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Probability is used to predict the likeliness of something happening. It is always given between 0 and 1. An experiment is a repeatable process that can have a number of outcomes An event is one single or multiple outcomes A sample space is the set of all possible outcomes For two events, E₁ and E₂, with probabilities P₁ and P₂ respectively: To find the probability of either E₁ or E₂ happening, add the two probabilities, P = P₁ + P₂ To find the probability of both E₁ and E₂ happening, multiply the two probabilities, P = P₁ x P₂ The sample space for rolling two fair six-sided dice and adding up the numbers that show would look like this: To work out the probability of getting a particular result, you count how many times the result occurs and divide by the total number, 36 (since 6² = 36). So to work out the probability of getting a 10, count the number of tens and divide by 36: 3/36 = 0.0833 Generally, give your answers as decimals to three significant figures Conditional Probability If the probability of an event is dependent on the outcome of the previous event, it is called conditional. Conditional probability is noted using a vertical line between the events: The probability of B occurring, given that A has already occurred is given by P(B|A) For two independent events: P(A|B) = P(A|B') = P(A) Experiments with conditional probability can be calculated using a two-way table/restricted sample space: Venn Diagrams A Venn diagram is used to represent events happening. The rectangle represents the sample space, and the subsets within it represent certain events. Set notation is used to describe events within a sample space: A ∩ B represents the intersection A ∪ B represents the union Adding a dash, ', means the compliment, or "not" The addition formula is very useful: P(A ∪ B) = P(A) + P(B) - P(A ∩ B) Mutually Exclusive Events As you can see from the Venn diagrams, mutually exclusive events do not intersect. This means there is no overlap, so: P(A or B) = P(A) + P(B) Independent Events When one event has no effect on the other, the two events are described as independent. Therefore: P(A ∪ B) = P(A) x P(B) Conditional Probability in Venn Diagrams You can find conditional probability easily from Venn diagrams using the multiplication formula: P(B|A) = P(B ∩ A) / P(A) Tree Diagrams Tree diagrams are used to show the outcomes of two or more events happening, one after the other. For example, if there are 3 red tokens and 7 blue tokens in a bag, and two are chosen one after the other without replacement (the first is not put back into the bag), a tree diagram can model this: When you have worked out the probability of each branch, add them together - if they sum to 1, it is correct. Conditional Probability in tree diagrams Tree diagrams show conditional probability in their second and third etc columns. The multiplication formula still applies: P(B|A) = P(B ∩ A) / P(A)
- Correlation & Regression
Bivariate data is data with two variables, and can be represented in a scatter diagram. We can describe the correlation between the two variables based on how much of a straight line the points on the diagram form. Correlation describes the nature of the linear relationship between two variables. A negative correlation occurs when one variable increases as the other decreases. A positive correlation occurs when both variables increase together. Causation The relationship can be described as causal if a change in one variable induces a change in the other. It is vital to remember that just because there may be a correlation, no matter how strong, between two variables, it does not mean the relationship is causal. Correlation does not imply causation You need to consider the context of the variables and use common sense to decide whether or not there is causation as well as correlation. Measuring Correlation The product moment coefficient, r, is a measure of strength for linear correlation between two variables. It takes values from -1 to 1, where If r = 1 the correlation is perfect and positive If r = 0 there is no correlation at all If r = -1 the correlation is perfect and negative You calculate the product moment coefficient using a stats-equipped scientific calculator. On a CASIO ClassWiz fx-991EX, to calculate the product moment coefficient, r: Click MENU Click 6: statistics Click 2: y=a+bx Input your data in the table Click AC Click OPTN Click 3: Regression Calc r is the product moment coefficient Linear Regression The line of best fit on a scatter diagram approximates the relationship between the variables. The most accurate form of line of best fit is the least squares regression line, which minimises the sum of the squares of the distances from each data point to the line. The regression line is plotted in the form y = a + bx Where b tells you the change in y for each unit change in x. If the correlation is positive, so is b, and vice versa. To calculate a and b, use your calculator and follow the steps above for the product moment coefficient. Independent & Dependent Variables The independent variable is the one that is being changed, the dependent variable is the one being measured and recorded. The independent variable should always be plotted on the x-axis The dependent variable should always be plotted on the y-axis You should only ever use the regression line to make predictions for the dependent variable Exponential Models Exponentials and logarithms can be used to model non-linear data that still has a clear pattern. If the equation is in the form y = axⁿ, a graph of log(y) against log(x) will give a straight line where log(a) is the y intercept and n the gradient. If the equation is in the form y = ab^x, a graph of log(y) against x will give a straight line where log a is the y intercept and log b the gradient.
