National 5 Mathematics Assessment Practice Topic 6: Graphs of quadratic functions and solving quadratic functions

Transcription

1 SCHOLAR Study Guide National 5 Mathematics Assessment Practice Topic 6: Graphs of quadratic functions and solving quadratic functions Authored by: Margaret Ferguson Heriot-Watt University Edinburgh EH14 4AS, United Kingdom.

2 First published 2014 by Heriot-Watt University. This edition published in 2016 by Heriot-Watt University SCHOLAR. Copyright 2016 SCHOLAR Forum. Members of the SCHOLAR Forum may reproduce this publication in whole or in part for educational purposes within their establishment providing that no profit accrues at any stage, Any other use of the materials is governed by the general copyright statement that follows. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, without written permission from the publisher. Heriot-Watt University accepts no responsibility or liability whatsoever with regard to the information contained in this study guide. Distributed by the SCHOLAR Forum. SCHOLAR Study Guide Assessment Practice Topic 6: National 5 Mathematics 1. National 5 Mathematics Course Code: C747 75

3 Acknowledgements Thanks are due to the members of Heriot-Watt University's SCHOLAR team who planned and created these materials, and to the many colleagues who reviewed the content. We would like to acknowledge the assistance of the education authorities, colleges, teachers and students who contributed to the SCHOLAR programme and who evaluated these materials. Grateful acknowledgement is made for permission to use the following material in the SCHOLAR programme: The Scottish Qualifications Authority for permission to use Past Papers assessments. The Scottish Government for financial support. The content of this Study Guide is aligned to the Scottish Qualifications Authority (SQA) curriculum. All brand names, product names, logos and related devices are used for identification purposes only and are trademarks, registered trademarks or service marks of their respective holders.

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5 1 Topic 1 Graphs of quadratic functions and solving quadratic functions Contents 6.1 Learning points Assessment practice

6 2 TOPIC 1. GRAPHS OF QUADRATIC FUNCTIONS AND SOLVING QUADRATIC FUNCTIONS By the end of this topic, you should have identified your strengths and areas for further revision. Read through the learning points before you attempt the assessments and go back to the Course Materials unit if you need more help. You should be able to: identify the shape, zeros and y-intercept of a quadratic function; determine the turning point and equation of the axis of symmetry of a quadratic function; recognize a quadratic function from its graph; determine the equation of a quadratic function from its graph; sketch a quadratic function; recognise and use function notation; solve a quadratic equation: graphically; by factorising; using the quadratic formula; identify and interpret the nature of the roots of a quadratic using the discriminant.

7 TOPIC 1. GRAPHS OF QUADRATIC FUNCTIONS AND SOLVING QUADRATIC FUNCTIONS Learning points Graphs of quadratic functions Features of quadratic functions The shape of the graph of a quadratic will be smiley if the x 2 term is positive e.g. y = 5x 2 looks like The shape of the graph of a quadratic will be sad if the x 2 term is negative e.g. y = x 2 looks like If the shape is smiley the nature will be a minimum. If the shape is sad the shape will be a maximum. The graph is symmetrical and the equation of the axis of symmetry will take the form x = a. The zeros or roots of a quadratic are the point(s) where the graph crosses the x-axis and will take the form (p,0). The y-intercept is the point where the graph crosses the y-axis and will take the form (0,c) where c can be identified from y = ax 2 + bx + c. Determining the equation of a quadratic function from its graph If the equation of the quadratic takes the form y = kx 2 : find the coordinates of a point on the graph (Note: you cannot use the origin); substitute the values of x and y into y = kx 2 ; calculate the value of k; state the equation of the function. If the equation of the quadratic takes the form y = k(x a) 2 + b: find the coordinates of the turning point from the graph; replace a with the x-coordinate; replace b with the y-coordinate; state the equation of the function Sketching the graph of a quadratic function Find the coordinates of the y-intercept If the equation takes the form ax 2 + bx + c = 0 then (0,c).

