Indices Surds Transformations Quadratics Circles

Approach to Indices ●

Look for common roots & possible simplifications of power expressions. e.g. 9 y  3 2 y =32y use brackets to make sure

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Simplify where possible using rules/identities. b

d

b

d

bd

e.g.

ax cx =acx x =acx remember, each multiplicative term "commutes"

e.g.

ab c =a bc and

1 −b =a b a 

If in doubt of an identity, or your working so far, substitute real numbers to check. Repeat the above steps until finished.

Approach to Surds ●

For simplification of surds of the form 1/a  b use





1 1 a− b a−  b = × = 2 a  b a  b a− b a −b this term = 1, so not changing value of expression! ●

Simplify any e.g.

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 expressions using  ab= a  b ,  a /b= a /  b and  x n =  x n

 8= 2×4= 2  4=2  2

Rearrange to the form ab  c unless otherwise specified. e.g.

a−  b a b = − ab ab ab

Equivalence between indices & surds ●

Equivalences between surds and indices can be worked out logically from the identities 1 1 −1 . a 1/ 2= a , a n =n a and a = a 3

e.g. e.g. ●

3

      

1 1 1 x = x  = 1/ 2 = = 3 x x x 3 3 1 1 1 = 4 = 1 /4 = x −1 /4 =x −3 / 4 4 3 x x x −3/ 2

−1/ 2 3

Either of the last two expressions are fine

These are the logical steps needed but once your confident you can go from step 1 to the end directly.

Approach to Quadratic Equations ●

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General form of quadratic equations is y=ax 2bx c this can be solved for −b±  b2−4ac using the formula x = or by completing the square. 2a b 2 b2 y=a x   − c To complete the square we write 2a 4a Note that what we have done is represent the original quadratic in the form y=a x b2 which is easy to solve, plus some "leftovers".

The significance of completing the square can be shown graphically. In the graph below are b 2 shown y =ax 2bx c and y =a x  , a and b are positive and c is negative. 2a

x=

−b 2a

2

x=

−b−  b −4ac 2a

2

x=

−

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y=0

−b  b −4ac 2a

b2 c 4a

It can be seen from this graph that completing the square provides an easy way to locate the minimum or maximum of a quadratic equation. The completed square graph is identical to 2 b the original quadratic, just shifted vertically by −c . 4a To fully understand this graph, first label the point at which each quadratic crosses the y -axis and produce your own graphs for a negative, and then for b negative.

Approach to Transformations ●

The graph below shows the main transformations performed on the equation f  x =2 x−1 for a=2.

Stretch by a centred on y=0

f  x =2 x−1 a f  x =4 x−2 f  x a=2 x1 f  xa =2 x3 f ax =4 x −1

Stretch by 1/a centred on x=0

Translate by a in y-direction

Translate by -a in x-direction

Approach to Circles ● ●

The equations representing a circle is usually in one of the two following forms,  x−a2 y−b2=r 2 or x 2 y 22gx2fyc=0 The former is the most useful representing a circle of radius r centred at (a,b). The latter can be rearranged to the former by completing the square for x and then y.

A

Chord

P Tangent Radius

B

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The properties of a circle are summarised in the diagrams above. ○ A triangle made with the diameter and an arbitrary point P on the circumference, is a right angled triangle at P. ○ The radius is always perpendicular to a tangent at the same point, e.g. B on the diagram. ○ A line taken from the circle centre, to an arbitrary chord will bisect that chord.

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Be familiar with these properties as they will come up in in questions in an indirect manner. E.g. being given several Cartesian points and being asked to form a circle from them.