N5 R1.2 and R1.3 Quadratics - Revision

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1 N5 R and R3 Quadratics - Revision This revision pack covers the skills at Unit Assessment and exam level for Quadratics so you can evaluate your learning of this outcome. It is important that you prepare for Unit Assessments but you should also remember that the final exam is considerably more challenging, thus practice of exam content throughout the course is essential for success. The SQA does not currently allow for the creation of practice assessments that mirror the real assessments so you should make sure your knowledge covers the sub skills listed below in order to achieve success in assessments as these revision packs will not cover every possible question that could arise in an assessment. Topic Unit Sub skills Questions Quadratic Graphs R Be able to recognise and determine the equation of a quadratic function from its graph in the form where Z y kx y x p and q k, p, q Sketch a quadratic function from an equation in the form y x mx n y x p q and Identify the nature, coordinates of the turning point and the equation of the axis of where or y k x p symmetry of a quadratic in the form q p, q Z k 1 1 and Sketch a quadratic function from an equation in the form y ax mbx n y k x p and q where k 1 or a b, m, n, p, q, Z 1 and Quadratic Equations R3 Be able to solve quadratic equations by factorising, graphically or using the quadratic formula Be able to use and interpret the discriminant in relation the roots of a quadratic 1 18 When attempting a question, this key will give you additional important information. Key Note Question is at unit assessment level, a similar question could appear in a unit assessment or an exam. Question is at exam level, a question of similar difficulty will only appear in an exam. * C The question includes a reasoning element and typically makes a question more challenging. Both the Unit Assessment and exam will have reasoning questions. If a star is placed beside one of the above symbols that indicates the question involves sub skills from previously learnt topics. If you struggle with this question you should go back and review that topic, reference to the topic will be in the marking scheme. Question should be completed without a calculator. Question should be completed with a calculator. Questions will be ordered by sub skill and typically will start of easier and then get more challenging. Some questions may also cover several sub skills from this outcome or even include sub skills from previously learnt topics (denoted with a *). Questions are gathered from multiple sources including ones we have created and from past papers. Extra challenge questions are for extension and are not essential for either Unit Assessment or exam preparation. JGHS - N5 R and R3 Revision

2 FORMULAE LIST The roots of ax bx c 0 are x b b ac a Sine rule: a sin A b sin B c sin C Cosine rule: a b c bccos A or b cos A c a bc Area of a triangle: 1 A absin C Volume of a sphere: V r 3 3 Volume of a cone: V 1 r 3 h Volume of a pyramid: V 1 3 Ah Standard deviation: x x x x s n, where n is the sample size. n 1 n 1 JGHS - N5 R and R3 Revision

3 Q Questions Marks 1 The graph opposite is of the form y kx. It passes through the point (0, 0) and (, 1). Evaluate the value of k and hence state the equation of the graph. The graph below is of the form y ax. Find the value of a. JGHS - N5 R and R3 Revision

4 3 The equation of the quadratic function whose graph is shown below is of the form y x a b are integers., where a and b Write down the values of a and b The equation of the quadratic function whose graph is shown below is of the form y x a b, where a and b are integers. Write down the values of a and b JGHS - N5 R and R3 Revision

5 Sketch the quadratic y x, indicating at least one coordinate other than the origin on your diagram Sketch the quadratic y (x 3) 1 indicating the turning point and where the graph crosses the y axis. 3 Sketch the quadratic x x 6 y indicating where the graph crosses the axes and state the coordinates of the turning point. 3 (a) Write y x 6x 11 in the form y ( x a) b. (b) Hence or otherwise sketch the quadratic y x 6x 10 indicating the turning point and where the graph crosses the y axis Sketch the quadratic y x x 15 indicating where the graph crosses the axes and state the coordinates of the turning point. (a) For the quadratic function y x 3 state (i) the y - intercept 1 (ii) the equation of the axis of symmetry 1 (iii) the coordinate of the turning point 1 (iv) whether this turning point is a maximum or minimum. 1 (b) Sketch the quadratic y x f ( x) 16 x state For the quadratic function (a) the y - intercept 1 (b) The equation of the axis of symmetry 1 (c) the coordinate of the turning point 1 (d) whether this turning point is a maximum or minimum. 1 JGHS - N5 R and R3 Revision