8 4 TOPIC 1. GRAPHS OF QUADRATIC FUNCTIONS AND SOLVING QUADRATIC FUNCTIONS If the equation takes the form y = (x m)(x n) then substitute x = 0 into the function to calculate the y-coordinate (0, m n) If the equation takes the form y = (x a) 2 + b then substitute x = 0 into the function to calculate the y-coordinate. Find the coordinates of the zeros or roots If the equation takes the form y = (x m)(x n) then the roots are (m,0) and (n,0). If the equation takes the form y = ax 2 + bx + c you must factorise the expression first. Find the equation of the axis of symmetry Find the value in the middle of the zeros by inspection or calculating the average of the roots m + n 2. State the equation in the form x = m + n 2. If the equation takes the form y = (x a) 2 + b then the equation of the axis of symmetry is x = a. Find the coordinates of the turning point The x-coordinate is the value in the middle of the roots. Substitute for x into the equation of the function to determine the y-coordinate. If the function is in the form y = (x a) 2 + b then the turning point is (a, b). Sketching the graph Identify the shape of the function. Identify the nature of the turning point. Draw a set of axes. Plot the points for the roots, turning point and y-intercept. (You may not always know the coordinates of the roots.) Bearing in mind the shape, sketch the graph. Write the coordinates beside the roots, turning point and y-intercept on your graph. Label your graph with its equation e.g. y = x 2 +2x 3 Function Notation A function can be expressed as an equation e.g. y = x 2 +6x 16 or in function notation e.g. f(x) = x 2 +6x 16. Functions have a domain the set of input values. Functions have a range the set of output values. A function is a rule which maps each input value to exactly one output value.

9 TOPIC 1. GRAPHS OF QUADRATIC FUNCTIONS AND SOLVING QUADRATIC FUNCTIONS 5 Solving quadratic equations Solving a quadratic equation graphically The zeros or roots of a quadratic are the point(s) where the graph crosses the x-axis and will take the form (p,0) and (q,0). The solution is x = p and x = q. Solving a quadratic equation by factorising A quadratic equation of the form y = ax 2 + bx + c must be factorised to take the form y = (x p)(x q). The roots are (p,0) and (q,0). To calculate the solution pull the brackets apart: x p = 0 and x q = 0. The solution is x = p and x = q. Solving a quadratic equation using the quadratic formula From a quadratic equation of the form y = ax 2 + bx + c you must identify the values of a, b and c. Substitute a, b and c into the quadratic formula. x = b± b 2 4ac 2a Remember to split the expression into: x = b+ b 2 4ac 2a x = b b 2 4ac 2a and Identifying and interpreting the nature of the roots of a quadratic using the discriminant From a quadratic equation of the form y = ax 2 + bx + c you must identify the values of a, b and c. The discriminant of a quadratic is b 2 4ac. If b 2 4ac > 0 (positive), the roots are real and distinct. There are two solutions. If b 2 4ac = 0, the roots are real and equal. There is one solution. There are two really but they are both the same. If b 2 4ac < 0 (negative), there are no real roots.

10 6 TOPIC 1. GRAPHS OF QUADRATIC FUNCTIONS AND SOLVING QUADRATIC FUNCTIONS There are no solutions. If b 2 4ac is positive and a square number, the roots are real and rational. If b 2 4ac is positive and not a square number, the roots are real and irrational.

11 TOPIC 1. GRAPHS OF QUADRATIC FUNCTIONS AND SOLVING QUADRATIC FUNCTIONS Assessment practice Make sure that you have read through the learning points or completed some revision before attempting these questions. Tailor your practice by choosing the most appropriate questions. The Theorem of Pythagorus: Questions 1 to 25 Vectors: Questions 26 to 47 Key point Note: None of these questions assess your reasoning skills. Assessment practice: Graphs of quadratic functions and solving quadratic functions Identifying features of a quadratic function A quadratic has the equation y = 3x 2 8x 6. Go online Q1: Identify the shape of the quadratic function. a) b) Q2: Identify the nature of the quadratic function. a) Maximum b) Minimum Q3: Identify the coordinates of the y-intercept. A quadratic has the equation y = (x 2)(x +6). Q4: What are the coordinates of the Zeros? Q5: What is the equation of the line of symmetry? Q6: What are the coordinates of the turning point?

