Page 19 - Focus SPM KSSM Tg 4.5 - Add Maths

Additional Mathematics SPM Chapter 2 Quadratic Functions

(c) The graph of f (x) intersects the x-axis at two This function has two equal roots and the graph
different points, therefore f (x) = 0 has two different touches the x-axis at one point.
roots.
(d) The graph of f (x) does not intersect or touches the
x-axis, therefore f (x) = 0 has no roots.

Try Question 3 in ‘Try This! 2.3’
Try Question 4 in ‘Try This! 2.3’
16
Determine the types of roots for the following 17
quadratic functions. Hence, sketch the position of the Given that the quadratic function f (x) = x + 2x + 1 – k
2
graph of the function with respect to the x-axis. intersects the x-axis at two different points. Find the
(a) f(x) = 3x – 4x – 2 range of values of k.
2
Form 4
(b) f(x) = –5x + 3x – 1
2
(c) f(x) = x – 8x + 16 Solution
2
f (x) = x + 2x + 1 – k
2
Solution a = 1, b = 2 and c = 1 – k
(a) f(x) = 3x – 4x – 2
2
a = 3, b = –4, c = –2 f (x) = 0 has two different roots when
b – 4ac . 0

2
b – 4ac = (–4) – 4(3)(–2) (2) – 4(1)(1 – k) . 0
2
2
2
= 40 . 0 4 – (4 – 4k) . 0
a = 3 . 0, therefore the graph of the function is in 4 – 4 + 4k . 0
the shape of . 4k . 0
This function has two different roots and the k . 0
graph intersects the x-axis at two points.
Try Questions 5 – 6 in ‘Try This! 2.3’
18
Given that the quadratic function
1
(b) f(x) = –5x + 3x – 1 f (x) = p x + (p – 1)x + — does not intersect the x-axis.
2
2 2
4
a = –5, b = 3, c = –1 Find the range of values of p.
b – 4ac = 3 – 4(–5)(–1)
2
2
= –11  0 Solution 1
2 2
a = –5  0, therefore the graph of the function is p x + (p – 1)x + — = 0
4
1
in the shape of . a = p , b = p – 1 and c = —
2
This function has no real roots and the graph does 4
not intersect the x-axis. f (x) = 0 does not have real roots when
b – 4ac  0
2
1
 
(p – 1) – 4(p ) —  0
2
2
4
p – 2p + 1 – p  0
2
2
–2p + 1  0
1  2p
(c) f(x) = x – 8x + 16 1
2
a = 1, b = –8, c = 16 —  p
2
1
b – 4ac = (–8) – 4(1)(16) p . —
2
2
= 0 2
a = 1 . 0, therefore the graph of the function is in Try Questions 7 – 10 in ‘Try This! 2.3’
the shape of .
42