6 1 Find the roots of the following quadratic functions by factorising. The first one is factorised for you. (a) x 1 x 3 y 1 (b) y x x (c) y x 9 (d) y x 8x 1 (e) y x x 5 (f) y x 9x Solve the following equations giving your answer to significant figures. (a) x 7x 0 (b) 3x 8x 5 0 (c) x 1 x 5x 5 1 Use the discriminant to determine the nature of the roots of these quadratics. (a) x 5x (b) x x (c) 9x x The curved part of the letter A in the Artwork logo is in the shape of a parabola. The equation of this parabola is y x 8 x (a) Write down the coordinates of Q and R. (b) Calculate the height, h, of the letter A. 3 JGHS - N5 R and R3 Revision

7 16 The diagram shows part of the graph of y 5 x x (a) Find the coordinates of A and B (b) Hence, find the maximum value of y 5 x x. 17 C The profit made by a publishing company of a magazine is calculated by the formula y x 10 x where y is the profit (in pounds) and x is the selling price (in pence) of the magazine. The graph below represents the profit y against the selling price x. Find algebraically the maximum profit the company can make from the sale of the magazine. JGHS - N5 R and R3 Revision

8 18 C The diagram below shows the path of a small rocket which is fired into the air. The height, h metres, of the rocket after t seconds is given by h( t) 16t t (a) After how many seconds will the rocket first be at a height of 60 metres? (b) Will the rocket reach a height of 70 metres? Justify your answer * Use the quadratic formula to find the exact values of the solutions to x x 1 0. Give your answer in its simplest form. 0 Two functions are given below f ( x) x x g( x) x 7 Find algebraically the values of x for which f ( x) g( x) 1 The height, W kilograms, of a giraffe is related to its age, M 1 W M M 7 months, by the formula At what age will a giraffe weigh 83 kilograms? JGHS - N5 R and R3 Revision

9 * A right angled triangle has dimensions, in centimetres, as shown Calculate algebraically the value of x. 5 3 * Triangles PQR and STU are mathematically similar. The scale factor is 3 and PR corresponds to SU. (a) Show that x 6x 5 0 (b) Given QR is the shortest side of triangle PQR, find the value of the side PR. 3 [END OF REVISION QUESTIONS] [Go to next page for the Marking Scheme] JGHS - N5 R and R3 Revision

10 Where suitable, you should always follow through an error as you may still gain partial credit. If you are unsure how to do this ask your teacher. Q Substitute suitable coordinate into equation Marking Scheme Evaluate k and state equation. 1 Substitute suitable coordinate into equation 1 1 k k 3 and 1 5 k 3 Evaluate a and state equation. a 5 and 1 State value of a 1 a State value of b 1 State value of a 1 b 3 a 1 State value of b b. y 3x y 5x JGHS - N5 R and R3 Revision

11 5 1 Correct quadratic shape passing through the origin 1 See diagram below Any suitable coordinate Eg (1, ), (, 16), (3, 36), (, 6) Example sketch. There is no need for a scale on the diagram to achieve full credit 6 1 y-intercept 1 ( 0, 8) Identify the turning point (3, 3 Sketch the quadratic indicating the turning point and y-intercept 3 1) There is no need for a scale on the diagram. Only the coordinates indicated. JGHS - N5 R and R3 Revision

12 7 1 Identify roots and indicate on diagram Identify turning point and indicate on diagram 3 Draw correct shape, indicating the y- intercept at (0. ) 1 1 ( 3 and, 6 16 ) There is no need for a scale on the diagram. 8 (a) 1 Start process 1 y x 3 Complete process y x 3 (b) 3 y-intercept 3 ( 0, 11) Identify the turning point ( 5 5 3, ) There is no need for a scale on the diagram. JGHS - N5 R and R3 Revision

13 9 1 factorise 1 y x 5x 3 3 Identify roots and indicate on diagram Identify turning point and indicate on diagram Draw correct shape, indicating the y- intercept at (0. ) 15 5 and 3 (1, 16 3 ) There is no need for a scale on the diagram. 10 (a) 1 y intercept 1 13 (ii) Equation of axis of symmetry (iii) 3 Coordinate 3 ( x 3 stated explicitly 3, ) (iv) Statement Minimum turning point (b) 5 sketch 5 11 (a) 1 y - intercept 1 1 (b) Equation of axis of symmetry x (c) 3 Coordinate 3 (, 16) (d) Statement Minimum JGHS - N5 R and R3 Revision

