SCHOLAR Study Guide National 5 Mathematics Assessment ... · Solving quadratic equations Solving a quadratic equation graphically • The zeros or roots of a quadratic are the point(s)

SCHOLAR Study Guide

National 5 MathematicsAssessment Practice 6:Graphs of quadratic functions and solv-ing quadratic functions (Topic 18-19)

Authored by:Margaret Ferguson

Heriot-Watt University

Edinburgh EH14 4AS, United Kingdom.

First published 2014 by Heriot-Watt University.

This edition published in 2017 by Heriot-Watt University SCHOLAR.

Copyright © 2017 SCHOLAR Forum.

Members of the SCHOLAR Forum may reproduce this publication in whole or in part for educationalpurposes within their establishment providing that no profit accrues at any stage, Any other use of thematerials 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 transmittedin 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 informationcontained in this study guide.

Distributed by the SCHOLAR Forum.

SCHOLAR Study Guide Assessment Practice: National 5 Mathematics

1. National 5 Mathematics Course Code: C747 75

AcknowledgementsThanks are due to the members of Heriot-Watt University's SCHOLAR team who planned and created thesematerials, and to the many colleagues who reviewed the content.

We would like to acknowledge the assistance of the education authorities, colleges, teachers and studentswho contributed to the SCHOLAR programme and who evaluated these materials.

Grateful acknowledgement is made for permission to use the following material in the SCHOLARprogramme:

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 aretrademarks, registered trademarks or service marks of their respective holders.

1

Topic 7

Graphs of quadratic functions andsolving quadratic functions

Contents7.1 Learning points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

7.2 Assessment practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 TOPIC 7. GRAPHS OF QUADRATIC FUNCTIONS AND SOLVING QUADRATICFUNCTIONS

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 CourseMaterials unit if you need more help.

Key point

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;

• recognize 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.

© HERIOT-WATT UNIVERSITY

TOPIC 7. GRAPHS OF QUADRATIC FUNCTIONS AND SOLVING QUADRATICFUNCTIONS

3

7.1 Learning points

Graphs of quadratic functionsFeatures of quadratic functions

• The shape of the graph of a quadratic will be smiley if the x2 term is positive

e.g. y = 5x2 looks like

• The shape of the graph of a quadratic will be sad if the x2 term is negative

e.g. y = − x2 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 willtake 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 = ax2 + bx + c.

Determining the equation of a quadratic function from its graph

• If the equation of the quadratic takes the form y = kx2:

◦ find the coordinates of a point on the graph (Note: you cannot use the origin);◦ substitute the values of x and y into y = kx2;◦ 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 ax2 + bx + c = 0 then (0,c).◦ 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 functionto 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 = ax2 + 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 theroots m + n

2 .

◦ State the equation in the form x = m + n2 .

◦ If the equation takes the form y = (x − a)2 + b then the equation of the axis of symmetryis 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 thecoordinates 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 = x2 + 2x − 3

Function Notation

• A function can be expressed as an equation e.g. y = x2 + 6x − 16 or in function notatione.g. f(x) = x2 + 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.



5

Solving quadratic equationsSolving a quadratic equation graphically

• The zeros or roots of a quadratic are the point(s) where the graph crosses the x-axis and willtake 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 = ax2 + bx + c must be factorised to take the formy = (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 = ax2 + bx + c you must identify the values of a,b and c.

• Substitute a, b and c into the quadratic formula.

• x = −b±√b2−4ac2a

• Remember to split the expression into:

◦ x = −b+√b2−4ac2a and

◦ x = −b−√b2−4ac2a

Identifying and interpreting the nature of the roots of a quadratic using the discriminant

• From a quadratic equation of the form y = ax2 + bx + c you must identify the values of a,b and c.

• The discriminant of a quadratic is b2 − 4ac.

• If b2 − 4ac > 0 (positive), the roots are real and distinct.

◦ There are two solutions.

• If b2 − 4ac = 0, the roots are real and equal.

◦ There is one solution.◦ There are two really but they are both the same.

• If b2 − 4ac < 0 (negative), there are no real roots.

◦ There are no solutions.

• If b2 − 4ac is positive and a square number, the roots are real and rational.

• If b2 − 4ac is positive and not a square number, the roots are real and irrational.



7.2 Assessment practice

Make sure that you have read through the learning points or completed some revision beforeattempting 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.

Go onlineAssessment practice: Graphs of quadratic functions and solvingquadratic functions

Identifying features of a quadratic function

A quadratic has the equation y = 3x2 − 8x − 6.

Q1: Identify the shape of the quadratic function.

a)

b)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Q2: Identify the nature of the quadratic function.

a) Maximumb) 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?



7

A quadratic has the equation y = x2 + 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)2 + 2.

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 = kx2.

Q14: What is the value of k?



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 = x2 + 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?

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .



