Core Maths C1. Revision Notes


 Dennis Skinner
 11 months 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
C1.1 Unit description. C1.2 Assessment information. Preamble. Examination. Formulae
Mathematics Unit C Core AS compulsory unit for GCE AS and GCE Mathematics, GCE AS and GCE Pure Mathematics C. Unit description Algebra and functions; coordinate geometry in the (, y) plane; sequences and
More informationThe NotFormula Book for C1
Not The NotFormula Book for C1 Everything you need to know for Core 1 that won t be in the formula book Examination Board: AQA Brief This document is intended as an aid for revision. Although it includes
More informationy 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 informationSolving Quadratic Equations
Workshop: Algebra 3 Solving Quadratic Equations Topics Covered: Quadratic Equation Quadratic Formula Completing the Square Sketching graphs of quadratic function by Dr.I.Namestnikova 1 Quadratic Equation
More informationFACTORING AND SOLVING QUADRATIC EQUATIONS
6 LESSON FACTORING AND SOLVING QUADRATIC EQUATIONS Solving the equation ax + bx + c = 0 1 Solve the equation x + 8x +15 = 0. The zero principle: When we solve quadratic equations, it is essential to get
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 informationAlgebra 1 Review for Algebra 2
for Algebra Table of Contents Section Topic Page 1.... 5. 6. Solving Equations Straightlined Graphs Factoring Quadratic Trinomials Factoring Polynomials Binomials Trinomials Polynomials Eponential Notation
More informationAlgebra Revision Sheet Questions 2 and 3 of Paper 1
Algebra Revision Sheet Questions and of Paper Simple Equations Step Get rid of brackets or fractions Step Take the x s to one side of the equals sign and the numbers to the other (remember to change the
More informationCore 1 Module Revision Sheet J MS. 1. Basic Algebra
Core 1 Module Revision Sheet The C1 exam is 1 hour 0 minutes long and is in two sections Section A (6 marks) 8 10 short questions worth no more than 5 marks each Section B (6 marks) questions worth 12
More informationTopic. Worked solutions 1 Progress checks. 1 Algebra: Topic 1 Revision of the basics. 1 (a) 3(2x + 5) = 6x + 15
1 Algebra: Topic 1 Revision of the basics Topic Worked solutions 1 Progress checks 1 (a) 3(x + 5) = 6x + 15 (b) (x + 6) = x 1 (c) 8(x + 3x + 4) = 8x + 4x + 3 (d) 6(3x 7x + 9) = 18x 4x + 54 = 4x + 54 (e)
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 informationOrder and Value. Order and Value. Types of Number. Types of Number. Types of Number. Types of Number. Basic Algebra. Basic Algebra.
Order and Value What is the general expression for writing numbers in standard form? 1 Order and Value The general expression for standard form is A 10 n, where 1 A 10 and n is an integer. 1 Types of Number
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 informationx 2 5x + 6 Definition 1.2 The domain of an algebraic expression is the set of values that the variable is allowed to take.
.4 Rational Epressions Definition. A quotient of two algebraic epressions is called a fractional epression. A rational epression is a fractional epression where both the numerator and denominator are polynomials.
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 informationMathematics Edexcel Advanced Subsidiary GCE Core 1 (6663) June 2010
Link to past paper on Edexcel website: http://www.edexcel.com These solutions are for your personal use only. DO NOT photocopy or pass on to third parties. If you are a school or an organisation and would
More informationSection P.9 Notes Page 1 P.9 Linear Inequalities and Absolute Value Inequalities
Section P.9 Notes Page P.9 Linear Inequalities and Absolute Value Inequalities Sometimes the answer to certain math problems is not just a single answer. Sometimes a range of answers might be the answer.
More informationQuestion: What is the quadratic formula? Question: What is the discriminant? Answer: Answer:
Question: What is the quadratic fmula? Question: What is the discriminant? Question: How do you determine if a quadratic equation has no real roots? The discriminant is negative ie Question: How do you
More informationUnit 1, Review Transitioning from Previous Mathematics Instructional Resources: Prentice Hall: Algebra 1
Unit 1, Review Transitioning from Previous Mathematics Transitioning from Seventh grade mathematics to Algebra 1 Read, compare and order real numbers Add, subtract, multiply and divide rational numbers
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 informationx 2 would be a solution to d y
CATHOLIC JUNIOR COLLEGE H MATHEMATICS JC PRELIMINARY EXAMINATION PAPER I 0 System of Linear Equations Assessment Objectives Solution Feedback To use a system of linear c equations to model and solve y
More informationPure Core 3. Revision Notes
Pure Core Revision Notes June 06 Pure Core Algebraic fractions... Cancelling common factors... Multipling and dividing fractions... Adding and subtracting fractions... Equations... 4 Functions... 5 Notation...
