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Question?Answer
If you can divide one number by another to get a positive integer, the second number is a factor of the first.
If you multiply 2 positive integers by one another the answer is a multiple of both numbers.
All positive integers can be written as a product of prime factors, which can be found through prime factor decomposition.
LCM stands for lowest common multiple.
HCF stand for highest common factor.
The reciprocal is a type of operation known as an inverse operation.
1/a is the reciprocal of a
An operation that reverses what has been done is called an inverse operation.
A x 10^-5 has the same product as A ÷ 10^5.
In fractions, if you multiply the denominator, multiply the numerator and vice versa.
When multiplying a fraction by a number, that number is multiplied by the numerator then divided by the denominator.
If the numerator is bigger than the denominator in a fraction the correct name for that fraction is an improper fraction, which is also known as a top- heavy fraction.
A decimal that doesn't keep going is known as a terminating decimal.
9 x 0.33(reoccurring)= 3
99 x 0.27(reoccurring)= 27
A calculation to increase 50 by 15% is 50 x 1.15
A calculation to increase 10 by 2.5% is 10 x 1.025.
A calculation to decrease 80 by 27% is 80 x 0.73
A calculation to decrease 50 by 40% is 50 x 0.6
A calculation to find the number when increased by 5% is 50, is 50 ÷ 1.05
If the cost of washing machine with 17.5% vat is £399.50, the equation to find the cost before vat is £399.50 ÷ 1.175, this type of operation is known as reversing or reversed percentages.
To find the Hcf, write the numbers as a product of their prime factors using prime factor decomposition, with the number of times you write each number in the multiplication equation depending on the least amount of times that it appears for a single number.
To find the Lcm, write the numbers as a product of their prime factors. The amount of times you write each prime factor into the multiplication equation depends on the largest amount of times they occur for a single number. Afterwards multiply.
A^-n in fraction form is 1/A^n which is the reciprocal of A^n.
(1/8)^-2 in fraction form= (8/1)^2.
N√b (as in 'b' to the 'nth') in the form of fractional indices= b^1/n.
The inverse operation for n√b (as in 'b' to the 'nth') is: (n√b)^n. For example 2√4=2 and (2√4)^2= 4.
The inverse of adding is subtracting.
The inverse of multiplying by is dividing.
When something is written in index notation it is written with indices.
When 'b' is to the power of fractional indices the calculation to solve the expression is 'b' to the root of the denominator, then to the power of the numerator.
If 'b' is the the power of a negative indice the answer is often(maybe always) equal to 'b' when it's turned into its reciprocal and put to the power of the indice multiplied by -1.
If there is an indice outside brackets EVERY-THING inside those brackets is put to the power of the indice.
The process of simplifying fractions is also sometimes known as cancelling.
0.nnn...(reccurring) (n>-1, n<10) is equivalent to the fraction: n/9.
To change a percentage into a fraction, write it over 100% then cancel common factors.
To change a fraction to a percentage, multiply it by 100%.
To change a percentage to a decimal, divide it by 100%.
To change a decimal to a percentage, multiply it by 100%.
To find 1 quantity as a percentage of another, divide the first by the second and multiply it by 100%.
One thing that compares the relative sizes of two or more quantities is ratio.
To find the value of '1' in a ratio: divide the total value by the sum of the parts of the ratio.
To write a ratio(with integers) in its simplest form(and have the numbers as integers): divide all numbers in the ratio by the hcf.
One method to estimate the result of a calculation is to approximate each number.
Numbers that can be written as fractions and have an ending are called rational numbers.
Numbers that aren't written as fractions with integers and keep on going are known as irrational number.
An irrational root (even when converted to a number) is a surd.
Is pi irrational or rational? Irrational.
Numbers
In 10^n, the amount of zeros after 1 is equal to n.
When multiplying two numbers in standard index form, multiply the simple numbers and add the indices (a*10^b x n*10^m = an*10^b+m), then check it's correct with the first number between 1 and 10.
When dividing two numbers in standard index form, divide the simple numbers and subtract the indices (a x 10^b / y x 10^z = a/y x 10^(b-z)), then check it's correct with the first number between 1 and 10.
10^-1= 0.1. 10^-2= 0.01.
0.00815 in standard form= 8.15 x 10^-3.
The lower bound of 'n' to the nearest 10 is n-5, while the higher bound is n+5.