9

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 − x2, find the value of f(4).

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Q24: g(x) = 3x2 − 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 = x2 + 3x + 2.

Use the graph to solve the equation y = x2 + 3x + 2.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .



Q27:

The diagram shows the graph of the function y = 6 − x − x2.

Use the graph to solve the equation y = 6 − x − x2.

Solving quadratic equations by factorising

Q28: Factorise x2 + 4x − 12.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Q29: Solve the equation x2 + 4x − 12.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Q30: Factorise 3 − 2x − x2.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Q31: Solve the equation 3 − 2x − x2.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Q32: Factorise 4x2 + 8x + 3.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Q33: Solve the equation 4x2 + 8x + 3.

Solving quadratic equations using the quadratic formula

Q34: Solve x2 + 6x + 2 = 0 using the quadratic formula, giving your answer correct to 1decimal place..

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Q35: Solve x2 − 6x − 8 = 0 using the quadratic formula, giving your answer correct to 1decimal place.



11

The discriminant

Q36: Determine the nature of the roots of 5x2 + 7x + 52.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Q37: Determine the nature of the roots of 3x2 + 5x − 2.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Q38: Determine the nature of the roots of 2x2 − 8x + 8.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Q39: Determine the nature of the roots of 5x2 − 7x − 4.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Q40: Determine the nature of the roots of 2x2 + 3x

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Q41: Determine the nature of the roots of 7x2 − 2

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Q42: Determine the nature of the roots of 12 − 5x + 3x2

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Q43: Determine the nature of the roots of 2 + x − 5x2

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Q44: Find the range of values of k such that the equation kx2 − 4x + 2 = 0, k �= 0, hasreal and distinct roots.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Q45: Find the range of values of q such that the equation 3x2 + 12x + q = 0, has real andequal roots.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Q46: Are the roots of the equation 3x2 + 2x − 5 = 0 are real and rational?

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Q47: Are the roots of the equation 5x2 − x − 7 = 0 are real and irrational?


12 ANSWERS: UNIT 2 TOPIC 6

Answers to questions and activities

Topic 6: Graphs of quadratic functions and solving quadratic functions

Assessment practice: Graphs of quadratic functions and solving quadratic functions (page6)

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 · 5Q9: Minimum

Q10: (-2·5,-20·25)

Q11: x = 4

Q12: (4,2)

Q13: Maximum

Q14:

Hints:

• 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


ANSWERS: UNIT 2 TOPIC 6 13

Q16:

Steps:

• What are the coordinates of the turning point? (3,2)

Answer: y = − (x − 3)2 − 2

Q17:

Hints:

• Factorise the expression.

Answer: (-3,0) and (-4,0)

Q18: (0,12)

Q19: x = − 3 · 5Q20: 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±√b2−4ac2a

• What is a? 1


14 ANSWERS: UNIT 2 TOPIC 6

• What is b? 6

• What is c? 2

• Substitute them into the quadratic formula and put it carefully into your calculator.

Answer: x = − 5 · 6 and x = − 0 · 4Q35:

Steps:

• x = −b±√b2−4ac2a

• 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

• b2 − 4ac

• What is the discriminant? −991

• Interpret this to identify the answer.

Answer: There are no real roots as b2 − 4ac < 0.

Q37: The roots are real and distinct as b2 − 4ac = 49 which is greater than 0.

Q38: The roots are real and equal as b2 − 4ac = 0.

Q39: The roots are real and distinct as b2 − 4ac = 129 which is positive.



Q42: There are no real roots as b2 − 4ac = − 119 which is less than 0.


Q44:

Hints:

• For real and distinct roots b2 − 4ac > 0.


ANSWERS: UNIT 2 TOPIC 6 15

• When a = k, b = − 4 and c = 2 we get,(−4)2 − 4 × k × 2 > 0

16 − 8k > 0

−8k > −16

k <−16

−8

k < ?

Answer: k < 2

Q45:

Hints:

• For real and equal roots b2 − 4ac = 0.

• When a = 3, b = − 4 and c = q we get,122 − 4 × 3 × q = 0

144 − 12q = 0

−12q = −144

q = ?

Answer: q = 12

Q46:

Hints:

• b2 − 4ac = 22 − 4 × 3 × (−5)

= 4 + 60

= 64

• The discriminant is positive and a square number (i.e. 82 = 64) so the roots of 3x2 + 2x − 5 =0 are real and rational.

Answer: Yes

Q47:

Hints:

• b2 − 4ac = (−1)2 − 4 × 5 × (−7)

= 1 + 140

= 141

• The discriminant is positive but not a square number so the roots of 5x2 − x − 7 = 0 arereal and irrational.

Answer: Yes


Documents

SCHOLAR Study Guide National 5 Mathematics Assessment ... · Solving quadratic equations Solving a quadratic equation graphically • The zeros or roots of a quadratic are the point(s)