Proof: We know the equation y=a(x¡h)^2 +k has vertex(h,k), so its line of symmetry
is x=h.
Expanding y=a(x¡h)^2 +k, we find y=a(x^2 ¡ 2 hx+h^2 )+k
) y=ax^2 ¡ 2 ahx+[ah^2 +k]:Comparing the coefficients ofxwith those of y=ax^2 +bx+c, we find ¡ 2 ah=b.) h=¡b
2 a, and so the line of symmetry is x=¡b
2 a.
Example 19 Self Tutor
Find the equation of the line of symmetry of y=2x^2 +3x+1:y=2x^2 +3x+1 has a=2, b=3, c=1) the line of symmetry has equation x=¡b
2 a=
¡ 3
2 £ 2
i.e., x=¡^34VERTEX
The vertex of any parabola lies on its line of symmetry, so itsx-coordinate will be x=¡b
2 a:
They-coordinate can be found by substituting the value forxinto the function.Example 20 Self Tutor
Determine the coordinates of the vertex of y=2x^2 +8x+3:y=2x^2 +8x+3 has a=2, b=8, and c=3,so¡b
2 a=
¡ 8
2 £ 2
=¡ 2
) the equation of the line of symmetry is x=¡ 2
When x=¡ 2 , y=2(¡2)^2 +8(¡2) + 3 = 8¡16 + 3 =¡ 5) the vertex has coordinates (¡ 2 ,¡5).Example 21 Self Tutor
For the quadratic function y=¡x^2 +2x+3:
a find its axes intercepts b find the equation of the line of symmetry
c find the coordinates of the vertex d sketch the function showing all important features.a When x=0,y=3, so they-intercept is 3 :
When y=0, ¡x^2 +2x+3=0
) x^2 ¡ 2 x¡3=0
) (x¡3)(x+1)=0
) x=3or¡ 1
) thex-intercepts are 3 and¡ 1.b a=¡ 1 , b=2, c=3)¡b
2 a=
¡ 2
¡ 2
=1
) the line of symmetry isx=1.442 Quadratic equations and functions (Chapter 21)IGCSE01
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Y:\HAESE\IGCSE01\IG01_21\442IGCSE01_21.CDR Monday, 27 October 2008 2:09:53 PM PETER