- Statistical Distributions
A random variable is a variable whose value is dependent on the outcome of a random event. This means the value is not known until the experiment is carried out, however the probabilities of all the outcomes can be modelled with a statistical distribution. There are multiple ways of writing out a distribution: The examples above are all the distribution of a fair, six-sided die. The probability of each outcome is the same, so it is called a discrete uniform distribution. The sum of all probabilities in a distribution must equal 1 The Binomial Distribution If you repeat an experiment multiples times (each time is known as a 'trial'), you can model the number of successful trials with the random variable, X. A binomial distribution is used when: There are a fixed number of trials, n There are only two possible outcomes (success or fail) The probability, p, of success is constant All trials are independent If the random variable X is distributed binomially with n number of trials and fixed probability of success p, it is noted as: X∼B(n, p) Generally, you use a calculator to calculate binomial problems, but you can also use the probability mass function: There are two forms of the binomial distribution: exact and cumulative: Exact Binomial Problems This is when a question asks you to find the probability of there being a specific number of successes out of the number of trials, n. For example, to find the probability of there being exactly 4 successes for the random variable X∼B(12, 0.4) use an exact binomial distribution. To find this on a CASIO ClassWix fx-991EX: Click MENU Click 7: Distribution Click 4: Binomial PD Click 2: Variable Input your values - in this example, x=4, N=12, p=0.4 Click = You should find the answer for this example is 0.213. This means there is a 21.3% chance of getting exactly four successful trials, out of a total 12 trials. Using the table function allows you to see the probabilities for multiple values of x. Cumulative Binomial Problems Generally, this is used more, and is used to find the sum of all the probabilities up to and including a certain value of x: P(X ≤ x) For example, if you want to find the probability of there being up to and including 4 successes (so there could be 0, 1, 2, 3 or 4 successes) for the random variable X∼B(12, 0.4), use the cumulative binomial distribution. This can be done in two ways: Tables There are tables with values for this, typically found at the back of formula books. These will have the most commons values for n, and some standard probabilities. Calculators To find this on a CASIO ClassWix fx-991EX: Click MENU Click 7: Distribution Click DOWN Click 1: Binomial CD Click 2: Variable Input your values - in this example, x=4, N=12, p=0.4 Click = Tables and calculators only ever give the probability for 'up to and including x', P(X ≤ x) Therefore, if you want other forms, such as P(X > a) you need to use the following functions: For P(X > a), use 1 - P(X ≤ a) For P(X < a), use P(X ≤ (a-1)) For P(X ≥ a), use 1 - P(X ≤ (a-1)) For P(X ≤ a), use P(X ≤ a) These rules work because the sum of all the probabilities equals 1. The Normal Distribution The normal distribution is used to model continuous random variables. These are variables that can take absolutely any value. The probability that the continuous random variable takes a particular specific value is always zero, but we can calculate the probability that it takes a value within a certain range. This is because continuous random variables have a continuous probability distribution: It is modelled as a curved graph The probability is the area under the curve The area under the curve can only be defined for ranges, as the area of an infinitely narrow line is zero Because the area under the graph is the probability, and the sum of all probabilities is 1, the area under the whole graph = 1 You can think of a continuous probability distribution as a histogram with an infinite number of infinitely narrow categories: The Normal Distribution A normal distribution is a continuous probability distribution that is bell-shaped and symmetrical about the mean. μ is the population mean, and is in the middle of the distribution σ is the standard deviation of the population σ² is the population variance The graph