2 Quadratic Functions
C H A P T ER
Topics
Key words
2.1 The quadratic function
intercept, maximum point, minimum point, quadratic function, root, turning point
2.2 Completing the square
coefficient, completing the square,
2.3 Domain and range
range
2.4 The discriminant
discriminant, real roots, soluble, square root
2.5 Intersecting graphs
chord, Intersection, tangent
2.6 Solving quadratic equations
difference of two squares, factors, quadratic equations, quadratic formula, surds
2.7 Quadratic inequalities
number line, quadratic inequality, range
In this chapter you will learn how to • Solve and manipulate quadratic functions and equations and how these relate to quadratic graphs and their maximum and minimum points. • Find the maximum or minimum value of the quadratic function f : x → ax2 + bx + c by sketching • Find the maximum or minimum value of the quadratic function f : x → ax2 + bx + c by completing the square • Use the maximum or minimum value of f(x), xx), where one of a, b or c is unknown to find work out f(x) xx) and sketch the graph or determine the range for a given domain. • Know the conditions for f(x) x = 0 to have (i) two different real roots, (ii) two equal roots, x) (iii) no real roots • Know how the number of real roots relate to a given line (i) intersecting a given quadratic curve twice, (ii) being a tangent to a quadratic curve, (iii) not intersecting a quadratic curve • Solve quadratic equations for real roots using (i) factorisation, (ii) the quadratic formula, (iii) completing the square • Solve a quadratic inequality and determine the domain for which certain inequalities are valid.
From your IGCSE Mathematics course you should be able to •
recognise, sketch and interpret graphs of quadratic functions
•
identify and interpret roots, intercepts and turning points of quadratic functions graphically
•
know the symmetrical property of a quadratic.
Starting point This topic extends what you have learnt about quadratic functions in IGCSE Maths. Before you move on to look in more detail at quadratic functions, review what you know already about graphs of quadratic functions. 1. Here is a table of values for the graph of y = x2 + x – 6 for values of x from –4 to 3 x
–4
–3
–2
–1
0
1
2
3
y
6
0
–4
6
–6
–4
0
6
a Sketch the graph. b On your graph show i
the value of the intercept on the y-axis
ii
the values of the roots
iii the line of symmetry of the graph and its equation
F
iv the coordinates of the minimum value c Factorise x2 + x – 6
Why this chapter matters What have the height of a ball thrown upwards, the shape of radio telescopes and the total resistance of two resistors placed in parallel got in common? They can all be modelled using quadratic functions. The shape of a quadratic curve is called a parabola. All parabolas are symmetric. Parabolas have the unique property that all light rays coming into a parabola parallel to the line of symmetry will reflect to the same point (called the focus). Also any light from the focus will be sent out parallel to the line of symmetry. This is why car headlights are parabolas. Quadratic functions can also be used to predict the selling price of a new product that will give the greatest profit based on predicted sales and production costs. This section will show you how to solve, sketch and manipulate quadratic functions and solve problems like those shown above.
Exploring the topics As you start to explore this topics, discuss possible answers to the following questions in pairs: a. Think of some examples of parabolas in architecture and transport. b. How could you use quadratic functions to map a journey up and back down a river? c. How might businesses use quadratic functions to plan sales?
Chapter 2: Quadratic Functions
3
Chapter 2 . Topic 1
2.1 The Quadratic Function All quadratic functions are of the form ax2 + bx + c, where a can take any positive or negative value except zero and b and c can take any positive or negative value including zero. When the graph of a quadratic function, y = ax2 + bx + c is drawn the graph will have a characteristic shape, called a parabola. You should already be able to identify the significant points of a quadratic graph. These are: the intercept on the y-axis, the intercepts on the x-axis, if any (if there are intercepts these are called the roots) and the maximum or minimum value (also called the turning point). All parabolas are symmetric and you should also be able to give the equation of the line of symmetry.
Special cases Use a graphical calculator or a graphing program on a computer to investigate the following situations. Look for a relationship between the significant points and the different values of a, b and c. a = 1, b = 0, c = 0, ie y = x2 1 2 x 2 You are advised to use integer values for the following investigations. a > 1, b = 0, c = 0, eg y = 2x2 or y = 6x2 or y =
a < 0, b = 0, c = 0, eg y = −3x2 or y = −x2 a = 1, b = 0, c ≠ 0, eg y = x2 + 2 or y = x2 − 1 a = 1, b ≠ 0, c = 0, eg y = x2 + 3x or y = x2 − 4x a ≠ 0, b ≠ 0, c = 0, eg y = 2x2 + x or y = −3x2 + 2x
Exercise 3A 1
Sketch the following graphs. Mark the values of the significant points and draw and label the line of symmetry. y = x2 − 3 c y = x2 + 4x a y = −x2 b d y = 2x2 − x e y = 3x2 − 1 f y = −2x2 + 5x
PS
4
2
Four sketches of graphs are shown. On each sketch the values of the roots are shown and the coordinates of the minimum or maximum points are given.