12 8 TOPIC 1. GRAPHS OF QUADRATIC FUNCTIONS AND SOLVING QUADRATIC FUNCTIONS A quadratic has the equation y = x 2 +5x 14. Q7: Identify the coordinates of the Zeros. Q8: Identify the equation of the line of symmetry. Q9: Identify the nature of the turning point. Q10: Identify the coordinates of the turning point. A quadratic function has the equation y = (x 4) Q11: Find the equation of the axis of symmetry of the parabola. Q12: Find the coordinates of the turning point. Q13: What is the nature of the turning point? Identifying equations of quadratic graphs The diagram shows the graph of a quadratic function of the form y = kx 2. Q14: What is the value of k?

13 TOPIC 1. GRAPHS OF QUADRATIC FUNCTIONS AND SOLVING QUADRATIC FUNCTIONS 9 The diagram shows the graph of a quadratic function of the form y = (x a) 2 + b. Q15: What is the equation of the quadratic in the form y = (x a) 2 + b? The diagram shows the graph of a quadratic function of the form y = (x a) 2 + b. Q16: What is the equation of the quadratic in the form y = (x a) 2 + b? Sketching the graph of a quadratic function A quadratic function has equation y = x 2 +7x +12. Q17: What are the coordinates of the roots? Q18: What are the coordinates of the y-intercept? Q19: What is equation of the axis of symmetry?

14 10 TOPIC 1. GRAPHS OF QUADRATIC FUNCTIONS AND SOLVING QUADRATIC FUNCTIONS Q20: What is the nature of the turning point? Q21: What are the coordinates of the turning point? Q22: What is the shape of the function? a) b) Function notation Q23: f(x) = 3 2x x 2, find the value of f(4). Q24: g(x) = 3x 2 2x, find the value of f( 1). Q25: h(x) = 7x 3, find x when f(x) = 11. Solving quadratic equations graphically Q26: The diagram shows the graph of the function y = x 2 +3x +2. Use the graph to solve the equation y = x 2 +3x +2.

15 TOPIC 1. GRAPHS OF QUADRATIC FUNCTIONS AND SOLVING QUADRATIC FUNCTIONS 11 Q27: The diagram shows the graph of the function y = 6 x x 2. Use the graph to solve the equation y = 6 x x 2. Solving quadratic equations by factorising Q28: Factorise x 2 +4x 12. Q29: Solve the equation x 2 +4x 12. Q30: Factorise 3 2x x 2. Q31: Solve the equation 3 2x x 2. Q32: Factorise 4x 2 +8x + 3. Q33: Solve the equation 4x 2 + 8x + 3. Solving quadratic equations using the quadratic formula Q34: Solve x 2 +6x +2=0using the quadratic formula, giving your answer correct to 1 decimal place.. Q35: Solve x 2 6x 8 = 0 using the quadratic formula, giving your answer correct to 1 decimal place.

16 12 TOPIC 1. GRAPHS OF QUADRATIC FUNCTIONS AND SOLVING QUADRATIC FUNCTIONS The discriminant Q36: Determine the nature of the roots of 5x 2 + 7x +52. Q37: Determine the nature of the roots of 3x 2 + 5x 2. Q38: Determine the nature of the roots of 2x 2 8x +8. Q39: Determine the nature of the roots of 5x 2 7x 4. Q40: Determine the nature of the roots of 2x 2 + 3x Q41: Determine the nature of the roots of 7x 2 2 Q42: Determine the nature of the roots of 12 5x +3x 2 Q43: Determine the nature of the roots of 2+x 5x 2 Q44: Find the range of values of k such that the equation kx 2 4x +2=0, k 0, has real and distinct roots. Q45: Find the range of values of q such that the equation 3x x + q = 0, has real and equal roots. Q46: Are the roots of the equation 3x 2 +2x 5 = 0 are real and rational? Q47: Are the roots of the equation 5x 2 x 7 = 0 are real and irrational?

