Name: Summer Review Packet for Students Entering IB MATH SL Year 2 Directions: Complete all pages in this review. Show all work for credit. You may get video help and instruction via YouTube for the topics in this review packet Due Date: 09/5/17 points will be lost for late submissions. If you get stuck: o Search YOUTUBE for the specific topics. o Visit hhsmathslyr2.wordpress.com some support videos will be uploaded during the summer months. o Contact either Ms. Devine or Ms. DeGrazia with any concerns: devinem@harrisoncsd.org or degraziab@harrisoncsd.org There will be a quiz on this material during the first week of school. Extra help provided during 1 st week. We look forward to working with you this school year! *NOTE- not all topics covered in SL Year 1 are represented in this packet. The material in this packet supports the new content that will be taught in the first semester of SL year 2.
Algebra Review Arithmetic Sequences have a common difference (increase/decrease through addition of a fixed amount). Here are formulas to use for Arithmetic Sequences: Geometric Sequences have a common ratio (Increase/decrease through multiplication by a fixed amount) Here are formulas to use for Geometric Sequences: 1) An arithmetic sequence has = 62 and = 670. Find the value of and the value of d. 2) A geometric sequence has and a common ratio of 3. Which term number will be the first to exceed 1,000,000?
Logarithms Example Evaluate My thinking The log is the number I must raise the 5 to in order to get 625. So, = 4 because 5 4 = 625. You must be able to change an equation form logarithmic to exponential form using the fact that: 3) Solve If no base is shown on a logarithm, the base is assumed to be 10. Natural logs have a base of e. 4) Solve Know your Log Rules! Here they are from you IB Math SL Formula Booklet! Tips for solving log equations: Short (single log) convert to exponential form. Long(multiple logs) convert to a single log of the same base on each side, drop the logs, solve then check 5) Solve for x: log x + log (x+1) = log2 6) Let and write the following in terms of x and y only. b)
Binomial Expansion 7) Find the coefficient of x 3 in the expansion of (2 x) 5. Functions Domain is the set of all input values (x-values) for a function. Generally it is limited by the following illegal operations: division by zero square roots of negative numbers Logs of zero or negatives Certain values for tan Range is corresponding set of output values (y-values) for a function. The graph and/or knowledge of asymptotes, maxima, minima, should help determine the range. X-Intercepts : The location where a function crosses the x-axis. Found by plugging in 0 for y (or f(x)) Y-Intercepts : The location where a function crosses the y-axis. Found by plugging in 0 for x Sometimes in IB SL you must sketch a grpah based on your calculator display and accurately show the appropriate domain, range, extrema, asymptotes, and axis-intercepts 8) Consider the following functions: f(x) = 2x 3 g(x) = 4 x h(x) = x 2-1 a) g( b) c) f -1 (x) d) Solve h(x) = 0
Families of functions KNOW THE SHAPES OF THE FOLLOWING Transformations For a function y = f(x) the following rules define certain transformations F(x-c) F(x) + d Kf(x) F(kx) Translate right c units Translate up d units Stretch vertically by a scale factor of k (Multiply y values by k) * Negative value means a reflection through the x-axis Stretch horizontally by a scale factor f 1/k (multiply x values by the reciprocal) * Negative value means a reflection through the y-axis
9) Consider the graph of f(x) shown below. ON the same grid, sketch the graph of y = -2f(x+3). State the transformations that are taking place. 10) A Point A(3,1) is on f(x). Write the coordinates of the image of A after f(x) under goes the transformation f(x+2) +4 11) The function g(x) = -f(x-3)+1. Give a full geometric description of the transformation that take place from f(x) to g(x). Quadratic Functions Quadratic graphs are parabolas. There are 3 popular forms to write them: 1) Vertex form y = a(x-h) 2 + k The vertex is at (h,k) The a-value determines the direction of opening and the steepness 2) Factored form y=a(x-p)(x-q) The values of p and q are the x-intercepts The vertex and the axis of symmetry lie halfway between the x-intercepts The a-value determines the direction of opening and the steepness The parabolas that do not have x-intercepts cannot be written in this form 3) Trinomial Form y=ax 2 +bx+c c is the y-intercept The a-value determines the direction of opening and the steepness Axis of symmetry is x= (IN YOUR FORMULA BOOKLET) Can be converted to vertex form through the process of complete the square 12) Algebraically express in the form, where h and k are to be determined
(b) Hence, or otherwise, write down the coordinates of the vertex of the parabola with equation. 13) A parabola has x-intercepts at -1 and 5 and has an equation y = a(x-m)(x-n). a) Write down the values of m and n. b) There is a minimum at (p,-36). Find the value of p. c) Hence find the equation of the parabola giving your final answer in trinomial form y=ax 2 +bx+c Solving a quadratic equation: For easy factorable quadratics, this is usually your best bet! 14) Try it! Solve for x: Discriminants In a quadratic equation, the discriminant tells us how many solutions exist (0,1, or 2).
15) The function has no x-intercepts. State the possible values of k. Logarithmic VS Exponential Graphs (THEY ARE INVERSES OF EACH OTHER)! 16)
Note/Key Ideas Answer These You should know how to: Use SohCahToa to solve for sides and angles of a right triangle. Know how to sketch a diagram and label the angle of elevation/depression correctly. Sketch a given angle onto the quadrants and find it s reference angle. Given an angle find the coterminal angle in standard position. The exact values of sine, cosine and tangent functions at 0, 30, 45,60 and 90 degrees. Understand that the values of the coordinates on the unit circle are equivalent to ( cos sin ) Know how to convert between radians and degrees using the conversion. *Or- redraw your UNIT CIRCLE!! 17) Evaluate: ( ) ( ) 18) If A is an obtuse angle in a triangle and sin A = 5, calculate 13 the exact value of sin 2A. Know how to simplify trig expressions in different forms.
Note/Key Ideas Need to know: Solve for all possible angles given a trig value. Ex1: Solve for Answer These 19) Solve the equation 2sinx - 1 = 0 on the interval. Clues given to us: 1) Value 2) Determine reference angle 3) Radian vs. Degree? 4) 20) Solve the equation for 5) Check the domain! It may help to draw a sketch Ex 2: Solve the equation for Double Angles: Ex 3: Solve the following equation for -180 < x < 180. cos(2x) = Clues given to us: 1) Replace given angle with A 2) Establish new domain 21) Solve the following equation for -180 < x < 180. cos(2x) = 3) Value 4) Determine reference angle 5) Radian vs. Degree? 6) 7) Solve for all possible angles within new domain 8) Solve for x that was replaced 9) Double check the domain! **Draw a sketch
Notes/Key Ideas Need to know: Sine Rule When the Sine rule is appropriate to use. How to use the sine rule to solve for angles and sides. TIP: 2 sides and 2 angles are involved. Answer These 22) NO CALC! Cosine Rule When the Cosine rule is appropriate to use. How to use the cosine rule to solve for angles and sides. TIP: 3 sides and 1 angle are involved. Area of triangles. Formula given in booklet. Watch out for the sides used vs angle C in formula. Sides used MATTER! Arc Length and Area of a sector Formulae given in booklet. ANGLE MUST BE IN RADIAN. MUST CONVERT IF IT IS NOT. Area of partial Sectors Sometimes worded as Area of the minor segment Rule of thumb: Area of sector MINUS Area of the triangle. 23) The diagram below shows a sector AOB of a circle of radius 15 cm and centre O. The angle at the centre of the circle is 2 radians. Diagram not to scale A B O (a) Calculate the area of the sector AOB. (b) Calculate the area of the shaded region.