2.1 The Quadratic Function
2.1 In each case work out the equation of the graph. a
b
y
0
0
6
y
x 0
–2
(–1, –1)
(–3, –9) c
d
y
0
4
0
x
0
y
(3, 9)
x
0
0
6
x
(2, –24) PS
3
Some properties of quadratic graphs are given. In each case use the information to work out the equation of the graph. a Roots 0 and 2. Minimum point (1, −1) b One root 0. Line of symmetry x = 5. Minimum point (5, −25) c One root −4. Line of symmetry x = −2. Minimum point (−2, −12) d Roots 0 and 6. Maximum point (3, 18)
The general case The general form of the quadratic function is ax2 + bx + c. Later you will see how to find the significant points algebraically. At the moment we are drawing the graphs using a table of values or a graphical plotting calculator or program. Use a graphical calculator or graphing program to draw the following graphs. y = x2 − 2x − 8
y = 2x2 − x − 1
y = x2 − 7x + 10
y = 6x2 + x − 2
Chapter 2: Quadratic Functions
5
In each case write down, the coordinates of the significant points and the equation of the line of symmetry. What is the relationship between the constant term c and the coordinate of the point where the graph intersects the y-axis? What is the relationship between the roots and the equation of the line of symmetry? What is the relationship between the sum of the roots and a and b? What is the relationship between the product of the roots and a and c?
Exercise 3B 1
The graph of y = x2 −3x − c crosses the x-axis at (4, 0) and (−1, 0). a Write down the equation of the line of symmetry. b Write down the coordinates of the intersection with the y-axis. c Work out the coordinates of the minimum point.
2
One root of the graph of y = x2 + 2x − 8 is 2. Work out the other root.
3
A quadratic graph has a line of symmetry x = 5 One root is 3.5 Work out the other root.
PS
4
One root of the graph y = x2 + bx − 7 is 1. Work out the value of b.
5
1 The graph of y = 2x2 + 5x − c crosses the x-axis at (−3, 0) and ( , 0). 2 a Write down the equation of the line of symmetry. b Write down the coordinates of the intersection with the y-axis. c Work out the coordinates of the minimum point.
6
One root of the graph of y = 3x2 − 5x − 2 is 2. Work out the other root.
PS
7
One root of the graph of y = 4x2 + x − c is 1. a Work out the other root. b Work out the value of c.
PS
8
One root of the graph y = 6x2 + bx − 2 is a Work out the other root b Work out the value of b.
6
2.1 The Quadratic Function
1 . 2
2.2 Completing the square (x + a)2 = x2 + 2ax + a2
Advice and Tips
Rearranging gives x + 2ax = (x + a) − a 2
2.2
2
2
When you have to square a bracket, always write down the bracket twice and then use whichever method you prefer for expanding two brackets, ie (x + a)2 = (x + a)(x + a)
Hence, for example x2 + 10x = (x + 5)2 − 25 and x2 + 10x + 9 = (x + 5)2 − 25 + 9 = (x + 5)2 − 16 b b So we can say that x2 + bx + c = (x + )2 − ( )2 + c 2 2 This is the method of completing the square.
Example 1 a Write the expression x2 + 6x − 16 in the form (x + p)2 − q b Use a graphical calculator or program to identify the minimum point of y = x2 + 6x − 16 a Using the rule above x2 + 6x − 16 = (x + 3)2 − 9 − 16 = (x + 3)2 − 25 b The minimum point is (−3, −25)
Example 2 a Write the expression x2 − 4x − 5 in the form (x + p)2 − q b Write down minimum point of y = x2 − 4x − 5 a Using the rule above x2 − 4x − 5 = (x − 2)2 − 4 − 5 = (x − 2)2 − 9 b The minimum point is (2, −9) Both examples above have a unit coefficient of x2. When a > 1 the method is a little more complicated.