is symmetrical about the mean The graph has a total area of 1 There are points on inflection at μ + σ and μ - σ If the continuous random variable X is distributed normally with population mean μ and standard deviation σ, it is noted as: X∼N(μ, σ²) All things in nature tend to be modelled with a normal distribution (hence the name), especially heights and lengths of members of a population. It is good to know how the data is spread across the graph: Around 68% of all data is within one standard deviation of the mean (between the two points of inflection) Around 95% of all data is within two standard deviations of the mean Around 99.7% of all data is within three standard deviations of the mean Example In the example above, the median is 180 cm, and the standard deviation is 16. Therefore, continuous random variable, X, is modelled: X∼N(180, 16²). Find P(170 < X < 190) To find this on a CASIO ClassWix fx-991EX: Click MENU Click 7: Distribution Click 2: Normal CD Input your values - in this example, lower=170, upper = 190, σ=16, μ=180 Click = You should find the answer for this example is 0.468. This means there is a 46.8% chance of someone's height being between 170 cm and 190 cm. If only one boundary is specified, e.g. P(X < 190) or P(X > 170), make the other boundary a ridiculously big negative or positive number. For P(X < 190), the upper boundary is 190, and make the lower one -9999999999999999 or something similar For P(X > 170), the lower boundary is 170 and make the upper one 9999999999999999 or something similar Inverse Normal You can also use the normal distribution backwards, to find limits from probabilities. This is done using the inverse normal distribution function. Continuing with the example above, where X∼N(180, 16²): find the value of a for which P(X < a) = 0.35. It is sometimes useful to represent this visually: To find this on a CASIO ClassWix fx-991EX: Click MENU Click 7: Distribution Click 3: Inverse Normal Input your values - in this example, area=0.35 σ=16, μ=180 Click = You should find the answer for this example is 174. This means there is a 35% chance of someone's height being between less than 174 cm Calculators only ever calculate the area to the left This means that if you want to find the value of a for which P(X > a) = 0.35, you need to input 0.65 (1-0.35) into your calculator. Standard Normal Normally distributed variables can be standardised using coding: The standard normal distribution has mean 0 and standard deviation 1 If X∼N(μ, σ²), it can be coded into Z∼N(0, 1²) using the equation Z = (X-μ) / σ Sometimes, the probability P(Z < a) is written as Φ(a) You use your calculator normally, just enter μ = 0, σ = 1. Finding μ and σ Often, you will not know either the mean or the standard deviation of a normal distribution and will have to find it. You will, however, be given a probability, so you can code it into the standard normal distribution and solve. For example, the random variable X ∼ N(μ, 3²). Given that P(X< 10) = 0.3, find the mean. Use the inverse normal to find the value for Z when p = 0.3: Z = -0.524, so rearrange to find μ (you know X=10 and σ = 3) (3)(-0.524) = 10 - μ μ = 13.572 = 13.6 If you know neither the mean nor standard deviation, but have two probabilities, set up simultaneous equations and solve these. Approximating a Normal Distribution Binomial distributions become difficult to work with when n is large. In these instances, if p is close to 0.5, the model can be approximated with a normal distribution: If n is large and p ≈ 0.5, then X∼B(n, p) can be approximated as Y∼N(μ, σ²) where μ = np and σ² =np(1-p) Remember to square root the variance The binomial distribution is discrete, whereas the normal distribution is continuous. This means you need to apply the continuity correction whenever you approximate a binomial normally. This means that you need to add or subtract 0.5 to account for rounding: P(X < a) ≈ P(Y < a+0.5) P(X ≤ a) ≈ P(Y < a+0.5) P(X = a) ≈ P(a-0.5 < Y < a+0.5) P(X ≥ a) ≈ P(Y > a+0.5) P(X > a) ≈ P(Y > a+0.5)
- Hypothesis Testing