More information1.1 Solving a Linear Equation ax + b = 0
1.1 Solving a Linear Equation ax + b = 0 To solve an equation ax + b = 0 : (i) move b to the other side (subtract b from both sides) (ii) divide both sides by a Example: Solve x = 0 (i) x = 0 x = (ii)
More informationObjectives. By the time the student is finished with this section of the workbook, he/she should be able
QUADRATIC FUNCTIONS Completing the Square..95 The Quadratic Formula....99 The Discriminant... 0 Equations in Quadratic Form.. 04 The Standard Form of a Parabola...06 Working with the Standard Form of a
More informationThe Quadratic Function
0 The Quadratic Function TERMINOLOGY Ais of smmetr: A line about which two parts of a graph are smmetrical. One half of the graph is a reflection of the other Coefficient: A constant multiplied b a pronumeral
More informationMathematics Higher Tier, Algebraic Fractions
These solutions are for your personal use only. DO NOT photocopy or pass on to third parties. If you are a school or an organisation and would like to purchase these solutions please contact Chatterton
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 informationg = l 2π g = bd b c d = ac bd. Therefore to write x x and as a single fraction we do the following
OCR Core 1 Module Revision Sheet The C1 exam is 1 hour 30 minutes long. You are not allowed any calculator 1. Before you go into the exam make sureyou are fully aware of the contents of theformula booklet
More informationEdexcel AS and Alevel Modular Mathematics
Core Mathematics Edexcel AS and Alevel Modular Mathematics Greg Attwood Alistair Macpherson Bronwen Moran Joe Petran Keith Pledger Geoff Staley Dave Wilkins Contents The highlighted sections will help
More informationEquations and Inequalities
Rational Equations Overview of Objectives, students should be able to: 1. Solve rational equations with variables in the denominators.. Recognize identities, conditional equations, and inconsistent equations.
More information5.5 The real zeros of a polynomial function
5.5 The real zeros of a polynomial function Recall, that c is a zero of a polynomial f(), if f(c) = 0. Eample: a) Find real zeros of f()= + . We need to find for which f() = 0, that is we have to solve
More informationDifferentiation  Past Edexcel Exam Questions
Differentiation  Past Edecel Eam Questions 1. (a) Given that y = 5 3 + 7 + 3, find i. ii. dy d d 2 y d 2. Question 2ai, 2aii  January 2005 [3] [1] 2. The curve C has equation y = 4 2 + 5, 0. The point
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 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 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 informationArithmetic Operations. The real numbers have the following properties: In particular, putting a 1 in the Distributive Law, we get
Review of Algebra REVIEW OF ALGEBRA Review of Algebra Here we review the basic rules and procedures of algebra that you need to know in order to be successful in calculus. Arithmetic Operations The real
More informationUnit 2: Checklist Higher tier (43602H)
Unit 2: Checklist Higher tier (43602H) recognise integers as positive or negative whole numbers, including zero work out the answer to a calculation given the answer to a related calculation multiply and
More informationFactors of 8 are 1 and 8 or 2 and 4. Let s substitute these into our factors and see which produce the middle term, 10x.
Quadratic equations A quadratic equation in x is an equation that can be written in the standard quadratic form ax + bx + c 0, a 0. Several methods can be used to solve quadratic equations. If the quadratic
More informationAppendix H: Interpreting and Working with Inequalities
Appendi H: Interpreting and Working with Inequalities The goal of interpreting inequalities The goal of solving an equation was usually to find one or more numbers that could be plugged into the equation.