14 1 13 (a) 1 State roots 1 3 and 1 (b) Factorise y x( x ) 3 State roots 3 and 0 (c) Factorise y x 3x 3 5 State roots 5 3 (d) 6 Factorise 6 y x 6x 7 State roots 7 (e) 8 Factorise 8 y x 5x 1 6 and 9 State roots 9 5 and 1 (f) 10 Factorise 10 y x 3x 6 11 State roots 11 3 and 6 Factorising must be used to solve these questions. The quadratic formula should not be used. (a) 1 Substitution 1 Evaluate discriminant Solve for each root 3 x 781 and x Round correctly x 8 and x 07 (b) 5 Substitution 5 6 Evaluate discriminant Solve for each root 7 x 0 53 and x Round correctly 8 x 0 5 and x 3 (c) 9 Write in standard form 9 5x x 3 0 or 10 Substitution Evaluate discriminant x 5x 1 Solve for each root 1 x 1 7 and x Round correctly 13 x 1 3 and x 0 7 For any solution that processes to a negative discriminant, only the first mark is available. JGHS - N5 R and R3 Revision

15 C (a) 1 Substitution Evaluate 13 3 State nature of roots 3 Since b ac 0 there are two real and distinct roots (b) Substitution 1 5 Evaluate State nature of roots 6 Since b ac 0 there are two real and equal roots (c) 7 Substitution Evaluate State nature of roots 9 Since b ac 0 there are two roots that are not real (a) 1 State coordinate of Q 1 (, 0) State coordinate of R (8, 0) (b) 3 Identify axis of symmetry 3 x 5 find height of maximum turning point y 9 5 State solution 5 5 (units) (a) 1 Factorise 1 y 5 x1 x State coordinates ( 1, 0) and (5, 0) (b) 3 Find axis of symmetry 3 x State maximum value Maximum value of 9 x 10 x 1 Valid strategy 1 0 Find roots x 0 and x 10 3 Axis of symmetry 3 x 70 y Find the coordinate of the maximum turning point Maximum profit of JGHS - N5 R and R3 Revision

16 18 C 19 * 0 (a) 1 Substitute height t t Rearrange and equate to zero Eg t 16t 60 0 t 6 t 10 3 Correct factorisation 3 0 Solve equation and state suitable solution t 6 t 10 (b) 5 Substitute value and equate to zero 5 t 16t Substitute into discriminant and evaluate and therefore the first time that the height is 60 metres is 6 seconds Make statement 7 Since b ac 0 this equation has no real solutions and therefore the rocket will never reach a height of 70 metres. 1 Substitute into formula 1 Process discriminant 3 simplify surd 3 Simplify solution x This question involves simplifying surds. Review Surds in Expressions and Formulae 1 if required. 1 Equate expressions 1 x x x 7 Make equation equal to zero x 6x 7 0 x 7 x 1 3 Factorise 3 0 State solutions x 1 and x 7 Non algebraic solutions are not suitable JGHS - N5 R and R3 Revision

17 1 * 3 * 1 Substitution M M 7 Equate to 0 M M 60 0 M 10 M 6 3 Factorisation 3 0 Both solutions and state only suitable solution M 10 M 6 and Therefore the solution is 10 months. 1 Valid strategy 1 x x 7 x 8 Expand brackets x 16 x 6 x x 1 x 9 3 Equate to zero 3 x x 15 0 x 5 x 3 Factorise 0 5 State both solutions and identify only valid solution. 5 x 3 and x 5 is 5 centimetres. therefore the length This question involves the use of Pythagoras, review Pythagoras in Relationships if required. (a) 1 Use similarity to equate 1 x x 5 Equate to zero x 6x 5 0 (b) 3 Factorise 3 x 1 x 5 0 Solve x and x 5 or 3x x 5 5 Use only suitable solution to find PR 5 x 5 is the only suitable solution therefore PR = 10 cm This question involves similar triangles. Review similar triangles in Relationships if required.. From 1 to there should be at least one line of working [END OF MARKING SCHEME] JGHS - N5 R and R3 Revision