Series Practice Problems Mr. W Name
IB Maths SL Sequence and Series Practice Problems Mr. W Name Remember to show all necessary reasoning! Separate paper is probably best. 3b 3d is optional! 1. In an arithmetic sequence, u 1 = and u 3 =
More informationGCSE MATHEMATICS. 43602H Unit 2: Number and Algebra (Higher) Report on the Examination. Specification 4360 November 2014. Version: 1.
GCSE MATHEMATICS 43602H Unit 2: Number and Algebra (Higher) Report on the Examination Specification 4360 November 2014 Version: 1.0 Further copies of this Report are available from aqa.org.uk Copyright
More informationPYTHAGOREAN TRIPLES KEITH CONRAD
PYTHAGOREAN TRIPLES KEITH CONRAD 1. Introduction A Pythagorean triple is a triple of positive integers (a, b, c) where a + b = c. Examples include (3, 4, 5), (5, 1, 13), and (8, 15, 17). Below is an ancient
More informationPolynomials Past Papers Unit 2 Outcome 1
PSf Polnomials Past Papers Unit 2 utcome 1 Multiple Choice Questions Each correct answer in this section is worth two marks. 1. Given p() = 2 + 6, which of the following are true? I. ( + 3) is a factor
More information5.1 Radical Notation and Rational Exponents
Section 5.1 Radical Notation and Rational Exponents 1 5.1 Radical Notation and Rational Exponents We now review how exponents can be used to describe not only powers (such as 5 2 and 2 3 ), but also roots
More informationThe Method of Partial Fractions Math 121 Calculus II Spring 2015
Rational functions. as The Method of Partial Fractions Math 11 Calculus II Spring 015 Recall that a rational function is a quotient of two polynomials such f(x) g(x) = 3x5 + x 3 + 16x x 60. The method
More informationKey Topics What will ALL students learn? What will the most able students learn?
2013 2014 Scheme of Work Subject MATHS Year 9 Course/ Year Term 1 Key Topics What will ALL students learn? What will the most able students learn? Number Written methods of calculations Decimals Rounding
More informationFACTORING QUADRATICS 8.1.1 and 8.1.2
FACTORING QUADRATICS 8.1.1 and 8.1.2 Chapter 8 introduces students to quadratic equations. These equations can be written in the form of y = ax 2 + bx + c and, when graphed, produce a curve called a parabola.
More informationAlgebra I Vocabulary Cards
Algebra I Vocabulary Cards Table of Contents Expressions and Operations Natural Numbers Whole Numbers Integers Rational Numbers Irrational Numbers Real Numbers Absolute Value Order of Operations Expression
More information3.2. Solving quadratic equations. Introduction. Prerequisites. Learning Outcomes. Learning Style
Solving quadratic equations 3.2 Introduction A quadratic equation is one which can be written in the form ax 2 + bx + c = 0 where a, b and c are numbers and x is the unknown whose value(s) we wish to find.
More informationSAT Math Hard Practice Quiz. 5. How many integers between 10 and 500 begin and end in 3?
SAT Math Hard Practice Quiz Numbers and Operations 5. How many integers between 10 and 500 begin and end in 3? 1. A bag contains tomatoes that are either green or red. The ratio of green tomatoes to red
More informationPartial Fractions. Combining fractions over a common denominator is a familiar operation from algebra:
Partial Fractions Combining fractions over a common denominator is a familiar operation from algebra: From the standpoint of integration, the left side of Equation 1 would be much easier to work with than
More informationZeros of a Polynomial Function
Zeros of a Polynomial Function An important consequence of the Factor Theorem is that finding the zeros of a polynomial is really the same thing as factoring it into linear factors. In this section we
More informationCAHSEE on Target UC Davis, School and University Partnerships
UC Davis, School and University Partnerships CAHSEE on Target Mathematics Curriculum Published by The University of California, Davis, School/University Partnerships Program 006 Director Sarah R. Martinez,
More informationAlgebra Practice Problems for Precalculus and Calculus
Algebra Practice Problems for Precalculus and Calculus Solve the following equations for the unknown x: 1. 5 = 7x 16 2. 2x 3 = 5 x 3. 4. 1 2 (x 3) + x = 17 + 3(4 x) 5 x = 2 x 3 Multiply the indicated polynomials
More informationAnswer Key for California State Standards: Algebra I
Algebra I: Symbolic reasoning and calculations with symbols are central in algebra. Through the study of algebra, a student develops an understanding of the symbolic language of mathematics and the sciences.
