3.6 Quadratic Functions, SPM Practice (Short Question)


Question 5:
Find the range of values of k if the quadratic equation 3(x2kx – 1) = kk2 has two real and distinc roots.

Solution:
3( x 2 kx1 )=k k 2 3 x 2 3kx3k+ k 2 =0 3 x 2 3kx+ k 2 k3=0 a=3,b=3k,c= k 2 k3 In cases of two real and distinc roots, b 2 4ac>0 is applied. ( 3k ) 2 4( 3 )( k 2 k3 )>0 9 k 2 12 k 2 +12k+36>0 3 k 2 +12k+36>0 k 2 +4k+12>0 k 2 4k12<0 ( k+2 )( k6 )<0 k=2,6



The range of values of k is 2<k<6.


Question 6:
Given that the quadratic equation hx2 – (h + 2)x – (h – 4) = 0 has real and distinc roots.  Find the range of values of h.

Solution:
The quadratic equation h x 2 ( h+2 )x( h4 )=0 has real and distinc roots. b 2 4ac>0 is applied. ( h2 ) 2 4( h )( h+4 )>0 h 2 +4h+4+4 h 2 16h>0 5 h 2 12h+4>0 ( 5h2 )( h2 )>0 The coefficient of  h 2  is positive,  the region above the x-axis should be shaded. ( 5h2 )( h2 )=0 h= 2 5 ,2



The range of values of h for ( 5h2 )( h2 )>0 is h< 2 5  or h>2.


Question 7:
The diagram below shows the graph of the quadratic function f(x) = (x + 3)2 + 2h – 6, where h is a constant.


(a) State the equation of the axis of symmetry of the curve.
(b) Given the minimum value of the function is 4, find the value of h.

Solution:
(a)
When x + 3 = 0
     x = –3
Therefore, equation of the axis of symmetry of the curve is x = –3.

(b)
When x + 3 = 0, f(x) = 2h – 6
Minimum value of f(x) is 2h – 6.
2h – 6 = 4
2h = 10
h = 5

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