quadratic theory strategies higher maths click to start
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Quadratic Theory
Strategies
Higher Maths
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Quadratic Theory Higher
Quadratic Theory
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Quadratic Theory Higher
Show that the line with equation
does not intersect the parabola
with equation
2 1y x
2 3 4y x x
Put two equations equal
Use discriminant
Show discriminant < 0
No real roots
Hint
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Quadratic Theory Higher
a) Write in the form
b) Hence or otherwise sketch the graph of
2( ) 6 11f x x x 2x a b
( )y f x
a) 2( ) ( 3) 2f x x
b) This is graph of2y x moved 3 places to left and 2 units up.
minimum t.p. at (-3, 2) y-intercept at (0, 11)
Hint
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Quadratic Theory Higher
2(1 2 ) 5 2 0k x kx k Show that the equation
has real roots for all integer values of k
Use discriminant (1 2 ) 5 2a k b k c k
2 24 25 4 (1 2 ) 2b ac k k k
2 225 8 16k k k 29 8k k
Consider when this is greater than or equal to zero
Sketch graph cuts x axis at8
and9
0k k
Hence equation has real roots for all integer k
Hint
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Quadratic Theory Higher
The diagram shows a sketch of a parabolapassing through (–1, 0), (0, p) and (p, 0).
a) Show that the equation of the parabola is
b) For what value of p will the line
be a tangent to this curve?
2( 1)y p p x x
y x p
a) ( 1)( )y k x x p Use point (0, p) to find k (0 1)(0 )p k p
p pk 1k ( 1)( )y x x p 2y x px x p 21y p p x x
b) Simultaneous equations 21x p p p x x
20 2p x x Discriminant = 0 for tangency 2p
Given , express in the form
Hint
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Quadratic Theory Higher
2( ) 2 8f x x x ( )f x 2x a b
2( ) ( 1) 10f x x
Hint
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Quadratic Theory Higher
For what value of k does the equation have equal roots? 2 5 ( 6) 0x x k
1 5 6a b c k Discriminant
2 4 25 4( 6)b ac k
0 25 4 24k
4 1k
1
4k
For equal rootsdiscriminant = 0
You have completed all 6 questions in this section
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Quadratic Theory Higher
30° 45° 60°
sin
cos
tan 1
6
4
3
1
2
1
23
2
3
2
1
21
21
3 3
Table of exact values
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Quadratic Theory Higher