More informationChapter 7  Roots, Radicals, and Complex Numbers
Math 233  Spring 2009 Chapter 7  Roots, Radicals, and Complex Numbers 7.1 Roots and Radicals 7.1.1 Notation and Terminology In the expression x the is called the radical sign. The expression under the
More informationStanford Math Circle: Sunday, May 9, 2010 SquareTriangular Numbers, Pell s Equation, and Continued Fractions
Stanford Math Circle: Sunday, May 9, 00 SquareTriangular Numbers, Pell s Equation, and Continued Fractions Recall that triangular numbers are numbers of the form T m = numbers that can be arranged in
More informationHigher. Polynomials and Quadratics 64
hsn.uk.net Higher Mathematics UNIT OUTCOME 1 Polnomials and Quadratics Contents Polnomials and Quadratics 64 1 Quadratics 64 The Discriminant 66 3 Completing the Square 67 4 Sketching Parabolas 70 5 Determining
More informationSection 1.1 Linear Equations: Slope and Equations of Lines
Section. Linear Equations: Slope and Equations of Lines Slope The measure of the steepness of a line is called the slope of the line. It is the amount of change in y, the rise, divided by the amount of
More informationAssessment Schedule 2013
NCEA Level Mathematics (9161) 013 page 1 of 5 Assessment Schedule 013 Mathematics with Statistics: Apply algebraic methods in solving problems (9161) Evidence Statement ONE Expected Coverage Merit Excellence
More informationSystems of Equations Involving Circles and Lines
Name: Systems of Equations Involving Circles and Lines Date: In this lesson, we will be solving two new types of Systems of Equations. Systems of Equations Involving a Circle and a Line Solving a system
More informationThe program also provides supplemental modules on topics in geometry and probability and statistics.
Algebra 1 Course Overview Students develop algebraic fluency by learning the skills needed to solve equations and perform important manipulations with numbers, variables, equations, and inequalities. Students
More informationSolutions of Linear Equations in One Variable
2. Solutions of Linear Equations in One Variable 2. OBJECTIVES. Identify a linear equation 2. Combine like terms to solve an equation We begin this chapter by considering one of the most important tools
More informationMPE Review Section III: Logarithmic & Exponential Functions
MPE Review Section III: Logarithmic & Eponential Functions FUNCTIONS AND GRAPHS To specify a function y f (, one must give a collection of numbers D, called the domain of the function, and a procedure
More information1.4 Compound Inequalities
Section 1.4 Compound Inequalities 53 1.4 Compound Inequalities This section discusses a technique that is used to solve compound inequalities, which is a phrase that usually refers to a pair of inequalities
More informationMark Scheme. Mathematics 6360. General Certificate of Education. 2006 examination June series. MPC1 Pure Core 1
Version 1.0: 0706 abc General Certificate of Education Mathematics 660 MPC1 Pure Core 1 Mark Scheme 006 examination June series Mark schemes are prepared by the Principal Examiner and considered, together
More informationx 2 + y 2 = 1 y 1 = x 2 + 2x y = x 2 + 2x + 1
Implicit Functions Defining Implicit Functions Up until now in this course, we have only talked about functions, which assign to every real number x in their domain exactly one real number f(x). The graphs
More informationPolynomial Degree and Finite Differences
CONDENSED LESSON 7.1 Polynomial Degree and Finite Differences In this lesson you will learn the terminology associated with polynomials use the finite differences method to determine the degree of a polynomial
More informationMath 115 Spring 2011 Written Homework 5 Solutions
. Evaluate each series. a) 4 7 0... 55 Math 5 Spring 0 Written Homework 5 Solutions Solution: We note that the associated sequence, 4, 7, 0,..., 55 appears to be an arithmetic sequence. If the sequence
More informationAlgebra and Geometry Review (61 topics, no due date)
Course Name: Math 112 Credit Exam LA Tech University Course Code: ALEKS Course: Trigonometry Instructor: Course Dates: Course Content: 159 topics Algebra and Geometry Review (61 topics, no due date) Properties
More informationALGEBRA 2: 4.1 Graph Quadratic Functions in Standard Form
ALGEBRA 2: 4.1 Graph Quadratic Functions in Standard Form Goal Graph quadratic functions. VOCABULARY Quadratic function A function that can be written in the standard form y = ax 2 + bx+ c where a 0 Parabola
More information2.3. Finding polynomial functions. An Introduction:
2.3. Finding polynomial functions. An Introduction: As is usually the case when learning a new concept in mathematics, the new concept is the reverse of the previous one. Remember how you first learned
More informationMath Review. for the Quantitative Reasoning Measure of the GRE revised General Test
Math Review for the Quantitative Reasoning Measure of the GRE revised General Test www.ets.org Overview This Math Review will familiarize you with the mathematical skills and concepts that are important
More informationYear 12 Pure Mathematics ALGEBRA 1. Edexcel Examination Board (UK)
Year 1 Pure Mathematics ALGEBRA 1 Edexcel Examination Board (UK) Book used with this handout is Heinemann Modular Mathematics for Edexcel AS and ALevel, Core Mathematics 1 (004 edition). Snezana Lawrence
More informationDefinition 8.1 Two inequalities are equivalent if they have the same solution set. Add or Subtract the same value on both sides of the inequality.