The lower bound of 'n' to the nearest 100 is n-50, while the higher bound is n+50
The lower bound of 'n' to the nearest 1000 is n-500, while the higher bound is n+500.
The lower bound for 'n' to b significant figures is the number that's smallest that would round up to 'n' when put to b significant figures and the higher bound is the number that everything smaller rounds down to 'n', when put to b significant figures.
Exponential describes a situation where the variable is in the index.
The opposite of exponential growth is known as exponential decay.
A good way to check an answer might be to go back to the starting number(s) using (an) inverse operation(s).
When dealing with interest the amount borrowed is known as the principal.
The formula to calculate simple interest when the principal(p), rate(r) and time/number of applications(t) are given is I= prt, where 'p', 'r' as well as 't' are the principal, rate(per x, converted to a decimal) and number of applications.
With simple interest, for example, if the rate is per annum, and the question is asking about payment for some months, 't' (the number of applications) equals the amount of months as a fraction over 12.
With simple interest, for example, if the rate is per annum and the applications is in weeks, write 't' (the number of applications) as the amount of weeks multiplied by 7/365 (or 1/52 if the question deserves it - assuming it's not a 'leap' year).
With simple interest, if one of the principal, interest, rate, or time is missing and the others are known: rearrange the formula, and then replace the letters with values.
In simple interests the formula for calculating repayments is repayment = P + I (principal + interest).
The formula to calculate future compound interests is: V= p(r+1)^t where p, r as well as t are principal(the initial amount that you borrow or deposit), the rate of interest (as a decimal) and number of applications (usually years).
The formula to calculate the the future value of P deposited for t at r p.a. compounded monthly is V= P(1 + r/12)^12t, if the compounding was (for example:) quarterly, the formula would be V= P(1 + r/4)^4t. (Assumably, the same technique can be applied to compounding reducing interests.)
B decreases by 5% each year. The calculation to find out how much will there be in 10 years time is: b x 0.95^10. The calculation to find out how much there was 5 years ago is: b ÷ 0.95^5.
The formula to calculate future reducing interests/depreciation is: V= p x (1-r)^t where p, r as well as t are price/previous value, reducing interest/depreciation rate and number of applications (usually years).
√bx√c= √(bc).
√b/√c= √(b/c).
(√b)^2= b.
When multiplying numbers with surds: put the number in front of the square root. If multiplying or dividing surds, some or all with values already multiplied by them: multiply or divide the value and surd parts seperately. b√c x d√e= bd√(ce). Some expressions may come in the form (√s+c) x (√p+t) = √(sp) + t√s + c√p + ct.
To make the number in the square root as small as possible find the largest factor that's a square number (y), then multiply the number in the square root by the square root of y and change the number within the radical symbol (n) to n/y.
To rationalise a Surd denominator which is √n, multiply the numerator and the denominator by √n. Conventionally, afterwards one would
To rationalise a Surd denominator which is b-√n: multiply both the numerator and the denominator by the denominator's complement (that being b+√n) which would multiply with the denominator to give a difference
The (proportion) unitary method can be used (for/with) proportions.
The (proportion) unitary method involves finding out what 1 unit is worth(u) then to find what n units is worth, multiply u by n.
If y is inversely proportional to x: y ∝ 1/x but if written in equation form then y= k/x.
If y is inversely proportional to the square of x: y ∝ 1/x^2, but if written in equation form then y= k/x^2.
If y is directly proportional to x: y ∝ x, but if written in equation form, then y= kx.
In direct and inverse proportions the constant is a number that is usually expressed in an equation as 'k'.
If y is proportional to the square of 'x' then y∝ x^2 or in equation form: y=kx^2.
If y is proportional to the square root of 'x' then y∝ √x or in equation form: y= k√x.
When data is plotted onto a linear graph, for y to be proportional to x (where k is a constant), y = kx, and would thus pass through the origin (0,0).
When factorising linear equations take out the largest common factors and put them outside the bracket(s).
When 2 brackets are next to each other: multiply them by each other. This can sometimes be done using(for order of multiplication) f.o.i.l, with those standing for: firsts, outsides, insides, lasts.
