E. Useful formulae

8/7/2019 E. Useful formulae

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Pearson Education New Zealand 2005Theta Mathematics NCEA Level 2

2.1 AlgebraManipulate algebraic expressions and solve equations

A linear equation: one that can be represented on a graph by a straight line.

A quadratic expression: ax2 + bx + c

A perfect square: (ax + b)2

Difference of two squares: (x + y)(x y) = x2 y2

Basic indices: bp (b is the base, p is the index or exponent).

A polynomial: p x a x a x a x a x a x ann

nn

nn( ) ...= + + + + + + 1 1 2 2 2 2 1 0

The numbers a a a a a an n n, , , ... , , 1 2 2 1 0 are called coefficients.

The highest power of x, which is n if an 0, is the degree of the polynomial.

a0 is the constant term.

A rational expression: a fraction where both the numerator and denominator are polynomials.

The quadratic formula

The two solutions of the quadratic equation ax2 + bx + c = 0 are:

x xb b aca

b b aca

= = + 2 24

24

2and

The discriminant of a quadratic equation ax2 + bx + c = 0 is the number:

b2 4ac

1 b2 4ac > 0 two real roots.

2 b2 4ac = 0 one repeated root.

3 b2 4ac < 0 no real roots.

Indices

x nxn

= 1 a ax x1

= x xp

q pq=

Logarithms

If , then

index form

b q q ppb

= =log ( )log form

b is called the base, p is called the logarithm, and q is the number.

We say logb(q)as log of q to base b.

1 log (ab) = log (a) + log (b) (When multiplying numbers, add their logarithms.)

2 log ab( ) = log (a) log (b) (When dividing numbers, subtract their logarithms.)

3 log (an) = n log (a) (When raising a number to a power, multiply the logarithm by that power.)

Changing the base of a logarithm:

log ( )log ( )

log ( )ab

b

a= 10

10

Useful formulae


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Theta Mathematics NCEA Level 2 Pearson Education New Zealand 2005

2.2 Non-linear graphsDraw and interpret straightforward non-lineargraphs

Useful formulae

Parabolas

Cubics

Hyperbolas


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Circles

Exponential graphs

Log graphs


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The rule for differentiation (finding derived

functions) of powers of x is:

f(x) = xn f(x) = n . xn 1

or

y = xn d

d

y

xnxn= 1

For multiples of powers of x:

f(x) = a.xn f(x) = a.n.xn 1

or

y = axn d

d

y

xanxn= 1

A function is said to be increasing when it has apositive gradient.

A function is said to be decreasing when it has anegative gradient.

A turning point on the graph of a function is thepoint where the function changes from beingincreasing to being decreasing, or from decreasingto increasing.

At a turning point the gradient is zero, i.e. f(x) = 0.

There are two kinds of turning point:

a maximum point (plural maxima) and a minimum point (plural minima).

2.3 Derivatives and integralsFind and use straightforwardderivatives and integrals

Useful formulae

Integration

x x cn xn

n

= ++

+d

1

1

ax x cn axn

n

= ++

+d

1

1

Rates of changed

d

y

xis the rate of change of y with respect to x.

Kinematics Velocity is the rate of change of distance with

respect to time.

v st

= dd

Acceleration is the rate of change of velocitywith respect to time.

a vt

= dd


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2.5 Statistical inference

The mean from a frequency table is:

xxf

f=

The range is the difference between the largest value and the smallest value.

The two quartiles, together with the median, separate a set of numbers into four quarters:

The lower quartile (LQ) is the value below which lie one-quarter of the measurements.

The upper quartile (UQ) is the value below which lie three-quarters of the measurements(or above which lie one-quarter).

The interquartile range is the difference between the two quartiles:

interquartile range = upper quartile lower quartile

The standard deviation of a set of numbers is:

sx x

n=

( )2

The standard deviation from a frequency table is:

f x x

f

( )

2

Statistical spreadsheet formulae

(for a set of data in cells A1 to A100)

Statistic Spreadsheet formula

mean =AVERAGE(A1:A100)

median =MEDIAN(A1:A100)

mode =MODE(A1:A100)

range =MAX(A1:A100)MIN(A1:A100)

lower quartile =QUARTILE(A1:A100,1)

upper quartile =QUARTILE(A1:A100,3)

interquartile range = QUARTILE(A1:A100,3)- QUARTILE(A1:A100,1)

standard deviation =STDEVP(A1:A100)

Random number formulae

=RAND() returns a random decimal between 0 and 1

=RANDBETWEEN(a,b) returns a random integer between a and b inclusive.Note: some spreadsheets do not support the RANDBETWEEN formula.

An alternative is to use =INT((ba+1)*RAND())+a

Useful formulae


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Venn diagrams

The union of two sets is the set containing theobjects or items in one, or the other, or both.

The symbol for union is .

