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Roots and the discriminant of a quadratic equation

ax^2 + bx + c = 0 ====> x = -b±#b^2 - 4ac 2ab^2 - 4ac = discriminant

Example 1

x^2 - x - 6 =0 ===> a=1, b=-1, c=-6

-b±#b^2 - 4ac ==> -(-1)±#(-1)^2 - 4(1)(-6) ==> 1±#25 ==> 1+5 and 1-5 ==> 3 and -2 2a 2(1) 2 2 2

==> 2 solutions => 2 different roots

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Completing the Square - Sketching Graphs

Example 1

y = x^2 - 8x + 6 => y = (x-4)^2 -16 + 6 => y = (x-4)^2 -10

let f(x) = x^2

f(x-4) = (x-4)^2 <-- 4 units right parallel to x axis

f(x-4) - 10 = (x-4)^2 - 10 <-- 10 units down

sub x = 0 to see where it crosses x value y = (0)^2 - 8(0) + 6 = 6

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Roots and the discriminant of a quadratic equation

Example 3

2x^2 + 2x + 1 = 0 ==> a=2, b=2, c=1

-b±#b^2 - 4ac => -2±#(2)^2-4(2)(1) => -2±#-4 => CAN'T SQUARE -VE => no roots/solution 2a 2(2) 4

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Roots and the discriminant of a quadratic equation

Example 2

x^2 - 10x + 25 = 0 ==> a=1, b=-10, c=25

-b±#b^2 - 4ac ==> -(-10)±#(-10)^2-4(1)(25) ==> 10±#0 ==> 10+0 and 10-0 ==> 5 and 5 2a 2(1) 2 2 2

1 solution = 1 root = equal roots ==> touches the curve

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Polynomials

3x^2 + 5x +2 ==> quadratic polynomial - degree 2

5x^3 - 2x^2 + 1 ==> cubic polynomial - degree 3 (don't need every term e.g. x^1 missing here)

7x^4 - 3x^3 + 4x^2 - 2x ==> quartic polynomial - degree 4

Example 1

If p = 2x^3 - 5x +1 , q = x^4 - x , write in descending powers of x: i) p-q ii) pq

i) p-q = 2x^3 - 5x + 1 - (x^4 - x) = 2x^3 - 5x + 1 - x^4 + x = - x^4 + 2x^3 - 4x + 1

ii) pq = (2x^3 - 5x +1)(x^4 - x) = 2x^7 - 2x^4 - 5x^5 + 5x^2 + x^4 - x = 2x^7 - 5x^5 - x^4 + 5x^2 - x

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Roots and the discriminant of a quadratic equation

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Translations

y = f(x-a) translates y = f(x) a units to the right

y = f(x+a) translates y = f(x) a units to the left

y = (x+2)^2 = f(x+2) = 2 units left y = (x-1)^2 = f(x-1) = 1 unit right

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Graphs

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Reflections

y = -f(x) reflects y= f(x) in the x axis

y = f(-x) reflects y = f(x) in the y axis

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Translations

y = f(x) - a translates y = f(x) a units down

y = f(x) + a translates y = f(x) a units up

y = f(x) = x^2 y = f(x) -3 = 3 down y = f(x) +1 = 1 up

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Transformations of Graphs Rules

TRANSLATIONS

y = f(x-a) translates y = f(x) a units to the right

y = f(x+a) translates y = f(x) a units to the left

y = f(x) - a translates y = f(x) a units down

y = f(x) + a translates y = f(x) a units up

REFLECTIONS

y = -f(x) reflects y = f(x) in the x axis

y = f(-x) reflects y = f(x) in the y axis

STRETCHES

y = af(x) stretches y = f(x) by a scale factor of a parallel to the y axis

y = f(ax) stretches y = f(x) by a scale factor of 1/a parallel to the x axis

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Stretches

y = af(x) stretches y = f(x) by scale factor a parallel to the y axis

y = f(ax) stretches y = f(x) by a scale factor 1/a parallel to the x axis

some invarient points*

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Inequalities - Double type

4 < 5x - 1 < x + 11

4 < 5x - 1 ////////// 5x - 1 < x + 11

5 < 5x /////////////// 5x < x + 12

1 < x ///////////////// 4x < 12

x > 1 ////////////////// x < 3

1 < x < 3

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Inequalities

if dividing by a negative number, reverse the inequality so it is true..

e.g. 2 < 6 ===> (divide by -2) ===> -1 > -3

if multiplying by a negative number, reverse the inequality for it to be true...