8 Inequalities Concepts: Equivalent Inequalities Linear and Nonlinear Inequalities Absolute Value Inequalities (Sections 4.6 and 1.1) 8.1 Equivalent Inequalities Definition 8.1 Two inequalities are equivalent
More informationExponential and Logarithmic Functions
Chapter 6 Eponential and Logarithmic Functions Section summaries Section 6.1 Composite Functions Some functions are constructed in several steps, where each of the individual steps is a function. For eample,
More informationChapter 4. Polynomial and Rational Functions. 4.1 Polynomial Functions and Their Graphs
Chapter 4. Polynomial and Rational Functions 4.1 Polynomial Functions and Their Graphs A polynomial function of degree n is a function of the form P = a n n + a n 1 n 1 + + a 2 2 + a 1 + a 0 Where a s
More information3 e) x f) 2. Precalculus Worksheet P.1. 1. Complete the following questions from your textbook: p11: #5 10. 2. Why would you never write 5 < x > 7?
Precalculus Worksheet P.1 1. Complete the following questions from your tetbook: p11: #5 10. Why would you never write 5 < > 7? 3. Why would you never write 3 > > 8? 4. Describe the graphs below using
More informationMaths Workshop for Parents 2. Fractions and Algebra
Maths Workshop for Parents 2 Fractions and Algebra What is a fraction? A fraction is a part of a whole. There are two numbers to every fraction: 2 7 Numerator Denominator 2 7 This is a proper (or common)
More informationIndiana State Core Curriculum Standards updated 2009 Algebra I
Indiana State Core Curriculum Standards updated 2009 Algebra I Strand Description Boardworks High School Algebra presentations Operations With Real Numbers Linear Equations and A1.1 Students simplify and
More information2013 MBA Jump Start Program
2013 MBA Jump Start Program Module 2: Mathematics Thomas Gilbert Mathematics Module Algebra Review Calculus Permutations and Combinations [Online Appendix: Basic Mathematical Concepts] 2 1 Equation of
More informationMATH 095, College Prep Mathematics: Unit Coverage Prealgebra topics (arithmetic skills) offered through BSE (Basic Skills Education)
MATH 095, College Prep Mathematics: Unit Coverage Prealgebra topics (arithmetic skills) offered through BSE (Basic Skills Education) Accurately add, subtract, multiply, and divide whole numbers, integers,
More informationALGEBRA. sequence, term, nth term, consecutive, rule, relationship, generate, predict, continue increase, decrease finite, infinite
ALGEBRA Pupils should be taught to: Generate and describe sequences As outcomes, Year 7 pupils should, for example: Use, read and write, spelling correctly: sequence, term, nth term, consecutive, rule,
More informationAnswer: The relationship cannot be determined.
Question 1 Test 2, Second QR Section (version 3) In City X, the range of the daily low temperatures during... QA: The range of the daily low temperatures in City X... QB: 30 Fahrenheit Arithmetic: Ranges
More informationAlgebra 2: Q1 & Q2 Review
Name: Class: Date: ID: A Algebra 2: Q1 & Q2 Review Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Which is the graph of y = 2(x 2) 2 4? a. c. b. d. Short
More information0.8 Rational Expressions and Equations
96 Prerequisites 0.8 Rational Expressions and Equations We now turn our attention to rational expressions  that is, algebraic fractions  and equations which contain them. The reader is encouraged to
More informationCopy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any.