When factorising quadratic equations where 'x^2' is being multiplied by more than 1: find numbers that multiply to give both the c[] and the c[] of x^2 'constant' and the coefficient of x^2, then expand to check if the answer's correct. If incorrect when the brackets have been expanded, then change positions, add or change (a) positive/negative symbols.
If factorising a quadratic equation where 'bx' is not present, then the expression might be the difference of 2 squares. The term 'bx' might be missing because it's (+/-) 0b which doesn't add value.
The values that a difference of two squares factorises to are usually composed of the starting values' square roots.
A technique while cancelling or simplifying algebraic fractions is to rewrite the terms using their prime factors and cancel repeating factors(if a denominator factor is cancelled, then an equal factor is cancelled on the numerator and vice versa.)
Algebraic fractions don't cancel when not all the terms in the numerator have been by, of a denominator, a factor multiplied. Nc+e/nc^2 doesn't cancel. To check if a factor of the denominator is multiplied by all other terms in the numerator, you may factorise.
If all the numerator or denominator terms cancel out in an algebraic fraction then instead of discarding it: write 1.
The common denominator contains all the factors from both denominators.
With fractions, before multiplying or dividing, it might be easier to c[] cancel.
It 6ts take 11d to collect n how long would it take for 12ts to do so. In fractions 12ts ÷ 6ts= 2/1. The inverse of 2/1 is 1/2. 11d x 1/2=
An arrangement of terms (and/or) numbers with symbols is an expression.
Something that connects two expressions containing variables, the value of one variable depending on the values of the others and has an equals sign is a formula.
In mathematics, a statement of equality - a mathematical statement that two expressions, usually divided by an equal sign, are of the same value is an equation.
An equation that's true no matter what values are chosen(example:a/2=a x 0.5) is an identity.
A relationship between 2 sets of values such that a value from the first set maps on to a unique value in the second set is a function.
√N^m as an expression including a 'fraction' is N^(m/2).
The cube root of n^m as a expression using a fraction is n^(m/3).
When trying to find an unknown value: simplify the given t[] and e[] terms and equation, then find the value using 'rearranging formulae'.
When solving an equation including algebraic terms: use 'r[] [] 'rearranging formulae' but if it reaches a quadratic (or higher polynomial) resulting in 0 (to find the solutions): factorise it, then single each section and put it equal to 0, then (assuming the sought value is of x) solve for x. (2x^2 - x factorises to x(2x-1), split it into x=0 and 2x-1=0, the solutions are x=0 and x=0.5.) Note: other forms of manipulation including the quadratic formula may be used.
When solving an equation with more than one term on a side whenever you multiply a fraction or term do the same to all other terms in the equation.
For solving Linear equations, a method that may be effective is using 'rearranging formulae'.
If y∝x, then y= kx.
In graphs if y∝1/'x'(inversely proportioned) the line in the top-right quadrant is shaped like the letter [] (but more curved) 'L', and the graph is symmetrical about the line y=-x, thus reflecting the more curved 'L' in the quadrant that's in the lower left.
In graphs if y∝1/'x'^2(inverse proportion) the lines shown are shaped(in the top-right quadrant) like the letter [] (but more curved) 'L' and is reflected through the line x=0.
If y∝f(x) then y= kf(x) and can be graphed that way.
{In graphs if y=√x, the graph is as half the line y=x^2, but rests on the line x=0; and y only takes the square root values that are positive.
'<' and '>' are types of inequalities.
For solving Linear inequalities an effective method may be what is done on one side is done on the other side(s), but if you multiply or divide by a negative number: flip the inequality sign(s).
In Linear Inequalities, if the term appears on multiple sides, you can split the inequality into [] parts (in an inequality with three sections) 2, with a side appearing in both of the newly created inequalities, that side being the one that was in the middle. After solving the seperate inequalities: combine them. x+5<3x-1<2x+1 splits to become x+5<3x-1 and 3x-1<2x+11. x+5<3x-1 turns to 3<x and 3x-1<2x+11 simplifies to x<12, thus a simplified x+5<3x-1<2x+11 is 3<x<12. Another method, instead of splitting the inequality is to do what is done on one side should be done on all other sides. (Assumably the method of splitting the inequality can be applied to situations where the inequality has more than three sections.)
In Quadratic-Inequalities the answer to 'x'^2≤n is: -√n≤x≤√n and the answer to 'x'^2≥n is; -√n≥x≥√n.