A hypothesis is a statement that has yet to be proved. In statistics, the hypothesis is about the value of a population parameter, and can be tested by carrying out an experiment or taking a sample of the population. The test statistic is the result of the experiment / the statistic generated from the sample. In order to perform a hypothesis test, two hypotheses are required: The null hypothesis, H₀ is the one you assume to be correct The alternative hypothesis, H₁ is the one you are testing for, to see if the assumed parameter is correct or not. A specific threshold for the probability of the test statistic must also be defined. If the probability of the test statistic is lower than this threshold, there is sufficient evidence to reject H₀. If it is above the threshold, there is insufficient evidence to reject H₀. This threshold is known as the significance level, and is typically set at 1, 5 or 10%. When ending a hypothesis test, you must conclude by saying whether or not there is sufficient evidence to reject H₀. Do not say accept or reject H₁ Critical Regions & Values If the test statistic falls within the critical region, there is sufficient evidence to reject H₀. The critical value is the first value to fall inside the critical region. The acceptance region is the set of values that are not in the critical region, so there is insufficient evidence to reject H₀. The actual significance level is the probability of incorrectly rejecting the null hypothesis. What this actually means is that: the actual significance level is the probability of getting the critical value One- and Two-Tailed Tests Hypothesis tests can be one-tailed or two-tailed. This refers to how many critical regions there are: For a one-tailed test, H₁: p < ... or H₁: p > ... and there is only one critical region For a two-tailed test, H₁: p ≠ ... so there are two critical regions, one on each 'tail' See the examples below. Hypothesis Tests on Binomial Distributions Often, hypothesis tests are carried out on discrete random variables that are modelled with a binomial distribution. One-Tailed Example A discrete random variable, X, is distributed as B(12, p). Officially, X is distributed with a probability of 0.45. However, there is a suspicion that the probability is, in fact, higher. Find, at the 5% significance level, the critical region and actual significance level of the hypothesis test that should be carried out. Write out the hypotheses & test statistic H₀: p = 0.45 H₁: p > 0.45 X∼B(12, p) Since we are only looking at whether or not the probability is more than 0.45, it is a one-tailed test. Therefore, look for the first value of X for which the cumulative probability is more than 0.95 (1 - 0.05, the 5% significance level) As you can see, the first value to have a cumulative probability of more than 0.95 is 8, so: The critical value is 8 The critical region is > 7 Find the actual significance level 1 - 0.964 = 0.036 0.036 = 3.6 % The actual significance level is 3.6% Write a conclusion If the experiment were repeated 12 times, and 8 or more of the 12 trials were successful, there would be sufficient evidence to reject H₀, suggesting the probability is indeed higher than 0.45 Two-Tailed Example a. A manufacturer of kebab-makers (a kebab-maker-maker, if you will) claims that just 25% of the kebab-makers he makes make low quality kebabs. At the 10% significance level, find the critical region for a test of whether or not the kebab-maker-maker's claim is true for a sample of 12 kebab-makers. Write out the hypotheses and test statistic H₀: p = 0.25 H₁: p ≠ 0.25 X∼B(12, p) We do not know if the probability could be more or less than 0.25, so the test is two tailed. Therefore, divide the significance level by two, and find the critical region. This will be any cumulative probability that is less than 0.05 or more than 0.95 Here you can see the critical region is in two parts, one at each 'tail' of the values; The critical region is X < 1, X > 5 b. A random sample of 12 kebab-makers is taken, and 5 are found to make low quality kebabs. Does this imply the kebab-maker-maker is lying? Method 1 See if 5 is in the critical region 5 is not > 5 not < 1 Conclude 5 does not lie within the critical region for this test (X < 1, X > 5), so there is insufficient evidence to reject H₀ - this implies the kebab-maker-maker is not lying. Method 2 Find the cumulative binomial probability when X=5 When X∼B(12, 0.25), P(X=5) = 0.946 Conclude P(X=5) = 0.945, which is not within the significance level for the