Core 1.docx

Core 1 Rules of Indices    

𝑎 𝑥 × 𝑎 𝑦 = 𝑎 𝑥+𝑦 𝑎 𝑥 × 𝑎 𝑦 = 𝑎 𝑥−𝑦 (𝑎 𝑥 )𝑦 = 𝑎 𝑥𝑦 1 𝑎−𝑥 = 𝑎𝑥



𝑎𝑥 = √𝑎

 

𝑎 = ( √𝑎)𝑥 = √𝑎 𝑥 𝑎0 = 1

1

𝑥 𝑦

𝑥

𝑦

𝑦

Types of Numbers

𝑎 𝑏



ℚ- Rational numbers can be written in the form

  

ℤ- Integers are any whole numbers ℕ- Natural numbers are any positive integer ℝ- Real numbers are any numbers that aren't imaginary

Surds  

√𝑎 × √𝑏 = √𝑎 × 𝑏 √𝑎 × √𝑎 = 𝑎



√𝑎 √𝑏

𝑎

= √𝑏

To rationalise the denominator 𝑎 √𝑏 𝑎 𝑏 + √𝑐

×

×

√𝑏

𝑎√𝑏 𝑏 𝑎𝑏 − 𝑎√𝑐 = 𝑏2 − 𝑐

=

√𝑏 𝑏 − √𝑐 𝑏 − √𝑐

Equations of a line   

𝑦 = 𝑚𝑥 + 𝑐 o 𝑚 is the gradient o 𝑐 is the y intercept 𝑦 − 𝑦1 = 𝑚(𝑥 − 𝑥1 ) o 𝑚 is the gradient o (𝑥1 , 𝑦1 )is a point on the line 𝑎𝑥 + 𝑏𝑦 + 𝑐 = 0 o 𝑎, 𝑏, and 𝑐 are all integers

Completing the square 𝑏 2 𝑏 2 𝑥 2 + 𝑏𝑥 + 𝑐 = (𝑥 + ) − ( ) + 𝑐 2 2 For equations such as 3𝑥 2 take the factor outside of the bracket

Sketching Graphs Consider SAXY  S- Shape  A- Asymptotes  X- X intercepts  Y- Y intercepts

Discriminant For 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 𝑏 2 − 4𝑎𝑐 If it is:  Greater than 0 o There are two real roots. 

Equal to 0 o There is one real root.



Less than 0 o There are no real roots.

Quadratic Inequalities ALWAYS DRAW THE GRAPH Solve the equation as if it is a quadratic equation, and plot it Find the x intercepts, and determine whether x should be greater than or less than those numbers Present on a numberline  A filled in circle means equal to  An empty circle means not equal to

Sketching Curves The graph of: 𝑦 = 𝑥2

𝑦 = 𝑥3

𝑦=

1 𝑥

Curve Transformations f(x) – original curve

f(x+a)- move left along x axis by a 10

10

8

8

6

6

4

4

2

2

0

0 -4

-2

0

2

4

f(x)+a – move up the y axis by a

-6

-4

-2

0

2

f(ax)- shrink in the x axis by a

12

10

10

8

8

6

6 4

4

2

2

0

0 -4

-2

0

2

4

af(x)- stretch in the y axis by a 20 15 10 5 0 -4

-2

0

2

4

-2

-1

0

1

2

Co-ordinate geometry 𝑚=  

 



𝑦2 − 𝑦1 𝑥2 − 𝑥1

𝑚 is the gradient (𝑥1 , 𝑦1 ) and (𝑥2 , 𝑦2 ) are points on the line Parallel lines- the same gradient Perpendicular lines- one gradient is the negative reciprocal of the other 𝐴𝐵 = √(𝑥1 − 𝑥2 )2 + (𝑦1 − 𝑦2 )2 𝐴𝐵 is the length of the line (𝑥1 , 𝑦1 ) and (𝑥2 , 𝑦2 ) are points on the line Pythagoras’ Theorem 𝑥1 + 𝑥2 𝑦1 + 𝑦2 Midpoint ( , ) 2 2 (𝑥1 , 𝑦1 ) and (𝑥2 , 𝑦2 ) are points on the line

Sequences   

𝑈𝑛 = 𝑈𝑛−1 + 𝑑 𝑈𝑛 = 𝑎 + (𝑛 − 1)𝑑 𝑈𝑛 is a term in the sequence 𝑎 is the first term 𝑑 is the common difference 𝑆𝑛 =



𝑆𝑛 is the sum of a sequence up to 𝑛

𝑛 (2𝑎 + (𝑛 − 1)𝑑) 2

𝐸𝑛𝑑 𝑣𝑎𝑙𝑢𝑒

∑ 

(𝑆𝑒𝑞𝑢𝑒𝑛𝑐𝑒)

𝑆𝑡𝑎𝑟𝑡𝑖𝑛𝑔 𝑣𝑎𝑙𝑢𝑒

∑(𝑆𝑒𝑞𝑢𝑒𝑛𝑐𝑒) means the sum of a sequence

Differentiation 𝑑𝑦 = 𝑛𝑥 𝑛−1 𝑑𝑥

For 𝑓(𝑥) the derivative is 𝑓′(𝑥) 1. For a gradient at a point substitute the x value into the differentiated equation 2. To find the coordinate where a gradient equals a number, make the differentiated equation equal to the number 3. To find the gradient where a curve meets a line find the coordinate where they meet and use method 1 𝑑2 𝑦

The second derivative is 𝑑𝑥 2 or 𝑓′′(𝑥)

Integration ∫(𝑥 𝑛 )𝑑𝑥 =

𝑥 𝑛+1 +𝑐 𝑛+1