CHAPTER 1: QUADRATIC FUNCTIONS AND EQUATIONS IN ONE VARIABLE
1.1.1 QUADRATIC EXPRESSIONS
Quadratic expressions are expressions which fulfill the following characteristics:
a) have only one variable
b) have 2 as the highest power of the variable
Quadratic expressions with three terms are expressions of the form ax2 + bx + c, where a 
0, b  0 and c  0
1. State whether each of the following expressions is a quadratic expression or not.
a) x2 + 4x – 7
b) 8d – 7
c) 4y – y2 + 10
d) 2p3 – 4p
e) 6m2
f)
4 − u2
2+u
1.1.2 QUADRATIC FUNCTIONS
A quadratic function is of the form y = ax2 + bx + c, where a, b and c are constants and a  0.
The value of c is the y-intercept of the graph
The graph of a quadratic function is in the
shape of a parabola
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1.1.3 THE EFFECT OF CHANGING THE VALUE OF a ON GRAPHS OF
QUADRATIC FUNCTIONS
The value of a
The value of b determines the position of the axis of symmetry
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The value of c determines the position of the y-intercept.
Exercise:
1. Sketch the effect of changing the value of a on the graphs of quadratic functions. Hence,
describe the changes.
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2. Sketch the effect of changing the value of b on the graphs of quadratic functions. Hence,
state the changing of axis of symmetry of the quadratic function.
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3. Sketch the effect of changing the value of c on the graphs of quadratic functions.
Hence, state the y-intercept of the quadratic function.
1.1.4 FORM QUADRATIC FUNCTIONS BASED ON REAL SITUATIONS
1. Form a quadratic equation based on the information given .
a) Ali is 8 years older than Ibrahim and the product of their ages is 240
Ans: x2 + 8x – 240 = 0
b) ABC is a right-angled triangle. It has an area of 10 cm2.
A
(x – 3)cm
B
2x cm
C
Ans : x2 – 3x – 10 = 0
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c) PQRS is a square with a circle inside it. The radius of the circle is 7 cm and the area of the
shaded region is 6x cm2
Ans: 2x2 – 3x – 77 = 0
d) PQRS is a rectangle. QA = SB = 3 cm and the area of PQRS is 54 cm2.
Q
P
A
(x + 5) cm
=
S
B
Ans : x2+ 13x -14 = 0
1.1.5 ROOTS OF A QUADRATIC EQUATION
1. Determine whether the given value is a root of the quadratic equation
a) x2 – 13x + 42 = 0 [ x = 6 ]
b) 5y2 -11y – 12 = 0 [ y = 3 ]
Ans : a) x = 6 is a root b) y = 3 is a root
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R
c) 3x2 – 7x – 6 = 0 [ x = 2 ]
d) y(y + 5) = 20
[ y = -5 ]
Ans : c) x = 2 is not a root d) y = -5 is not a root
2. Solve the following quadratic equations.
a) x2 – 9x = 0
b) 2d2 – 6d = 0
Ans : a) x = 0, x = 9 b) d = 0. d = 3
d) n2 – 81 = 0
c) 5h = 10h2
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Ans : c) h = 0, h = 2, d) n = 9, n = -9
e) 2x2 – 72 = 0
f) 3p2 = 75
Ans : e) x = 6, x = -6 f) p = 5, p = -5
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3. Solve the following quadratic equations.
a) x2 + 13x – 30 = 0
b) y2 – 5y + 6 = 0
Ans : a) x = 2, -15 b) y = 2, 3
c) n2 – 18 = 3n
d) 4q2 + 3q = 10
5
Ans : c) n = 6, -3 d) q = 4, -2
e) 45 – 4x = x2
f) 2y2 = 5y + 63
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Ans : e) x = 5,-9 f) y = - 2, 7
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4. ABCD is a rectangle. Given the area of the rectangle is 110 cm2.
a) Write a quadratic equation in terms of x
Ans : x2 + 7x -98 = 0
b) Hence, solve the equation for the value of x.
Ans : x = 7
D
A
(x + 3) cm
B
C
(x + 4) cm
5.
P
PQR is a right-angled triangle. Based on the diagram given,
a) write a quadratic equation in terms of x
b) hence, find the length, in cm, of PR
Ans :a) 7x2 – 26x – 45 = 0
b) 13 cm
(3x – 2) cm
x cm
Q
(x + 7) cm
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1.1.6 SKETCH GRAPHS OF QUADRATIC FUNCTIONS
1. Sketch the graph for each of the following quadratic functions.
a) y = 2x2
b) y = - 3x2
c) y = x2 + 16
d) y = 9 – x2
e) y = (x + 1)(x – 5)
f) y = 10 – 3x – x2
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SPM PRACTICE
1. A ball is thrown from a building to the ground. The height, h, in meters, of the ball at time t
seconds after the throw is h = 10 + t – 2t2. How many seconds does the ball take to reach the
ground?
Ans : 2.5 seconds
2. Solve the following quadratic equation
x=
x(3x − 2)
=5
x+2
Ans :
10
, −1
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3. An aquarium has a height of (x + 5) cm, width of x cm and length of 60 cm. The total
volume of the aquarium is 30 000 cm3. The aquarium will be filled fully with water.
Calculate the value of x.
Ans : 20 cm
4. A (p + 1) metres long ladder reaches a height of (2p – 5) metres when it is leaned against a
wall. Given that the distance between the foot of the ladder and the wall is p metres, find the
value of p.
Ans : p = 4
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5. Two boys are playing volleyball at a beach. The height, h in meters, of the ball at t seconds
after being hit is h = -5t2 + 11t + 1. Find the possible times the ball will reach the height of 3
metres.
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Ans : t = , 2
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6. The following diagram shows a piece of square cardboard PQRS with length 6x cm. A
semicircle KLM with diameter 14 cm is cut out from the square cardboard and the remaining
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area is 241x cm. Using  = , calculate the length, in cm, of the cardboard.
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Ans : 42 cm
P
Q
L
S
R
K
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M
7. The following diagram shows the cross-section of a ladder leaning against a wall and
touching a cupboard under it. Given the height and the depth of the cupboard are x m and
m respectively, find the value of x.
Ans : x =
5
2
8. The shaded region in the diagram below is the remaining shape when a semicircle with
diameter 28 cm is cut and removed from a piece of rectangular cardboard.
Given the area of the shaded region is 61x cm2. Calculate the length, in cm, of PQ.
Ans : 21 cm
13
5
x