Page 17 - Ranger SPM 2022 - Additional Mathematics

Additional Mathematics SPM Chapter 2 Quadratic Functions

Solution Example 13
(a) f(x) = 2x – 6x + 3 If the graph of f(x) = 2x + (1 – p)x + 8,
2
2
where p is a constant, does not intersect the
b – 4ac = (–6) – 4(2)(3) x-axis, find the range of values of p.
2
2
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= 12  0
This function has two different real Solution
Form 4 a  0, so the shape of the graph f(x) f(x) = 2x + (1 – p)x + 8 The graph does not
roots.
2
b – 4ac  0
2
intersect the x-axis.
and it intersects the x-axis at two
is
2
points. (1 – p) – 4(2)(8)  0
1 – 2p + p – 64  0
2
p – 2p – 63  0
2
x (p + 7)(p – 9)  0
+ +
x
2
(b) f(x) = –9x + 12x – 4 –7 – 9
b – 4ac = 12 – 4(–9)(–4)
2
2
= 0 –7  p  9
This function has two equal real roots.
a  0, so the shape of the graph f(x) Example 14
is and it intersects the x-axis at one
point. Express the quadratic function
x f(x) = 3 x – 3  2 – 147 in intercept form,
4
2
f(x) = a(x – p)(x – q) where a, p and q are
constants and q  p. State the values of
a, p and q.
Solution
Example 12 9

f(x) = 3 x – 3x +  – 147
2
If the graph of f(x) = 2x + 2x + 1 – k, 4 4
2
2
where k is a constant, intersects the x-axis = 3x – 9x – 30 General form
2
at two different points, find the range of = 3(x – 3x – 10)
values of k. = 3(x + 2)(x – 5) Intercept form
Compare with f(x) = a(x – p)(x – q).
Solution
Thus, a = 3, p = –2, q = 5
f(x) = 2x + 2x + 1 – k The graph intersects
2
b – 4ac  0 the x-axis at two
2
different points.
2 – 4(2)(1 – k)  0 Example 15
2
4 – 8 + 8k  0 Express f(x) = –2x – 10x – 11 in the form
2
8k  4 f(x) = a(x – h) + k where a, h and k are
2
k  1 constants. Hence, determine the values of
2 a, h and k.
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