Algebra 2  Chapter Prerequisites Vocabulary Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any. P1 p. 1 1. counting(natural) numbers  {1,2,3,4,...}
More informationCONTENTS. Please note:
CONTENTS Introduction...iv. Number Systems... 2. Algebraic Expressions.... Factorising...24 4. Solving Linear Equations...8. Solving Quadratic Equations...0 6. Simultaneous Equations.... Long Division
More informationIndices and Surds. The Laws on Indices. 1. Multiplication: Mgr. ubomíra Tomková
Indices and Surds The term indices refers to the power to which a number is raised. Thus x is a number with an index of. People prefer the phrase "x to the power of ". Term surds is not often used, instead
More informationThnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks
Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks Welcome to Thinkwell s Homeschool Precalculus! We re thrilled that you ve decided to make us part of your homeschool curriculum. This lesson
More informationMTN Learn. Mathematics. Grade 10. radio support notes
MTN Learn Mathematics Grade 10 radio support notes Contents INTRODUCTION... GETTING THE MOST FROM MINDSET LEARN XTRA RADIO REVISION... 3 BROADAST SCHEDULE... 4 ALGEBRAIC EXPRESSIONS... 5 EXPONENTS... 9
More informationFlorida Math for College Readiness
Core Florida Math for College Readiness Florida Math for College Readiness provides a fourthyear math curriculum focused on developing the mastery of skills identified as critical to postsecondary readiness
More informationALGEBRA REVIEW LEARNING SKILLS CENTER. Exponents & Radicals
ALGEBRA REVIEW LEARNING SKILLS CENTER The "Review Series in Algebra" is taught at the beginning of each quarter by the staff of the Learning Skills Center at UC Davis. This workshop is intended to be an
More informationAlgebra. Exponents. Absolute Value. Simplify each of the following as much as possible. 2x y x + y y. xxx 3. x x x xx x. 1. Evaluate 5 and 123
Algebra Eponents Simplify each of the following as much as possible. 1 4 9 4 y + y y. 1 5. 1 5 4. y + y 4 5 6 5. + 1 4 9 10 1 7 9 0 Absolute Value Evaluate 5 and 1. Eliminate the absolute value bars from
More informationLAKE ELSINORE UNIFIED SCHOOL DISTRICT
LAKE ELSINORE UNIFIED SCHOOL DISTRICT Title: PLATO Algebra 1Semester 2 Grade Level: 1012 Department: Mathematics Credit: 5 Prerequisite: Letter grade of F and/or N/C in Algebra 1, Semester 2 Course Description:
More informationMESSAGE TO TEACHERS: NOTE TO EDUCATORS:
MESSAGE TO TEACHERS: NOTE TO EDUCATORS: Attached herewith, please find suggested lesson plans for term 1 of MATHEMATICS Grade 11 Please note that these lesson plans are to be used only as a guide and teachers
More informationFACTORING QUADRATICS 8.1.1 through 8.1.4
Chapter 8 FACTORING QUADRATICS 8.. through 8..4 Chapter 8 introduces students to rewriting quadratic epressions and solving quadratic equations. Quadratic functions are any function which can be rewritten
More informationUnit 6: Polynomials. 1 Polynomial Functions and End Behavior. 2 Polynomials and Linear Factors. 3 Dividing Polynomials
Date Period Unit 6: Polynomials DAY TOPIC 1 Polynomial Functions and End Behavior Polynomials and Linear Factors 3 Dividing Polynomials 4 Synthetic Division and the Remainder Theorem 5 Solving Polynomial
More informationALGEBRA 2 CRA 2 REVIEW  Chapters 16 Answer Section
ALGEBRA 2 CRA 2 REVIEW  Chapters 16 Answer Section MULTIPLE CHOICE 1. ANS: C 2. ANS: A 3. ANS: A OBJ: 53.1 Using Vertex Form SHORT ANSWER 4. ANS: (x + 6)(x 2 6x + 36) OBJ: 64.2 Solving Equations by
More informationHFCC Math Lab Beginning Algebra 13 TRANSLATING ENGLISH INTO ALGEBRA: WORDS, PHRASE, SENTENCES
HFCC Math Lab Beginning Algebra 1 TRANSLATING ENGLISH INTO ALGEBRA: WORDS, PHRASE, SENTENCES Before being able to solve word problems in algebra, you must be able to change words, phrases, and sentences
More informationUnderstanding Basic Calculus
Understanding Basic Calculus S.K. Chung Dedicated to all the people who have helped me in my life. i Preface This book is a revised and expanded version of the lecture notes for Basic Calculus and other
More informationSimplifying Exponential Expressions
Simplifying Eponential Epressions Eponential Notation Base Eponent Base raised to an eponent Eample: What is the base and eponent of the following epression? 7 is the base 7 is the eponent Goal To write
More informationTSI College Level Math Practice Test
TSI College Level Math Practice Test Tutorial Services Mission del Paso Campus. Factor the Following Polynomials 4 a. 6 8 b. c. 7 d. ab + a + b + 6 e. 9 f. 6 9. Perform the indicated operation a. ( +7y)
More informationMathematics programmes of study: key stage 4. National curriculum in England
Mathematics programmes of study: key stage 4 National curriculum in England July 2014 Contents Purpose of study 3 Aims 3 Information and communication technology (ICT) 4 Spoken language 4 Working mathematically
More informationZeros of Polynomial Functions
Zeros of Polynomial Functions The Rational Zero Theorem If f (x) = a n x n + a n1 x n1 + + a 1 x + a 0 has integer coefficients and p/q (where p/q is reduced) is a rational zero, then p is a factor of
More information