test. Therefore, there is insufficient evidence to reject H₀ - this implies the kebab-maker-maker is not lying. Hypothesis Tests on Normal Distributions You can carry out hypothesis tests on the mean of a normally distributed random variable by looking at the mean of a random sample taken from the overall population. To find the critical region or critical value, you need to standardise the test statistic: Then, you can use the percentage points table to determine critical regions and values, or you can use the inverse normal distribution function on a scientific calculator. Example The kebabs that the kebab-maker makes have diameter D, where D is normally distributed with a mean of 4.80 cm. The kebab-maker is cleaned, and afterwards a 50 kebabs are made and measured, to see if D has changed as a result of the cleaning. D is still normally distributed with standard deviation 0.250 cm. Find, at the 5% significance level, the critical region for the test. Write out your hypotheses H₀: μ = 4.8 H₁: μ ≠ 4.8 Assume H₀ is true: Sample mean of D, Ď ∼ N(4.8, 0.25²/50 ) Code data: Z = (Ď - 0.48) / (0.25/√50) Z ∼ N(0, 1) The test is two tailed, so area on each side should be 0.025 (half of 5%): Decode, using ±1.96 (Ď - 0.48) / (0.25/√50) = -1.96 Ď - 0.48 = -0.0693 Ď = 0.411 (Ď - 0.48) / (0.25/√50) = 1.96 Ď - 0.48 = 0.0693 Ď = 0.549 Conclude The critical region is when the sample mean is smaller than 0.411 or larger than 0.549 Hypothesis Tests for Zero Correlation You can determine whether or not the product moment coefficient, p, of a sample indicates whether or not there is likely to be a linear relationship for the wider population using a hypothesis test. Use a one-tailed test if you want to test if the population p is either > 0 or < 0 Use a two tailed test if you want to see that there is any sort of relationship, so p ≠ 0 The critical region can be determined using a product moment coefficient table. It depends on significance level and sample size. To calculate the product moment coefficient of the sample, use your calculator (see notes sheet on regression & correlation).
- Indices & Algebraic Methods
When working with indices, there are eight laws that must be followed: These can be used to factorise and expand expressions. Surds Surds are examples of irrational numbers, meaning they do not follow a repeating pattern but go on forever, uniquely. Pi is the most common example of an irrational number, but surds are slightly different - they are the square roots of non-square numbers. √4 = 2 4 is a square number, so gives a rational square root √2 = 1.4142... 2 is not a square number, so its square root is irrational Like with indices, there are rules that apply to surds: These can be used to rationalise denominators: For fractions in the form 1 / √a, multiply both numerator and denominator by √a For fractions in the form 1 / (a + √b), multiply both numerator and denominator by (a - √b) For fractions in the from 1 / (a - √b), multiply both numerator and denominator by (a + √b) This is known as the conjugate pair (switching the sign of the denominator) Algebraic Fractions To simplify algebraic fractions, factorise whatever can be factorised so that parts of the numerator and denominator can cancel: Multiplication To multiply fractions, any common factors can be cancelled before multiplying the numerators and denominators. Division To divide fractions, multiply the first fraction by the reciprocal of the second fraction (flip the second fraction). Addition & Subtraction To add or subtract one fraction from another, a common denominator must be found. Long Division of Polynomials Polynomials are expressions that contain only rational numbers, positive indices and/or variables in the numerators. 