1.1 Laws of exponents Conversion between exponents and logarithms Logarithm laws Exponential and logarithmic equations 10
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1 CNTENTS Algebra Chapter Chapter Chapter Eponents and logarithms. Laws of eponents. Conversion between eponents and logarithms 6. Logarithm laws 8. Eponential and logarithmic equations 0 Sequences and series. Arithmetic sequences and series 7. Geometric sequences and series. Sigma notation 8 The binomial theorem. Pascal s triangle. Binomial theorem Functions and equations Chapter Chapter 5 Chapter 6 Introduction to functions. Finding function values 0. Domain and range. Composite functions. Inverse functions 6 5. Quadratic functions and their graphs 5 5. Reciprocal functions and rational functions Graphs of inverse functions Transformations of graphs 7 Eponential and logarithmic functions 6. Eponential and logarithmic functions Applications 9
2 Circular functions and trigonometr Chapter 7 Chapter 8 Chapter 9 Radian angle measure and sectors 7. Definition of a radian Length of an arc and area of a sector 0 Trigonometric functions 8. Sine, cosine, and tangent functions 0 8. Trigonometric identities 8. Graphs of trigonometric functions 8 8. Applications Trigonometric equations Trigonometr of non-right angled triangles 9. The cosine rule 5 9. The sine rule Area of a triangle Applications to geometric problems 65 Vectors Chapter 0 Chapter Vector algebra 0. Basic vector concepts 7 0. Algebraic operations of vectors in component form Scalar products 9 Vector equations of lines. Vector equation of a line 0. Angle between two lines 09. Relationships between two lines
3 Calculus Chapter Chapter Chapter Chapter 5 Differential calculus. Derivatives of functions. Differentiation rules 9. Higher order derivatives 7 Applications of differentiation. Tangents and normals. Increasing and decreasing functions 7. Local maimums and minimums of functions 5. Concavit of functions 5.5 Graphical behaviour of functions 6.6 ptimization 70.7 Kinematics 7 Integral calculus. Indefinite integrals 8. Integration b inspection and substitution 87. Definite integrals 9 Applications of integration 5. Areas under curves Volumes of revolution 5. Kinematics and combined integration problems 9 Statistics and probabilit Chapter 6 Statistics measures 6. Basic concepts 0 6. Measures of central tendenc for ungrouped data 6. Measures of central tendenc for grouped data 6. Quartiles and percentiles 6.5 Measures of dispersion 6.6 Transformations on data sets Frequenc curves and histograms Bo and whisker plots 8
4 Chapter 7 Chapter 8 Chapter 9 Bivariate statistics 7. Linear correlation of bivariate data Use of the equation of the regression line for prediction purposes 6 Probabilit 8. Introduction to probabilit Combined events Venn diagrams and tree diagrams 7 8. Conditional probabilit and independent events 79 Probabilit distributions 9. Discrete random variables 9 9. Binomial probabilit distribution Normal distribution 0 Mathematical smbols and notation 5 Practice test Practice test Answers and markschemes can be downloaded for free at
5 C h a p t e r 5 In Paper of the IB SL eam, ou are epected to know the properties of the graphs of some basic functions. In Paper, ou ma also need to use a graphic displa calculator (GDC) to analse the graphs of more complicated functions. Basic terminolog: -intercept: The intersection of the graph of f ( ) and the -ais -intercept: The intersection of the graph of f ( ) and the -ais Asmptote: A straight line whose distance to the graph of f ( ) tends to zero (that is, the graph of f ( ) gets closer and closer to the asmptote.) -intercept -intercept asmptote EXAM TIP The -intercept and -intercept can usuall be found b a GDC in Paper. 5. Quadratic functions and their graphs General form The general epression of a quadratic function is f ( )= a + b + c, where a 0. Its graph is in the shape of a parabola. The orientation of the parabola is determined b the sign of the coefficient a of the term. The lowest or highest point on the parabola is called the verte (plural: vertices ). If a > 0, the parabola opens upward. e.g. f ( )= + If a < 0, the parabola opens downward. e.g. f ( )= verte verte 5 Functions and equations
6 From the graphs above, ou can see that the graph of a quadratic function is smmetric about a vertical straight line through the verte. The equation of the line is given b: = b a This is given in our formula booklet This vertical line is called the ais of smmetr. Verte form B completing the square, we can rewrite the quadratic function f ( )= a + b + c in verte form. If f ( )= a + b + c f = a h + k, where, then f ( ) can be written as ( ) ( ) ( hk, ) are the coordinates of the verte of the graph of f. ( h,k) EXAM TIP Completing the square is an essential skill for the IBDP SL eam. Candidates are epected to be able to change quadratic functions from one form to another. Eample 5- Find the coordinates of the verte of the graph of a = + 6 b = + 5 Solution a = + 6 = ( + ) 9 Divide the -coefficient b Cop the constant term Square the and subtract = ( + ) 0 Verte is at (, 0). Chapter 5 5