17 ANSWERS: TOPIC 6 13 Answers to questions and activities 6 Graphs of quadratic functions and solving quadratic functions Assessment practice: Graphs of quadratic functions and solving quadratic functions (page 7) Q1: a) Q2: b) Minimum Q3: (0,-6) Q4: (2,0) and (-6,0) Q5: x = 2 Q6: (-2,-16) Q7: (-7,0) and (2,0) Q8: x = 2 5 Q9: Minimum Q10: (-2 5,-20 25) Q11: x = 4 Q12: (4,2) Q13: Maximum Q14: Hint: Identify the coordinates of a point on the graph but not (0,0) and substitute into the equation. Answer: k = 3 Q15: Steps: What are the coordinates of the turning point? (-6,2) Answer: y = (x + 6) 2 +2 Q16: Steps:

18 14 ANSWERS: TOPIC 6 What are the coordinates of the turning point? (3,2) Answer: y = (x 3) 2 2 Q17: Hint: Factorise the expression. Answer: (-3,0) and (-4,0) Q18: (0,12) Q19: x = 3 5 Q20: Minimum Q21: (-3 5,-0 25) Q22: a) Q23: f(4) = 21 Q24: f( 1) = 5 Q25: x = 2 Q26: x = 2 and x = 1 Q27: x = 3 and x = 2 Q28: (x +6)(x 2) Q29: x = 6 and x = 2 Q30: (x + 3)(1 x) Q31: x = 3 and x = 1 Q32: (2x + 3)(2x + 1) Q33: x = 0 5 and x = 1 5 Q34: Steps: x = b± b 2 4ac 2a What is a? 1 What is b? 6

19 ANSWERS: TOPIC 6 15 What is c? 2 Substitute them into the quadratic formula and put it carefully into your calculator. Answer: x = 5 6 and x = 0 4 Q35: Steps: x = b± b 2 4ac 2a What is a? 1 What is b? 6 What is c? 8 Substitute them into the quadratic formula and put it carefully into your calculator. Answer: x = 7 1 and x = 1 1 Q36: Steps: What is a? 5 What is b? 7 What is c? 52 b 2 4ac What is the discriminant? 991 Interpret this to identify the answer. Answer: There are no real roots as b 2 4ac < 0. Q37: The roots are real and distinct as b 2 4ac = 49 which is greater than 0. Q38: The roots are real and equal as b 2 4ac = 0. Q39: The roots are real and distinct as b 2 4ac = 129 which is positive. Q40: The roots are real and distinct as b 2 4ac = 9 which is greater than 0. Q41: The roots are real and distinct as b 2 4ac = 56 which is greater than 0. Q42: There are no real roots as b 2 4ac = 119 which is less than 0. Q43: The roots are real and distinct as b 2 4ac = 41 which is greater than 0. Q44: Hint: For real and distinct roots b 2 4ac > 0.

20 16 ANSWERS: TOPIC 6 When a = k, b = 4 and c = 2 we get, ( 4) 2 4 k 2 > k > 0 8k > 16 k < 16 8 k <? Answer: k < 2 Q45: Hint: For real and equal roots b 2 4ac = 0. When a = 3, b = 4 and c = q we get, q = q = 0 12q = 144 q =? Answer: q = 12 Q46: Hint: b 2 4ac = ( 5) = = 64 The discriminant is positive and a square number (i.e. 8 2 = 64) so the roots of 3x 2 +2x 5 = 0 are real and rational. Answer: Yes Q47: Hint: b 2 4ac = ( 1) ( 7) = = 141 The discriminant is positive but not a square number so the roots of 5x 2 x 7 = 0 are real and irrational. Answer: Yes