3x + 5 and 3x² + 5x + 7 are examples of polynomials 3/x, √x and 5x-² are not polynomials Polynomials can be divided by (x ± p), where p is a constant, using long division: The Factor Theorem The example above divides perfectly - it does not have a remainder. This means that (2x+1) must be a factor of the initial expression we divided it into. There is a quicker way to check this: the factor theorem. The factor theorem states that if f(x) is a polynomial, then: if f(p) = 0, then (x-p) is a factor of f(x) if (x-p) is a factor of f(x), then f(p) = 0 The Remainder Theorem This can be used to find the remainder of a long division, without actually doing the division. If (x-a) is not a factor of f(x), then the remainder is given as f(a) Partial Fractions If a fraction has two or more distinct factors in its denominator, it can be separated into partial fractions. Two Linear Factors Three Linear Factors This method cannot be used if two of the factors are the same (repeated) Repeated Factors Improper Partial Fractions An improper algebraic fraction is a fraction where the numerator has an equal or higher power to the denominator. These must first be converted into proper fractions before they can be expressed as partial fractions. There are two methods of doing this: Use algebraic long division and add the remainder divided by the divisor to the quotient Multiply by the divisor and add the remainder You can find the remainder using the remainder theorem. Mathematical Proof A mathematical proof is a logical argument to show that a conjecture (a mathematical statement) is always true. Typically, a proof begins with a theorem (a pre-established fact). There are a number of requirements for a valid mathematical proof: All information and assumptions being used must be stated Every step must be shown explicitly Every step must lead on logically from the previous step All cases must be covered A proof must always end in a statement of proof. To prove an identity, like (a+b)(a-b) ≡ a² - b², you start with the expression on one side, and manipulate it algebraically until it is exactly the same as the other side. Again, the requirements above apply. Proof by Deduction Proof by deduction means starting from a fact or definition and using logical steps to prove the conjecture. Conjecture: The product of two odd numbers is also odd Proof: If a and b are integers, then 2a+1 and 2b+1 are definitely odd integers (2a+1)(2b+1) = 4ab + 2a + 2b +1 = 2(2ab+a+b) + 1 This is in the same form as 2m+1, the standard form for an odd number Therefore, the product of two odd numbers is always odd Proof by Exhaustion This involves breaking a proof into smaller proofs and dealing with these all individually. Since it requires every possible instance to be calculated, it is impossible for an infinite range (such as the example above) but can only be used on smaller scales between set limits. Conjecture: The sum of perfect cubes between zero and 100 is a multiple of 10. Proof: The only perfect cubes between zero and 100 are 1, 8, 27 and 64 1 + 8 + 27 + 64 = 100 100 = 10(10) Therefore, the sum of cubes between zero and 100 is a multiple of 10. Proof by Counter-Example Perhaps the simplest form of mathematical proof (though often the most frustrating), proof by counter-example works by finding a single occurrence when the conjecture is false. If the conjecture is false once, it is always false. Conjecture: All even multiples of five are also multiples of 4 Counter-Example: 5x2 = 10 10 is even 10 is not divisible by 4 Therefore, not all even multiples of five are also multiples of four Proof by Contradiction Proof by contradiction works by first assuming the conjecture is untrue. Then, you show through logical steps that this is impossible, and so you conclude that the initial conjecture is in fact true. The contradiction can either be with the initial assumption, or something else that is known to be true. Conjecture: √2 is irrational Assumption: √2 is not irrational Proof by contradiction: Rational numbers can be expressed in the form a/b, where a and b have no common factors So, √2 can be written as a/b: √2 = a/b Squaring both sides gives 2 = a²/b² This can be rearranged to give a² = 2b² This means that a² must be even, and so a is also even, and can be expressed as 2n (where n is an integer) Therefore, (2n)² = 2b² This equals 4n² = 2b², which cancels to 2n² = b² This shows that b² is also even, and so is b If a and b are both even, they have a common factor, 2 This contradicts the statement that a and b have no common factors, so √2 must be irrational
- Sequences & Series