7 b = + 5 = Divide the -coefficient b Cop the constant term Square the and subtract = + Verte is at,. If the coefficient of is not, we have to perform a couple of etra steps to obtain the verte form: Eample 5- Find the coordinates of the verte of the graph of the function f ( )= + +. Solution Factor out the leading coefficient Verte is at (, 9). f ( ) = + + = 6 = ( ) 9 = ( ) 9 = ( ) +9 Complete the square inside the square brackets using the same technique as in Eample 5- Multipl b to eliminate the square brackets 5 Functions and equations
8 The epression under the radical sign b ac is called the discriminant and is usuall denoted b Δ. Its value enables us to find the number of real solutions to the corresponding quadratic equation. Discriminant Number of real solutions Tpical graph Δ = b ac > 0 Δ = b ac = 0 Two distinct real solutions (two -intercepts) ne real solution (one -intercept) (This is also called a repeated root) CMMN MISTAKE Don t confuse the discriminant and the quadratic formula. The discriminant onl tells ou how man solutions there are; the quadratic formula gives the actual values. Δ = b ac < 0 No real solutions (no -intercepts) Eample 5-8 How man real solutions do the following equations have? a + + = 0 b + 5= 0 c = 0 Solution a a =, b =, c = Hence = b ac = ( )( ) = 7 < 0 no real solutions You don t have to write down the values of a, b and c, but writing them can help minimize mistakes, especiall with more complicated situations. Chapter 5 59
9 Step A vertical stretch b a scale factor of : = ( ) ( ) is multiplied b to get ( ). This is a vertical stretch b a factor of = + Step A vertical translation of units upward: ( ) is added to the whole epression ( ). This is a vertical shift up units Warm-up Eercise 5D The graph of = ( ) f is shown below. = f ( ) Sketch the graph of each of the following: a = f ( ) + b = f ( + ) c = f ( ) d = f ( ) e = f ( ) f = f ( ) Chapter 5 77
10 The graph of = f ( ) is shown below. = f ( ) Sketch the graph of each of the following: a = f ( ) + b = f ( ) c = f ( ) d = f ( + ) e = f ( ) f = f ( ) The graph of = f ( ) is shown below: = f ( ) Sketch the graph of the following functions. a = f ( ) b = f ( + ) c = f ( ) d = f ( ) e = f ( ) f = f ( ) Describe eactl the two transformations required to obtain the graph of each of the following from the graph of = f. ( ) a = f ( ) + b = f ( ) c = f ( ) + d = f ( + ) 78 Functions and equations
11 5 Epress the function g( ) in terms of f ( ) where the graph of = ( ) obtained b appling the following transformations to the graph of = f ( ). a Translate b the vector. b Move downward units and then move to the right b units. c Move to the left units and then stretch verticall b a factor of. g can be d Reflect about the -ais and then stretch horizontall b a factor of 5. e Stretch horizontall b a factor of and then stretch verticall b a factor of. f Reflect about the -ais and then move downward b units. g Move upward b 7 units and then reflect about the -ais. Eam Practice 5D Paper The graph of = f ( ), = f ( ) + on the same aes., is shown below. Sketch the graph of 5 = f ( ) The graph of = f ( ) is transformed into the graph of f ( ) full geometric description of this transformation. = 5 +. Give a Chapter 5 79
12 The graph of = ( ) f ( ) f,, is shown below. Sketch the graph of = on the same aes. = f ( ) The graph of = ( ) f,, is shown below. The point ( ) A, lies on the graph, = is a vertical asmptote, and = 0 is a horizontal asmptote. = f ( ) A = a Write down the equation of the new -asmptote if f ( ) is translated units to the left. b Sketch the graph of = f ( + ) on the same aes. c The point A on the graph of f is mapped to the point A on the graph of = f ( + ). Find the coordinates of A. 5 The graph of = ( ) = f ( ) f,, is shown below. Draw the graph of on the same aes. Mark A and B, the image of A and B respectivel, on our graph, together with their coordinates. A(, ) = f ( ) B, 80 Functions and equations
13 Summar Quadratic functions The graph of a quadratic function looks like this: = = b a or p + q ( ) = f ( ) p c q verte( h, k) We can epress the quadratic function in:. General form f ( )= a + b + c The parabola opens upward if a > 0 and the parabola opens downward if a < 0. The -intercept is c. b The ais of smmetr can be found b =. a f = a h + k. Verte form (completed square form) ( ) ( ) The point ( hk, ) is the verte of the graph of f. The ais of smmetr is = h.. Intercept form f ( ) = a( p)( q) The -intercepts of the graph of f are p and q. p+ q The ais of smmetr can be found using =. The number of real roots can be determined b the discriminant Δ = b ac. If Δ > 0 two distinct real roots. If Δ = 0 one root (repeated root, equal roots). If Δ < 0 no real roots. Inverse functions The graph of the inverse function f ( ) f ( ) about the line =. is the reflection of the graph Chapter 5 8
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