Page 26 - Focus SPM KSSM Tg 4.5 - Add Maths

Additional Mathematics SPM Chapter 2 Quadratic Functions

SPM Highlights 2. The diagram below shows the sketch of the graph
for f(x) = 5 + 3x – 2x .
2
f(x)
The curve of the quadratic function f (x) = 3(x – p) + 4q
2
intersects the x-axis at points (0, 0) and (4, 0). The line 5 f(x) = 5 + 3x – 2x 2
y = –12 touches the minimum point of the curve.
(a) Find the values of p and q. 0 x
(b) Hence, sketch the graph of f(x) for –1 < x < 5.
(c) If the graph is reflected on the x-axis, write the
equation of the curve. Analyse and make generalisations on the shape and
position of the graph of the given functions when
Solution compared to the values of a, b and c of the following
(a) f (x) = 3(x – p) + 4q. functions. Hence, sketch the graphs of the functions.
2
The minimum point is (p, 4q), such that p is the (a) f(x) = 5 – 3x – 2x 2
midpoint of 0 and 4. (b) f(x) = –1 + 3x – 2x 2 Form 4
(c) f(x) = 2x – 3x – 5
2
p = 4 + 0 = 2
2 3. Based on the graph for each of the following
4 q = –12 quadratic functions, state the types of roots of the
q = –3 functions.
(a) (b) f(x)
(b) f(x) f(x)
x
f(x) = 3(x – p) + 4q x
2

x (c) f(x) (d)
–1 0 1 2 3 4 5 f(x)
y = –12
–12 x

(c) f(x) = –3(x – 2) + 12 x
2
(e) f(x) (f) f(x)

x
x
Try This! 2.3

1. The diagram below shows the sketch of the graph 4. Determine the types of roots for each of the following
f(x) = x – 3x – 4. quadratic functions. Hence, sketch the position of
2
the graph of the function with respect to x-axis.
f(x) (a) f(x) = 4x – 2x – 5 (b) f(x) = –2x + 4x + 1
2
2
2
f(x) = x – 3x – 4 (c) f(x) = 3x – 4x + 2 (d) f(x) = 2 + 6x – 3x 2
2
(e) f(x) = 4x – 12x + 9 (f) f(x) = –1 + 4x – 4x 2
2
x
0
5. The graph of the quadratic function
–4 f(x) = 2x – 5x + p + 3 intersects the x-axis at two
2
different points. Find the range of values of p.
Analyse and make generalisations on the shape and 6. The graph of the quadratic function
2
position of the graph of the given functions when f(x) = qx + 2qx – 6 + q intersects the x-axis at two
compared to the values of a, b and c of the following different points. Find the range of values of q.
functions. Hence, sketch the graphs of the functions. 7. Show that the quadratic function f(x) = 3x – 4x + k + 1
2
(a) f(x) = x + 3x – 4 1
2
(b) f(x) = x + 3x + 2 does not intersect the x-axis for k . .
2
3
(c) f(x) = 3x – 3x – 4
2
8. The graph of the quadratic function
f(x) = x – 6x + 2p – 3 does not intersect the x-axis.
2
Find the range of values of p.
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