In Quadratic-Inequalities the answer to 'x'^2<n is: -√n<x<√n and the answer to 'x'^2>n is; -√n>x>√n.
Before plotting Inequalities on graphs: si[] simplify using 'rearranging formulae'.
Quadratic Inequalities may be simplified using 'rearranging formulae'.
If you are with a graph, with a quadratic curve f(x) and a line y=n, to find the boundaries of the quadratic inequality with reference to 'n', find the places where f(x) and y = n meet.
In graphs the roots are the places where the line crosses the 'x' axis.
To solve 'linear programming' using graphs: to start turn any inequality into an equals sign, then have the equations graphed. Then it is conventional (but perhaps not always neccesary) to shade in the areas that are not needed/valid.
When solving simultaneous Linear and Quadratic equations(with one of each) for x, the solutions amount to 2.
To graphically solve simultaneous equations with one linear and one quadratic equation(with terms y and x): if not already change the subject of each equation to 'y', then draw the graphs of each equation. The solution of the simultaneous equations can be found at the intersecting points of the two graphs. (This technique can be applied to situations with more than two equations.)
If an equation includes 'x'^3 then it would be Cubic.
Just as a quadratic equation may have two real roots, so a cubic equation has possibly 3, but unlike a quadratic equation which may have no real solution, a cubic equation always has at least one real root. The amount or repeated roots in a cubic function can be up to three.
In solving Cubic equations apart from 'manipulation', it is possible to use an a[] method approximation method using [] and [] trial and improvement, which can be refined to different degrees of accuracy. The objective of the trial and improvement method is to find a value of x that makes the expression of f(x) as close as possible (usually) to 0. This value is called a [] of the equation root--.
{The formula for Geometric sequence is a x r^n-1: where 'a' is the first number and 'r' is the ratio.}
The line; 'y'=mx+c is straight.
If a line is diagonally to the left, then let the change in 'x' be negative.
Lines are parallel if they have equal gradients.
A line that is straight at an angle of 90° to a given plane or a vertical line is (to it) perpendicular.
Multiplying the gradients of 2 perpendicular lines gives -1.
In distant/time graphs, the higher the gradient: the quicker the movement. In velocity/time graphs, the higher the gradient:
In velocity/time graphs: if the speed increases steadily (changes by an equal amount in every equal time period), this is called constant acceleration and if the speed decreases, it is called negative acceleration.
When drawing a/'x' graphs, it is convention to leave out the 'x' value of 0.
A trigonometric graph that seems to have a shape like a 's' shape, tilted about 90°, has the x-axis cutting through it and has a bump that is cut through (about the middle) by the y-axis is the graph y=cos 'x'°, a similar graph that doesn't have the y-axis cutting through a bump about the middle is the graph y=sin x°.
Percentage change= change/original x 100%.
The lower bound for 'b' to nearest 'n' is b-(n/2)and the higher bound is b+(n/2).
When asking questions for handling data, avoid asking open questions.
With 'handling data': the set of individuals or items of interest are known as the population. Things that distort data and give an unfair representation of a population are known as bias. Two of the ways that bias can arrive is from the asked/examined population or the style of the question like asking a population a question that is leading.
With 'handling data': a sample is a small part of the population. Samples may be used as they can be both quicker and cheaper. If the variations or composition of a population is known then those variations should be added to the Sample.
If n represents a number then Systematic sampling is picking out every nth item. This can be unrepresentative though if the order is not random.
Sampling that requires people to fit a certain criteria is known as Quota sampling and may be used for things like market research.
With Cluster sampling, the 'population' is divided into small groups called clusters. One or more Clusters are chosen using random sampling. This can lead to bias if the clusters are all different.
Data that a one collects is known as [] data Primary data while Secondary data is data that is from an external source and collected by others.
When Sampling (in data handling), try to start with enough p[] (and/or) s[] data for sampling. primary -- secondary
To cover the possible answers: a questionnaire should give sufficient choices.
In terms of clarity, the information in a questionnaire should be clear.
In terms of length, generally answers as well as questions in a questionnaire should be kept short. Answers should also be capable of being easily analysed. Between Closed and Open questions, in questionnaires, generally you should have closed questions.
In a questionnaire, there shouldn't be biased questions and questions should be relevant to the survey.
When collecting data by observation; consider the importance of both the time and place. Also consider the importance of the length of time.