More informationof a straight line graph as a rate of equation of a circle change at a point, on a curve, exponentials
MATHEMATICS GRADE PROGRESSION SUMMARY 9 Apply and interpret limits Find the equation of a tangent Interpret the gradient of accuracy to a curve at a given point of a straight line To manipulate surds Recognise
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 informationName Date Block. Algebra 1 Laws of Exponents/Polynomials Test STUDY GUIDE
Name Date Block Know how to Algebra 1 Laws of Eponents/Polynomials Test STUDY GUIDE Evaluate epressions with eponents using the laws of eponents: o a m a n = a m+n : Add eponents when multiplying powers
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 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 informationHelp Your Child With Higher Maths
Help Your Child With Higher Maths Help Your Child With Higher Maths Introduction We ve designed this booklet so that you can use it with your child throughout the session, as he/she moves through the Higher
More informationMath 1051 Solutions to Review Problems
Math Solutions to Review Problems Homework is to be handed in for grading according to the schedule at the end of the syllabus. Each homework assignment is worth points, which will be awarded as follows:
More informationPrep for College Algebra
Prep for College Algebra 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
More informationONLY FOR PERSONAL USE. This digital version of the DictaatRekenvaardigheden  Algebraic Skills is for personal use because of copyright.
ONLY FOR PERSONAL USE This digital version of the DictaatRekenvaardigheden  Algebraic Skills is for personal use because of copyright. c Algebraic Skills Department of Mathematics and Computer Science
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 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 informationCOGNITIVE TUTOR ALGEBRA
COGNITIVE TUTOR ALGEBRA Numbers and Operations Standard: Understands and applies concepts of numbers and operations Power 1: Understands numbers, ways of representing numbers, relationships among numbers,
More information9.3 Solving Quadratic Equations by the Quadratic Formula
9.3 Solving Quadratic Equations by the Quadratic Formula OBJECTIVES 1 Identify the values of a, b, and c in a quadratic equation. Use the quadratic formula to solve quadratic equations. 3 Solve quadratic
More informationDirections Please read carefully!
Math Xa Algebra Practice Problems (Solutions) Fall 2008 Directions Please read carefully! You will not be allowed to use a calculator or any other aids on the Algebra PreTest or PostTest. Be sure to
More informationPrep for Intermediate Algebra
Prep for Intermediate Algebra This course covers the topics outlined below, new topics have been highlighted. You can customize the scope and sequence of this course to meet your curricular needs. Curriculum
More informationSUMMER MATH PACKET. AP Calculus AB Or AP Calculus BC Required
SUMMER MATH PACKET AP Calculus AB Or AP Calculus BC Required MATH SUMMER PACKET INSTRUCTIONS Attached you will find a packet of eciting math problems for your enjoyment over the summer. The purpose of
More informationThe xintercepts of the graph are the xvalues for the points where the graph intersects the xaxis. A parabola may have one, two, or no xintercepts.
Chapter 101 Identify Quadratics and their graphs A parabola is the graph of a quadratic function. A quadratic function is a function that can be written in the form, f(x) = ax 2 + bx + c, a 0 or y = ax
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 informationPrentice Hall Mathematics, Algebra 2, Indiana Edition 2011
Prentice Hall Mathematics, Algebra 2, Algebra 2 C O R R E L A T E D T O from March 2009 Algebra 2 A2.1 Functions A2.1.1 Find the zeros, domain, and range of a function. A2.1.2 Use and interpret function
More informationChapter 5: Complex Numbers
Algebra and Trigonometry Honors Chapter 5: Comple Numbers Name: Teacher: Pd: COMPLEX NUMBERS In this part of the unit we will: Define the comple unit i Simplify powers of i Graphing comple numbers on the
More informationWhat you can do  (Goal Completion) Learning
What you can do  (Goal Completion) Learning ARITHMETIC READINESS Whole Numbers Order of operations: Problem type 1 Order of operations: Problem type 2 Factors Prime factorization Greatest common factor
More informationChapter One TRIG Targets
Chapter One TRIG Targets 1. Add angles in degrees & minutes. 2. Find measures of coterminal angles. 3. Convert measures of angles between minutes & decimals. 4. Find missing values using trig/pythagorean
More information4. Curriculum content
Candidates may follow either the Core Curriculum or the Extended Curriculum. 1 Number Core curriculum Notes Link within 1.1 Vocabulary and notation for different sets of numbers: natural numbers k, primes,
More informationMATH PLACEMENT TEST STUDY GUIDE
MATH PLACEMENT TEST STUDY GUIDE The study guide is a review of the topics covered by the Columbia College Math Placement Test The guide includes a sample test question for each topic The answers are given
More informationALGEBRA 2 SCOPE AND SEQUENCE. Writing Quadratic Equations Quadratic Regression
ALGEBRA 2 SCOPE AND SEQUENCE UNIT 1 2 3 4 5 6 DATES & NO. OF DAYS 8/229/16 19 days 9/199/30 9 days 10/310/14 9 days 10/2010/25 4 days 10/2712/2 17 days 12/512/16 10 days UNIT NAME Foundations for
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 informationAlgebra 2 and Trigonometry
Algebra 2 and Trigonometry Text: Amsco Algebra II and Trigonometry Text/Workbook Topic Textbook 1. The Integers Alg II/Trig A. Adding and Subtracting Polynomials 1.3 B. Multiplying Polynomials 1.5 C. Factoring
More informationAbsolute Value Inequalities and Equations
Absolute Value Inequalities and Equations I. Equations:. ISOLATE the absolute value. epression involving = constant 2. Rewrite as TWO equations: epression involving = constant or epression involving =
More informationIdentify examples of field properties: commutative, associative, identity, inverse, and distributive.
