A-level Mathematics/OCR/C3/Formulae

By the end of this module you will be expected to have learnt the following formulae:

Transformations of Graphs edit

Reflection edit

  1.   is a reflection of   through the x axis.
  2.   is a reflection of   through the y axis.
  3.   is a reflection of   when y < 0, through the x-axis.
  4.   is a reflection of   when x < 0, through the y-axis.
  5.   is a reflection of   through the line y = x.
    Note:   exists only if   is bijective, that is, one-to-one and onto.

Stretching edit

  1.   is stretched toward the x-axis if   and stretched away from the x-axis if  . In both cases the change is by a units.
  2.   is stretched away from the y-axis if   and stretched toward the y-axis if  . In both cases the change is by b units.

Translations edit

  1.   is a translation of f(x) by h units to the right.
  2.   is a translation of f(x) by h units to the left.
  3.   is a translation of f(x) by k units upwards.
  4.   is a translation of f(x) by k units downwards.

Natural Functions edit

  1.  
  2.  , where y(t) is the final value,   is the initial value, k is the growth constant, t is the elapsed time.
  3.  , k for calculations involving half-lives.

Trigonometry edit

Reciprocal Trigonometric Functions and their Inverses edit

  •  
  •  
  •  
  •  
  •  

Angle Sum and Difference Identities edit

  •  
  •  
  •  

Note: The sign   means that if you add the angles (A+B) then you subtract in the identity and vice versa. It is present in the cosine identity and the denominator of the tangent identity.

Double Angle Identities edit

  •  
  •  
  •  

Combination of Trigonometric Functions edit

Using radians r = amplitute α = phase.  

 

where

 

 

where

 

Differentiation edit

  • If  , then  
  • If  , then  
  • If  , then  
  • If  , then  
  •  
  • If  , then  
  •  

Integration edit

  •  
  •  

For volumes of revolution:

  •  
  •  

Numerical Methods edit

Simpson's Rule  

where  and n is even