When collecting data by experiment: consider whether the experiment test either the concept or hypothesis and have there been sufficient experiments producing enough results to reflect what is happening.
Two way tables may be used to display 2 sets of information.
Pie charts can be used to display a type of data known as categorical data. If you know the amount of degrees a section contains but want to find the value divide the amount of degrees by 360° then multiply that by the total.
If you know the value of a sector but want to find the amount of degrees: divide the value by the total, then multiply by 360°.
A time series is made up of a type of data known as numerical data recorded at intervals of time and plotted as a line graph.
To plot data unto a Stem and Leaf tables; divide each number into the tens and units. In the Left column place the figures for the tens but don't repeat any numbers and in the right column place the figures for units in front of the corresponding, previously aligned digit of the tens. All the numbers in both columns should be placed in order. This method may be adapted for larger numbers. Example: 15|3=153.
In a cummulative frequency graph - you can find out the number of occurrences of a result in accordance to an inequality(such as less than: <n) by analysing the cumulative frequency.
When plotting a stem and leaf table, to possibly help improve understanding, write a key.
If data is grouped; as a replacement of mode, there is the 'modal class', which is the group with the highest frequency.
With Stem and leaf tables; if there are 3 columns, often there is one in the middle containing the figures in the tens and the other 2 which are located on either side contain the figures representing the units.
[] ratios may be defined as angles of any size using coordinates Trigonometric--. A circle hs radius 1 unit. The point P with coordinates (x,y) moves round the circumference of the circle. OP makes an angle θ with the positive x-axis. As P rotates anticlockwise, the angle increases. For any angle θ, the sine, cosine and tangent can be given by the coordinates of P: (keeping in mind that P has coordinates (x,y)) sin θ = y/1 = y, cos θ= x/1 = x and tan θ = y/x.
A line that the distance between it and the curve tends to zero as they head to infinity is the asymptote. Often the two asymptote for Y=1/x are y=0 and x=0. The three types of asymptotes are horizontal, vertical and oblique. The asymptotes can approach from any side and may actually (possibly many times)c... ross over , and even move away and again come back.
When 2 chords intersect inside a circle; the products of the lengths of both parts of the chords are equal.
{To calculate the radius of a circle, given an arc or segment with known width('w') and height('h'), the formula is radius(r)= (4h^2+w^2)/8h.}
The convertion from Fahrenheit to Celsius is Celsius= 5F-160/9 or 5/9(f-32).
In centimeters, 1 inch is 2.54cm, 1 foot is 30.48cm and 1 yard is 91.44cm.
In kilometers: 1 mile= 1.6(093), in pounds: 1 kg = 2.2. In litres: 1 imperial gal= 4.54609. In pints: 1 litre= 1.75.
In inches: 1 meter= 39.37.
{A step graph is a special type of [] graph line graph that is made up of lines in several horizontal (either named) intervals or 'steps'. The end of the previous horizontal step often lies on the same vertical line for the start of the next step.}
{When two events, c and d, are dependent, the probability of both occurring is: P(c and d)= P(c) x P(d|c). Using 'rearranging formulae' P(d|c)= P(c and d )/P(c).}
For mutually exclusive events P(c) + P(d)=P (c or d).
With mutually exclusive events; P(c and d)= 0.
With dependent events; if C and D (probabilities of 'c' and 'd' respectively) are the events with t as the total plus n the value taken away each time. p(C and C) = c x []/[] c-n/ t-n. While p(C and D) = c/t x d-n/t-n. These formulas can be used for more event (d,c,d,d; c,c,c,c; etc).
{A statistical chart consisting of data points plotted on a simple scale, typically using filled in circles and may be used as an alternative to the bar chart, in which dots are used to show the values associated with categorical variables is known as a dot plot as described by Cleveland.
The point exactly between other points is known as the midpoint. For example: you may need to find a line that bisects a given line segment. The midpoint(p) formula is: (^x1 + ^x2)/2, (^y1 + ^y2)/2. The partially super-scripted which precede '1' represent the first set of coordinates, while the ones preceding '2' represent the coordinates at the other side of the line.
Probability of an independent event happening=[]/[] ways/possible outcomes. So the probability of rolling '2' on a dice= 1(ways)/6(possible outcomes). Independent events are events unaffected by previous events.