Topic: Expressions and Operations ALGEBRA II  STANDARD AII.1 The student will identify field properties, axioms of equality and inequality, and properties of order that are valid for the set of real numbers
More informationQuadratic Functions and Parabolas
MATH 11 Quadratic Functions and Parabolas A quadratic function has the form Dr. Neal, Fall 2008 f () = a 2 + b + c where a 0. The graph of the function is a parabola that opens upward if a > 0, and opens
More informationI used math language and/or math notation throughout my work.
Progressive Learning Scale Rubric*: Algebra 2 with Trig Learning Targets Page 1 Level Problem Solving Reasoning and Proof Communication Connections Representation 1 (Beginner) Makes an effort No or little
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 informationBasic Trigonometry SOHCAHTOA =
Trigonometry Triangle ABC Sine Rule = = can be used upside down if trying to find an angle Important that side a is opposite angle A and side b opposite angle B and side c opposite angle C Cosine Rule
More informationMethods to Solve Quadratic Equations
Methods to Solve Quadratic Equations We have been learning how to factor epressions. Now we will apply factoring to another skill you must learn solving quadratic equations. a b c 0 is a seconddegree
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 informationEdexcel Core Maths C1 January Minutes
Edexcel Core Maths C1 January 2005 90 Minutes 1 2 1. (a) Write down the value of 16. Answer: 4 [ Question says write down the answer so no working out is expected. x 1/2 should be recognised as x ] 3 2
More informationFind the length of AB, leaving your answer in surd form. (2) (c) Find the value of t. (1) Find the area of triangle ABC. (2) (Total 8 marks) 5 x.
. (a) Find an equation of the line joining A (7, 4) and B (, 0), giving your answer in the form ax + by + c = 0, where a, b and c are integers. () (b) Find the length of AB, leaving your answer in surd
More informationReview for College Algebra Midterm 1
Review for College Algebra Midterm 1 1.2 Functions and graphs Vocabulary: function, graph, domain, range, VerticalLine Test. Determine whether a given correspondence/mapping is a function. Evaluate a
More informationAlgebra II Pacing Guide First Nine Weeks
First Nine Weeks SOL Topic Blocks.4 Place the following sets of numbers in a hierarchy of subsets: complex, pure imaginary, real, rational, irrational, integers, whole and natural. 7. Recognize that the
More informationYear 11 Scheme of work Autumn Term Half term 1 Foundation level
Year 11 Scheme of work Autumn Term Half term 1 Foundation level Topic Title: Transformations 5 lessons Routemap: 3 Year Foundation Pre requisite knowledge: From the Key stage 3, students will already have
More informationLesson 71. Roots and Radicals Expressions
Lesson 71 Roots and Radicals Epressions Radical Sign inde Radical Sign n a Radicand Eample 1 Page 66 #6 Find all the real cube roots of 0.15 0.15 0.15 0.15 0.50 (0.50) 0.15 0.50 is the cube root of 0.15.