Example 3 a Write the expression 2x2 + 4x − 1 in the form r(x + p)2 − q b Use a graphical calculator or program to identify the minimum point of y = 2x2 + 4x − 1 1 ] 2 3 Now complete the square for the expression inside the bracket 2[(x + 1)2 − ] 2 Now expand the bracket 2(x + 1)2 − 3
a First factorise out the coefficient of x2 2x2 + 4x − 1 = 2[x2 + 2x −
b The minimum point is (− 1, −3)
Chapter 2: Quadratic Functions
7
Example 4 a Write the expression 6x2 − x − 2 in the form r(x + p)2 − q b Write down the minimum point of y = 6x2 − x − 2 1 1 x− ] 6 3 1 2 49 Now complete the square for the expression inside the bracket 2[(x − ) − ] 144 12 1 2 49 Now expand the bracket 6(x − ) − 12 24 1 49 b The minimum point is ( , − ) 12 24
a First factorise out the coefficient of x2 6x2 − x − 2 = 6[x2 −
Exercise 3C 1
Write each of the following in the form (x − p)2 + q x2 + 8x − 1 c x2 − 4x + 2 a x2 + 6x − 7 b d x2 + 2x − 3 e x2 − 6x + 10 f x2 − 3x + 1
2
Write each of the following in the form r(x − p)2 + q 3x2 + 6x + 1 c 2x2 − 8x + 5 a 2x2 + 4x − 5 b d 5x2 + 20x − 7 e 2x2 − 6x + 1 f 2x2 − 5x − 3
3
Write down the minimum points of the following graphs (use your answers to questions 1 and 2 to help you) x2 − 4x + 2 c x2 − 6x + 10 a x2 + 6x − 7 b d 2x2 + 4x − 5 e 3x2 + 6x + 1 f 2x2 − 5x − 3
Progression checklist I can find the trigonometrical function for angles of any size I can draw graphs of y = a sin (bx) + c y = a cos (bx) + c y = a tan (bx) + c I can use trigonometrical graphs to solve problems I know the special relationships between the trigonometrical functions I can solve simple trigonometrical equations I can prove simple trigonometrical identities
8
2.2 Completing the square
2.2 Chapter review π π ) = 1 for 0 ⩽ x ⩽ 3 4
1
Solve 2cos2 (3x −
2
Show that (sec2 θ −1) + (cosec2 θ −1) = sec θ ⋅ cossec θ
3
y
(4 marks) (5 marks)
5 4 3 2 1 0
0
p 2
p
3p 2
2p
x
The graph shows the graph of y = a + bsincx Find the value of a, b and c 4
Solve the following equations (i) 4sin2x + 5cos2x = 0
5
(3 marks)
0° ⩽ x ⩽ 180°
0° ⩽ x ⩽ 360° (ii) cot2y + 3cosec y = 3 π 1 (iii) cos(z + ) = − 0 ⩽ z ⩽ 2p 4 2 (i) Prove that sec2 x + cosec2 x = sec2 x · cosec2 x
(3 marks) (3 marks) (4 marks) (4 marks)
(ii) Hence or otherwise solve sec2 x + cosec2 x = 4tan2 x 90° ⩽ x ⩽ 270° 6
(4 marks)
(i) State the period of sin2x
(1 mark)
(ii) State the amplitude of 1 + 2cos3x
(1 mark)
(iii) Sketch the graph of a y = sin2x 0° ⩽ x ⩽ 180° b y = 1 + cos3x
0° ⩽ x ⩽ 180°
(4 marks)
(iv) State the number of solutions of sin2x − 2cos3x = 1
0° ⩽ x ⩽ 180°
(1 mark)
Chapter 2: Quadratic Functions
9
Answers to practice questions π π , 6 3 2 LHS = tanq + cotq →
1 0,
3 a = 3, b = 2, c = 4
1 → RHS sinθ ⋅ cos θ
5π 13π , 4 i 64.3°, 154.3° ii 90°, 194.5°, 345.5° iii 12 5 1 → RHS ii 135°, 225° 5 i LHS → sin2 x ⋅ cos 2 x 6 i 180° ii 2 iii
iv 3
7 i 1 + 2 2 ii 10 − 4 2 8 LHS →
sinθ → RHS sinθ ⋅ cos θ
9 a 15°, 45°, 75°, 105° b 71.6°, 153.4°, 251.6°, 33.4° c 10 y
−π π , 6 2
1
0
90
180
270
360
x
11 i LHS →
2 2 → → RHS 2 sin2 x (1 − cos x )
ii 30°, 150°, 210°, 330°
12 i LHS →
sin2 x → RHS cos x ⋅ sinx
ii 54.7°, 125.3°, 234.7°, 305.3°
13 a = 3
b=2
c=4
14 a 121°, 301° 15 a LHS → 16 LHS →
(sin2 θ + cos2 θ ) → RHS (sinθ + cos θ )
7π 23π 31π b 36 , 36 , 36 −12 5 b sin x = cos x = 13 13
sin2 θ (1 − cos2 θ ) → RHS cos2 θ
17 a i 705°, 120° ii 19.5°, 160.5° b
10
Answers to practice questions
π , 0.927rad 2