If an event has outcomes affected by previous outcomes, then the event is dependent.
To find the probability of multiple independent events, multiply the probabilities. P(a and b)= P( a) x P(b).
If a set of events cover all the possible outcomes (e.g for numbers 1 to 4: p(odd or even)), then the set is exhaustive. With exhaustive events: p(a or b)= p(a)+p(b)-p(a and b). If there is no overlap between a set of events (e.g for numbers 1 to 4: p(odd or even)), then the events are mutually exclusive. With mutually exclusive events, p(a or b)= p(a)+p(b).
{With dependent events; P(A and B and C) = P(A) x P(B|A) x P(C|A and B).}
The quadratic formula: x= (-b±√b^2-4ac)/2a.
The factorisation of the quadratic expression: x²+bx+c is (x+o)(x+p), where p and o multiply to make c, while if added would equate to b.
{(#G2)With step graphs: the y axis contains the 'constant'(like time) and the x-axis contains the variable (the r()
{A type of line graph that is usually used to represent situations that involve sudden jumps across intervals, such as cost of postage, car parking fees or telephone rates is a step graph. When a gap or jump appears in a graph, 'we say the graph is discontinuous'.}
It is a horizontal asymptote when: as f(x) goes to infinity (or to -infinity) then the curve approaches some fixed 'constant' value of y. It is a Vertical asymptote when: as f(x) approaches some constant value "c" on the axis of x(from the left or right) then the curve goes towards either infinity (or -infinity). It is an Oblique asymptote when: as f(x) goes to infinity (or to -infinity) then the curve goes towards a line defined by the equation y=ax+c (note: the line would be horizontal if a is 0).
When asking questions for handling data: three things that questions shouldn't be (in alphabetical order) are b[], l[] and o[] biased, leading and open.
A mathematical phrase that is a representation of a value - variables and/or numerals that appear alone or in combination with operators (i.e. 7+3/2, 14x, 16p^3+5) is an expression.
If a question asks how much is someone's repayment per time period (for example: per month), the repayment = (Principal + Interest) / number of payments. Joe borrows £5000 from a bank to buy a car. The bank charges 14% p.a simple interest on the loan. Calculate his monthly repayment if he repays the loan in 3 years. P + I = 7100, monthly repayment = []/[] 7100/(3 x 12) = 7100/36.
Linear programming problems can be broken up into three parts: represent the data using inequalities, showing the inequalities as regions on a diagram, using the diagram to find answers to your problems.
When trying to solve quadratic inequalities where the solution is given in two sections (such as x>a,x<b and x>=a,x<=b), the shading should be done to the line's side.
With quadratic inequalities, when shading to the side of the parabola, shade downwards from the x-axis if the quadratic is negative, and if it's positive shade upwards from the x-axis.
To graphically solve quadratic inequalities: let on one side of the inequality be the number 0, then change the inequality to an equation with the subject (for convention) y. (If one is not already drawn, draw a parabola that passes through the vertex and the x-intercepts. If the inequality involved '...or equal to' use a s[] line solid line, but if doesn't use a line with gaps (a series of dashes: ---). If the inequality contained y>(=) or 0>(=) shade between the [] and the []-[] vertex and the x-axis, and if the inequality contained y<(=) or 0<(=), shade to the sides of the roots - away from the v[] vertex. (The shaded area is the solution region.) (Such graphing is the case for positive quadratics).
The inequality ax+c<(=)/>(=)0 is associated with the two-variable linear graph with equation y=ax+c.
If the inequality was in the form y/0 >(=)..., then the solution is the region in which the graph is [] (or [] to) the []-[] below or equal to the x-axis, whereas if the inequality was in the form y/0 <(=)..., then the solution is the region in which the graph is [] (or [] to) the []-[] above (or equal to) the x-axis.
The inequality f(x)<(=)/>(=)0 is associated with the two-variable linear graph with equation y=f(x)
A Stratified random sample is obtained by: separating a population into strata. Then, the [] is chosen to reflect the properties of these strata(subgroups) sample--. For example, if the population contained three times as many people under 25 as over 25, then the samle should contain three times as many people that are under 25. the sample should also be large enough for the results to be s[] significant.