A sequence is a list of numbers with a particular relation; a series is the sum of such a list of numbers. Sequences are sometimes a,so referred to as progressions. Arithmetic Arithmetic Sequences An arithmetic sequence has a constant defined distance between terms, e.g. 1, 3, 5 7, 9 etc. The first term is 1 and the common difference is +2. The common difference can be positive (the sequence is increasing) or negative (the series is decreasing). To calculate the nth term, u(n), of an arithmetic sequence: where a is the first term and d the common difference. Arithmetic Series An arithmetic series is the sum of all numbers in an arithmetic sequence. The sum of the first n terms is given by: where a is the first term, d the common difference, and l the last term. Geometric Geometric Sequences A geometric sequence is a sequence where there is a common ratio, not a common difference. This means the relationship between the numbers is a multiplication, not addition/subtraction. For example, the sequence 2, 4, 8, 16, 32 etc. is geometric - each term is multiplied by 2. The formula for the nth term of a geometric sequence is: where a is the first term and r is the common ratio. Geometric Series A geometric series is the sum of the first n terms of a geometric sequence. The sum of the first n terms is given by: where a is the first term and r the common ratio. The common ration cannot be 1, else each number would be the same and the sum is just n x a. Sum to Infinity As n tends towards infinity, the sum of the series is called the sum to infinity. If a series is getting bigger, its sum tends to infinity, e.g. the series 2 + 4 + 8 + 16 + 32 etc. This happens when r > 1, and the series is known as divergent. If a series is getting smaller, its sum tends to a finite value, e.g. the series 2 + 1 + 1/2 + 1/4 + 1/8 etc. This happens when r < 1, and the series is known as convergent. When a series is convergent, we can calculate the fixed value that its sum tends to, its sum to infinity: Sigma Notation The Greek capital letter sigma, ∑, is used to note sums. Limits are shown above and below the ∑ to tell you from which term to which term to sum, followed by an expression. This is the function used to calculate every term in the sequence: There are standard results for this you can substitute into other series: If the series is given in the form of an expression, but there are too many terms to write out, just write out the first few to find the first term and common difference/ratio. Then use standard arithmetic/geometric sum equations. Recurrence Relations A recurrence relation defines each term of a sequence as a function of the previous term. This means you need to know at least one term in a sequence to work forwards or backwards from. A recurrence relation is noted as: An example of how to calculate a sequence from a recurrence relation is: There are three forms a sequence can take, depending on the recurrence relation: A sequence is increasing when u(n+1) > u(n) A sequence is decreasing when u(n+1) < u(n) A sequence is periodic if the terms repeat in a regular cycle: u(n+k) = u(n) where k is the order of the sequence. A sequence can also take none of these three forms. 1, 3, 5, 7, 9... is increasing 6, 4, 2, 0, -2... is decreasing 1, 3, 5, 1, 3, 5, 1, 3, 5... is periodic with an order of 3 1, 7, 4, -9, 12... is none of the above.
- Quadratic & Simultaneous Equations and Inequalities
The standard format of a quadratic expression is ax² + bx + c There are three ways of solving quadratic equations: Factorising Factorise a quadratic in the form ax² + bx + c = 0, and set each bracket to equal 0 to find the values of x (the roots). The Quadratic Equation The (b² - 4ac) inside the square root is known as the discriminant, and is used to show how many roots a quadratic has: b² - 4ac > 0: The quadratic has two distinct roots b² - 4ac = 0: The quadratic has one repeated root b² - 4ac < 0: The quadratic has no real roots Completing the Square More commonly, quadratics are in their standard form. In this case, this version is used: Simultaneous Equations Linear Simultaneous Equations There are two ways of solving linear simultaneous equations: elimination and substitution. For example, solve the following simultaneous equations: x + 3y = 11 4x - 5y = 10 Elimination Multiply the first equation by 4 4x + 12y = 44 4x - 5y = 10 Subtract 17y = 34 y = 2 Substitute this into equation 1 x + 6 = 11 x = 5 Substitution Rearrange the first equation to make x the subject x = 11 - 3y Substitute this into the