e.g 2 < 6 ===> (multiply by -2) ===> -4 > -12

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Series Sigma Notation

Example 1

rth term in a sequence = 12/r 1st term => r = 1 => 12/1 = 12 2nd term => r = 2 => 12/2 = 6 3rdterm => r = 3 => 12/3 = 4 4th term => r = 4 => 12/4 = 3

Series... 12 + 6 + 4 + 3 = 25 ==> sigma notation just means add

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Sequences and Series

Recurrance Relationships

Example 1

6, 13, 27, 55

U1 = 6, U2 = 13, U3 = 27, U4 = 55 ==> Un+1 = 2Un + 1

Exam question

Write down the first five terms in the sequence Un+1 = 3Un^2 - 9 where U1 is 2.

when n = 1 => U2 = 2(U1)^2 - 9 = 3(2)^2 - 9 = 3 when n = 2 => U3 = 2(U2)^2 - 9 = 3(3)^2 - 9= 18 when n = 3 => U4 = 2(U3)^2 - 9 = 3(18)^2 - 9 = 963 when n = 4 => U5 = 2(U4)^2 - 9 =3(963)^2 - 9 = 2782098

sequence: 2, 3, 18, 963, 2782098

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Arithmetic Sequences and Series - Progression

Example 1

6, 8, 10, 12 ==> common difference (d) = +2, first term (a) = 6 11, 8, 5, 3 ==> d = -3, a = 11

In general...

1st, 2nd, 3rd, 4th, 9th a, a+d, a+2d, a+3d.... a+8d

nth term = a+(n-1)d

Example 2: For the sequence 5, 9, 13, 17... what is the 21st term?

a = 5, d = +4 n = 21

a+(n-1)d = 5+(21-1)x4 = 5+(20x4) = 5+80 = 85

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Series Sigma Notation 2

Example 1 3(1)^2 + 3(2)^2 + 3(3)^2 + 3(4)^2 = 3 [ (1)^2 + (2)^2 + (3)^2 + (4)^2]

===> = 3 [1+4+9+16] = 90

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Coordinate Geometry

GRADIENT

* Gradient of a Line Segment Gradient of line AB = 2 ======> Gradient of line CD = -5

GRADIENT = CHANGE IN Y CHANGE IN X

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Arithmetic Sequences and Series - Progression

SUM

Sn = n/2[2a+(n-1)d] = n/2[a+l] where a = first term, d = common difference, l = last term

Example 1: 5, 9, 13, 17 => a = 5, d = 4, n = 21

Sn = n/2[2a+(n-1)d] = 21/2[2(5)+(21-1)x4] = 21/2[10+(20x4)] = 21/2[90] = 21x45 = 945

Sn = n/2[a+l] = 21/2[5+85] = 21/2[90] = 21x45 = 945

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Coordinate Geometry

PERPENDICULAR LINES

* the gradient of a perpendicular line is the negative reciprocal of the first line *

grad of blue line = 2 grad of green line = -1/2

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Coordinate Geometry

PARALLEL LINES GRADIENT

* if two lines are parallel, their gradients are equal *

gradient of the green line = 2 gradient of the red line = 2

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Coordinate Geometry - Distance between 2 points

Example 1

AB^2 = AC^2 + BC^2 AB^2 = 8^2 + 6^2 AB^ = 64 + 36 = 100 AB = #100 = 10

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Coordinate Geometry - y = mx + c

Example 1: y = 2x + 1

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Coordinate Geometry - Equation of Parallel Line

Example 1: Find the equation of a line parallel to the line 3y - 2x - 6 = 0 and passing through the

point (-1,-2)

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Coordinate Geometry - Midpoint of a line segment

Example 1

Midpoint of AB = (-2+3, 4+-2) = (1/2, 1) 2 2

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Coordinate Geometry - Intersections

Intersection of 2 straight lines:

SOLVE SIMULTANEOUSLY => get the point of intersection

Intersection of a line and U shaped curve (parabola):

MAKE THEM EQUAL EACHOTHER => get equation => solve = 2 values of x and 2 of y

Intersection of a tangent to curve (parabola):

MAKE THEM EQUAL EACHOTHER => b2-4ac = 0 => solve => 1 root

Intersection of a line and hyperbola (1/x curves):

MAKE THEM EQUAL EACHOTHER => solve equation

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Coordinate Geometry - Equation of perp. bi sector

Example 1: Find the equation of the perpendicular bisector of the line joining the points A (3,5)and B (-2,-1)

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Differentiation and Integration29 of 31

Nature of Intersection of a line and curve - roots

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Differentiation - Equations of Tangents and Normal