A stratified random sample is obtained by: seperating the population into appropiate strata (e.g. by age), finding out what proportion of the [] is in each stratum population-- and selecting a [] from each stratum sample-- in proportion to the [] [] stratum size. This can be done by random sampling hence the technique is known as stratified random sampling. The formula for selecting a sample from each stratum in proportion to the stratum size can be given as: Sample size for each subgroup = size of subgroup/size of population x size of whole sample.
The data handling cycle: (1)Specify the p[] and p[] problem and plan, (2)collect data from a variety of sources, (3)p[] and r[] data process and represent data, (4)i[] and d[] data interpret and discuss data, (5)[] results evaluate--.
To get the centre of rotation that is used to rotatate shape ABC to shape A'B'C', draw the [] [] perpendicular bisectors of at least two out of AA', BB' and CC'. The perpendicular bisectors intersectat the centre of rotation.
A diagonal that cuts through 2 parallel lines is called an Intersecting transversal.
When a line cuts through 2 parallel lines the 2 bunches of angles are Equal.
When a line that's not perpendicular cuts through 2 parallel lines there are only 2 different angles. One is acute while the other is obtuse.
An alternate angle is in a ‘z’ shape where the inside angles are equal. Alternate angles can also be in a ‘z’ shape but backwards.
Supplementary angles are a ‘c’ or ‘u’ shape where the inside angles add up to 180 degrees.
Supplementary loosely translated means; Add up to 180.
Corresponding angles are in a ‘f’ shape where the 1st and 3rd inside angle are equal and possibly probably obtuse.
Diagonally opposite angles in a rhombus are Equal.
It 6ts take 11d to collect n how long would it take for 12ts to do so? In fractions 12ts ÷ 6ts= 2/1. The inverse of 2/1 is 1/2. 11d x 1/2= 5.5d.
To solve simultaneous Linear and Quadratic equations; first eliminate either 'x' or 'y' by substitution. Then if it's not already turn the equation to a Quadratic equation using 'rearranging formulae'. The Quadratic equation can be solved through Quadratic factorisation or completing the square. Then use the discovered value(s) to find the value(s) of the remaining variable.
In Cubic equations apart from manipulation it is possible to use an approximation method of trial and improvement, which can be refined to different degrees of accuracy. The object of trial and improvement is to systematically find a value of 'x' which makes the expression as close as desirable to 0. This value is called the root.
To construct a perpendicular bisector move the compass to the end of the line, set it over [] way along the line half way-- and draw an arc the size of at least about half a circle, then repeat from the other side. Afterwards draw a line through both of the [] points of the arcs intersection--. Things that may be labelled are the two intersecting points of the arcs and the right angle(s) formed.
To bisect an angle: place the compass on the vertex and draw an arc that crosses both sides of the angle. At this point, you may change the compass' length. At each point where the arc crossed the sides of the angle, draw an arc that is between the two sides of the angle. To bisect the angle, from the crossing point of those two arcs: draw a line to the vertex. (Note: unless it is stated that it's allowable, don't change compass length).
To construct a triangle with sides of 6cm, 5cm and 4cm: first draw the side of length 6cm (AC). If AB is the side of length 5cm: after that move the compass to A and with the compass opened to a radius of 5cm, draw an arc [] the line above--, if CB is the side of length 4cm: after that move the compass to C and with the compass opened to a radius of 4cm, construct an [] above the line arc--. Join each end of the line to the point where the arcs cross.
To construct a triangle with angles; c°, d° and a side of 6cm: draw the 6cm line. You may label the ends of the line. From each end, measure either one of the angles and mark it with a dot. With lines connect the lines to the dots. The lines should keep going until you have drawn a triangle. In a test, unless asked to do otherwise, if you have any, leave the construction lines.
To construct a triangle with sides of a cm (the longest side), b cm and an angle of c°: draw the longest side (a cm). From the the longest side (a cm), from the end of the line, draw an angle of c°. From the edge of the line at an angle of c° draw a line of length b cm. To finish it join up all the sides. In test you generally should leave the construction lines.
With union and intersection symbols - with exhaustive events: p(a ∪ b)= p(a)+ p(b)-p(a ∩ b), with mutually exclusive events: p(a ∪ b)= p(a) + p(b).
The union symbol is ∪ and the intersection symbol is ∩.
To graph Quadratic inequalities: find the solution(s). The solutions are also where the line crosses the x-axis (the roots). Put x=0 into the equation to find the y-intercept.
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