The intersection of two sets is the set containingthe objects or items in bothonly.

The symbol for intersection is .

For complementary events: P(A) + P(A) = 1

For mutually exclusive events:P(A B) = P(A) + P(B)

For intersecting events:P(A B) = P(A) + P(B) P(A B)

2.6 ProbabilitySimulate probability situations and apply thenormal distribution

1 Using a long-run relative frequency approach, the probability of an event is given by:

number of times the event occurredtotal numbber of observations

2 When outcomes are equally likely the probability of an event is:

number of outcomes favourable for the eventttotal number of outcomes

Useful formulae

Expected value

When a process, involving individual outcomes witha fixed probability (p), is repeated a fixed number oftimes (n) the expected value is given by:

Expected value = n p or np

Normal distribution

Any data that is normally distributed fits certainproportions:

68% of the data lies within 1 standard deviation(either side) of the mean. It is likely or probablethat the data will be in this region.

95% of the data lies within 2 standard deviations(either side) of the mean. It is verylikely or veryprobable that the data will be in this region.

99% of the data lies within 3 standard deviations(either side) of the mean. It is almostcertain thatthe data will be in this region.


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The standard normal distribution

An ordinary normal curve has a mean of and standard deviation (sd) of .

The standard normal curve has a mean of 0 and a standard deviation (sd) of 1.

To convert an ordinary normal measurement (x) to a standard normal measurement (z), use the equation:

zx

=

Spreadsheet formulae for probability

Required value Spreadsheet formula

1. Probability, where Z is standard normal with mean 0 and sd 1:

P(Z < k) =NORMSDIST(k)P(Z > k) =1NORMSDIST(k)

P(a < Z < b) =NORMSDIST(b) =NORMSDIST(a)

2. Probability, where X is standard normal with mean and sd :

P(X < k) =NORMDIST(k,,,TRUE)

P(X > k) =NORMDIST(k,,,FALSE) or=1NORMDIST(k,,,TRUE)

P(a < X < b) =NORMDIST(b,,,TRUE) NORMDIST(a,,,TRUE)

3. Inverse normal values given probability, mean and sd:

Determine a such that P(X < a ) = k =NORMINV(k,,)


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2.7 SequencesSolve straightforward problems involving sequences

Sigma notation

t t t t tii

n

n= = + + + +1 1 2 3

....

Arithmetic sequences

First term = a

Common difference = d

General term: tn= a + (n 1)d

Sum of first n terms: S a n dnn= + [ ]2

2 1( )

Alternative formula for sum of n terms: S a lnn

= +2 ( )(where a is the first term and l is the last term)

Geometric sequences

First term = a

Common ratio = r

General term: tn = a rn 1

Sum of first n terms: Sna r

r

n

=

( )1

1or Sn

a r

r

n

=

( )1

1

Sum to infinity: S ra

r = <


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2.8 Trigonometry problemsSolve trigonometry problems requiringmodelling of practical situations

Right-angled trigonometry

sin( )A o

h=

cos( )A a

h=

tan( )A o

a=

s = h sin(A)

c = h cos(A)

s = c tan(A)

Area of a triangle:

Area C= 12

ab sin

The sine rule:

a b csin sin sinA B C

= = or sin sin sinAa

Bb

Cc

= =

The cosine rule:

a2 = b2 + c2 2bc cos A or cos Ab c a

bc= +

2 2 2

2

Useful formulae


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Radian measure

1 radian = the angle formed in a sector with the arc length the same as the radius

Conversions between degrees and radians:1180

= and radians = 180

Arc length formula: s = r

where s = arc length

r = length of radius

= angle at centre of centre, measured in radians

Sector area formula: Area = 12

2r

Segment area formula:

12

12

12

2 2 2r r r = ( )sin( ) sin( )

Note: is measured in radians


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2.9 Trig graphs and equationsSolve straightforward trigonometricequations

The three main trig graphs are those of y = sin(x), y = cos(x), and y = tan(x).

Note:

The y-values for the sin and cos graphs always lie between 1 and 1.

tan(x) is undefined for the values 90o and 270o. The dotted lines on the diagram are called asymptotes.Here they are lines that the graph is very close to, but never actually touches.

Period and amplitude

Using quadrants to solve trig equations

To solve simple trig equation like these:

sin(x)= k cos(x)= k tan(x)= k

follow the steps below.

Step 1 kwill either be positive or negative. This tells youwhich two quadrants the solutions are in (by recallingthe sin, cos or tan graphs).

Step 2 Use your calculator: take the decimal k (if it is negativemake it positive), and calculate inverse sin, cos, ortandepending on which of the three equations it is.This always gives you a Q1 angle.

Step 3 To find:

a Q1 anglekeep the angle from above

a Q2 anglesubtract it from 180 a Q3 angleadd it to 180 a Q4 anglesubtract it from 360.

Useful formulae

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