More informationDefinition of Subtraction x  y = x + 1y2. Subtracting Real Numbers
Algebra Review Numbers FRACTIONS Addition and Subtraction i To add or subtract fractions with the same denominator, add or subtract the numerators and keep the same denominator ii To add or subtract fractions
More informationContents. Number. Algebra. Exam board specification map Introduction Topic checker Topic checker answers. iv vi x xv
Contents Exam board specification map Introduction Topic checker Topic checker answers iv vi x xv Number The decimal number system Fraction calculations Fractions, decimals and percentages Powers and roots
More informationDetermine whether the sequence is arithmetic, geometric, or neither. 18, 11, 4,
Over Lesson 10 1 Determine whether the sequence is arithmetic, geometric, or neither. 18, 11, 4, A. arithmetic B. geometric C. neither Over Lesson 10 1 Determine whether the sequence is arithmetic, geometric,
More information1.6 Powers of 10 and standard form Write a number in standard form. Calculate with numbers in standard form.
Unit/section title 1 Number Unit objectives (Edexcel Scheme of Work Unit 1: Powers, decimals, HCF and LCM, positive and negative, roots, rounding, reciprocals, standard form, indices and surds) 1.1 Number
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 information1. (a) Find an equation of the line joining A (7, 4) and B (2, 0), giving your answer in the form ax + by + c = 0, where a, b and c are integers.
1. (a) Find an equation of the line joining A (7, 4) and B (2, 0), giving your answer in the form a + by + c = 0, where a, b and c are integers. (b) Find the length of AB, leaving your answer in surd form.
More informationSurds and Indices. Calculators may NOT be used for these questions.
Paper Reference(s) 666/0 Edexcel GCE Core Mathematics C Advanced Subsidiary Surds and Indices Calculators may NOT be used for these questions. Information for Candidates A booklet Mathematical Formulae
More informationSolving Equations 1.0
Solving Equations 1.0 MTH 18 Jane Long, Stephen F. Austin State University February 16, 2010 Solving Equations 1.0 Introduction: This activity is meant to give you additional practice solving equations.
More informationYear 8 Maths Spring 1 Level Ladder
Year 8 Maths Spring 1 Level Ladder Topic: Algebra All students are expected to master at least the Level 4 content by the end of the half term. Check Arbor or ask your child what their current working
More informationSolving Linear and Quadratic Equations
Solving Linear and Quadratic Equations Mathematics Help Sheet The University of Sydney Business School Introduction to Functions Linear Functions A linear function possesses the general form, y = ax +
More informationActually, if you have a graphing calculator this technique can be used to find solutions to any equation, not just quadratics. All you need to do is
QUADRATIC EQUATIONS Definition ax 2 + bx + c = 0 a, b, c are constants (generally integers) Roots Synonyms: Solutions or Zeros Can have 0, 1, or 2 real roots Consider the graph of quadratic equations.
More information1.1. Basic Concepts. Write sets using set notation. Write sets using set notation. Write sets using set notation. Write sets using set notation.
1.1 Basic Concepts Write sets using set notation. Objectives A set is a collection of objects called the elements or members of the set. 1 2 3 4 5 6 7 Write sets using set notation. Use number lines. Know
More informationChapter 2 Analysis of Graphs of Functions
Chapter Analysis o Graphs o Functions Chapter Analysis o Graphs o Functions Covered in this Chapter:.1 Graphs o Basic Functions and their Domain and Range. Odd, Even Functions, and their Symmetry.. Translations
More informationTesting Center Student Success Center x x 18 12x I. Factoring and expanding polynomials
Testing Center Student Success Center Accuplacer Study Guide The following sample questions are similar to the format and content of questions on the Accuplacer College Level Math test. Reviewing these
More informationExceeding minimum expected standard at end of KS2. Below minimum expected standard at end of KS2. At minimum expected standard at end of KS2.
Pupils will be following stages according to their prior attainment at primary school. A maximum of 13 mastery indicators each year are chosen to represent the most important skills that students need
More informationradical sign 625, both 1 5 and ,
6 Reteaching Roots and Radical Epressions For any real numbers a and b and any positive integer n, if a raised to the nth power equals b, then a is an nth root of b. Use the radical sign to write a root.
More informationAlgebra C/D Borderline Revision Document
Algebra C/D Borderline Revision Document Algebra content that could be on Exam Paper 1 or Exam Paper 2 but: If you need to use a calculator for *** shown (Q8 and Q9) then do so. Questions are numbered
More informationMath Analysis Chapter 2 Notes: Polynomial and Rational Functions
Math Analysis Chapter Notes: Polynomial and Rational Functions Day 1: Section 1 Comple Numbers; Sections  Quadratic Functions 1: Comple Numbers After completing section 1 you should be able to do the
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 information