To find the inverse of a quadratic function, swap x and y and complete the square.
Algebraic Fractions
Simplifying
To simplify
- Look for common factor in the numerator and denominator
- Sometimes we have to factorise first
- If we have fractions in the numerator or denominator, multiply to remove the fraction
Multiplying and Dividing
- Multiply the numerators and multiply the denominators.
- Cancel where necessary
- Where one fraction is divided by another, invert/flip the second fraction and multiply.
Adding and Subtracting
- To add or subtract fractions, they must have the same denominator
- Be careful when subtracting (especially where there are double negatives)
- When the denominators have no common factor, their product gives the new denominator (multiply the numerator accordingly)
- It may be necessary to factorise the denominators first to spot the common factors
Writing Improper Fractions as Mixed Numbers
- Improper fractions, where the numerator is larger than the denominator can be written as mixed numbers
- The same can be done with algebraic fractions, where the numerator can order that is equal to or larger than the denominator
- The remainder is written as a fraction
- There are two methods for doing this
- Polynomial long division
- Remainder theorem
- Remainder has to have a lower power than the divisor
Let F(x) be a polynomial
Then F(x) = Q(x) * divisor + remainder
Sketching Graphs of y= 1/f(x)
We can sketch a graph of y=1/f(x) if we already know the graph of y = f(x)
Where y=f(x) crosses the x axis, y=1/f(x) has vertical asymptotes.
1/f(x) does not cross x axis if f(x) has no asymptotes
f(x) = 1/f(x) where f(x) = ±1
(Where y values are ±1, both graphs pass through the same point
Minima on f(x) become maxima on 1/f(x) and vice versa
Positive values of f(x) remain positive on 1/f(x)
Negative values of f(x) remain negative on 1/f(x)
If f(x) -> ±∞ then 1/f(x) -> 0
Trigonometry
secant
cosecant
cotangent
sec θ = 1 / cos θ θ ≠ ∏/2 + n∏ where n Є Z
cosec θ = 1 / sin θ θ ≠ n∏ where n Є Z
cot θ = 1 / tan θ θ ≠ ∏/2 + n∏ where n Є Z
cot θ = cos θ / sin θ
2 More Pythagorean Identities
tan2θ + 1 Ξ sec2θ
cot2θ + 1 Ξ cosec2θ
Numerical Method
Some equations cannot be solved using algebraic methods. For these we have to use numerical or graphical methods. Only approximate answers are found
Solving Equations Graphically
Approximations of the roots f(x) = 0 can be found by drawing f(x) and finding where it crosses the x axis
Inverse Trigonometric Equations
y = sin x is a many to one function, so we limit its domain to -∏/2 ≤ x ≤ ∏/2 . This is called the principal argument of sin x
For y = cos x, domain is 0 ≤ x ≤ ∏
For y = tan x, domain is ∏/2 ≤ x ≤ ∏/2 .
The inverse function of y = sin x is y = arcsin x
The inverse function of y = cos x is y = arccos x
The inverse function of y = tan x is y = arctan x
Trig Identities
sin(x y) = sin x cos y cos x sin y
cos(x y) = cos x cosy sin x sin y
tan(x y) = (tan x tan y) / (1 tan x tan y)
Fixed Point Iteration
In fixed point iteration, a single value estimate is found rather than the interval in which the root lies. This is done by generalising a sequence of numbers which converge to the root.
- Rearrange f(x) = 0 as x=g(x)
-
Use this as the basis for our iterative formula xn+1 = g(xn)
-
To start, substitute x0
Different rearrangements of f(x) may produce different roots
x0 needs to be close to the root
Sometimes the sequence won't converge to a root
Modulus Function
The modulus of a number a is written as |a| and is read mod a
It is its positive numerical value
The modulus function of f(x) is defined as:
f(x) f(x)≥0
|f(x)| =
-f(x) f(x)<0
The modulus is also known as the absolute value and is usually the [Abs] button on your calculator.
Graphs of the Modulus Function y = |f(x)|
- Draw the graph without the modulus
- Reflect any parts below the x axis in the x axis
Sharp corners, called cusps, are created. Leave them alone
Graphs of the Function y = f(|x|)
To sketch this
Draw the graph of y = f(x) for x ≥ 0
Reflect this in the y axis
Solving Equations Using the Modulus
When solving equations involving a modulus, roots can come from either the reflected part of the graph. Draw a sketch to see where the solutions lie before solving the equation.
Trigonometric Equations
It would never be a problem to divide by cosθ to get tanθ as the values of θ that make cosθ=0 could never be a solution anyway
Transformations of Graphs
f(x + a) horizontal translation of -a units a
f(x) + a vertical translation of a units a
f(kx) stretch by scale factor 1/k in the x direction k
kf(x) stretch by scale factor k in the y direction k
f(-x) reflection in y axis
-f(x) reflection in x axis
The Double Angle Formulae
Let B = A in the addition formulae for sin(A + B), cos(A + B) and tan(A + B)
Sin(2A) = sinAcosA + sinAcosA
= 2sinAcosA
Cos(2A) = cosAcosA - sinAsinA
= 2cos2A - 1
= 1 - 2sin2A
Tan(2A) = tanA + tanA
1 - tanAtanA
= 2tanA
1 - tan2A
Can calculate sin(3A) by expanding sin(2A + A)
((sin(3A) = 3sinA – 4sin3A))
Expressing aSinθ + bCosθ as a Single Sin or Cos
aSinθ + bCosθ Ξ Rsin(θ + c)
Rsinc = b
Rcosc = a
R = √a2 + b2
R is the min/max value (with -R being the max/min value)
Differentiation
We call (function)n a function of a function.
Chain Rule
dy dy du
dx du dx
When y = [f(x)]n
dy f'(x) n [f(x)]n-1
dx
dy 1
dx dy/dx
The Product Rule
If y = f(x)g(x)
dy v du u dv
dx dx dx
The Quotient Rule
If y = f(x)
g(x)
dy v du u dv
dx dx dx
v2
Differentiating ef(x)
If y = ef(x)
dy f'(x)ef(x)
dx
Differentiating ln(x)
If y = ln(x)
dy 1
dx x
If y = ln(f(x))
dy f'(x)
dx f(x)
∫x-1 dx = ln(x)
Differentiating Trig Functions
If y = sinx
dy cosx
dx
If y = cosx
dy -sinx
dx
If y = tanx
dy 1 _ sec2x
dx cos2x
If y = secx
dy sinx _ secxtanx
dx cos2x
If y = cossecx
dy cosx -cosecxcotx
dx sin2x
If y = cotx
dy -1 _ -cosec2x
dx sin2x
