# Core 1 Revision

Some basic points, examples and equations for some topics in Core 1.

Some of the symbols and text may not be shown, sorry!

- Created by: Rajvir
- Created on: 17-05-14 18:47

## 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**

## Roots and the discriminant of a quadratic equation

ax^2 + bx + c = 0 ====> x = **-b±****√b^2 - 4ac**

** 2a**

b^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

## 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

## 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

## Roots and the discriminant of a quadratic equation

## 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

## Graphs

## 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

## 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

## 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 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*

## 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

## 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

## 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

## 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

## 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

3rd term *=*> r = 3 => 12/3 = 4

4th term => r = 4 => 12/4 = 3

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

## 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

## 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

## 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

## 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**

## 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

## 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

## Coordinate Geometry - y = mx + c

*Example 1*: y = 2x + 1

## 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

## Coordinate Geometry - Midpoint of a line segment

*Example 1*

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

2 2

## 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)

## 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)

## 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

## Nature of Intersection of a line and curve - roots

## Differentiation and Integration

## Differentiation - Equations of Tangents and Normal

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