second equation and solve 4(11 - 3y) - 5y = 10 44 - 12y - 5y = 10 44 - 17y = 10 -17y = -34 y = 2 Substitute this into the rearranged equation 1 x = 11 - 3(2) x = 5 Quadratic Simultaneous Equations Two simultaneous equations, one linear and one quadratic, can have up to two pairs of solutions. Don't get confused between the tow pairs! You always use the substitution method above to solve quadratic simultaneous equations - rearrange the linear equation and sub into the quadratic: Solve the simultaneous equations, x + 2y = 3, and x² +3xy = 10 Rearrange linear equation to make x the subject x = 3 - 2y Substitute into quadratic equation & solve (3 - 2y)² + 3y(3 - 2y) = 10 9 - 12y + 4y² + 9y - 6y² = 10 9 - 3y - 2y² = 10 2y² + 3y +1 = 0 (2y +1)(y +1) = 0 y = -1/2, -1 Find the corresponding x values x = 3 - 2(-1/2) x = 4 x = 3 - 2(-1) x = 5 Get the pairs together correctly: x = 4, y = -1/2 and x = 5, y = -1 Graphing Simultaneous Equations The solutions to a pair of simultaneous equations represents the intersections between their graphs. For a linear and quadratic pair of simultaneous equations, you can use the discriminant of the substituted equation (the linear equation substituted into the quadratic equation) to show whether or not there are any solutions, and if so, how many. Inequalities There is certain notation for inequalities on a number line: Linear inequalities are rearranged to make the variable the subject. To solve a quadratic inequality: rearrange so that to the right of the inequality sign is 0 solve the remaining quadratic on the left Sketch this equation roughly to see where the roots are and if it is positive or negative Identify the correct section. Regions on Graphs It is possible to show regions closed off by one or multiple lines on a graph. Again, there is certain notation to be aware of: Dashed lines do not include the curve Solid lines do include the curve Shaded areas represent the defined region
- Graphs, Functions & Transformations
When sketching graphs, it is important to clearly show and label any coordinate-axis intercepts (y-intercepts and roots) as well as any stationary points (e.g. turning points). Linear Graphs The general from for a linear graph is y = mx + c, where m is the gradient and c the y-intercept. Gradient is found as rise/run: This equation can be rearranged to give an alternate equation for a line, which is more useful when you know two points and need to know the line connecting them. y2 - y1 = m(x2 - x1) To find the length of a section of line, use Pythagoras' Theorem. Two parallel lines have an equal gradient, so will never meet. Two perpendicular lines have gradients that are each other's negative reciprocal, and so they do cross. This means that the product of their two gradients equals -1 Quadratic Graphs The general form of a quadratic expression is ax² + bx + c. All quadratic graphs are parabola-shaped, symmetrical about one turning point (this can be a maximum or minimum): For quadratics in the form ax² + bx + c, c is the y-intercept. Completing the square gives the coordinates of the turning point: When f(x) = a(x + p)² + q, the turning point is at (-p, q) The discriminant tells you how many roots there are, so how many times the graph crosses the x-axis. Cubic Graphs The general form for a cubic expression is ax³ + bx² + cx + d, and can intercept the x-axis at 1,2 or 3 points. If you do not know the coefficient, then you can find out which way the graph goes by seeing what happens as x tends to ±∞: If as x → ∞, y → ∞ and x → -∞, y → -∞, the graph is positive If as x → ∞, y → -∞ and x → -∞, y → ∞, the graph is negative Cubic graphs can have just 1 or 3 distinct roots, 1 distinct root with a repeated root, or 1 triple repeated root. A triple repeated root occurs when the graph has just one stationary point, and this is on the x-axis A distinct root occurs when the graph crosses the x-axis A repeated root occurs when the graph touches the x-axis but does not cross it To sketch a cubic, you need to know the roots. If it is given in the form ax³ + bx² + cx + d, you need to factorise it first. This will tell you how many roots it has, and where they are. Then, testing to see what happens as x tends to ±∞ shows the shape. Quartic Graphs The standard form for a quartic function is ax⁴ + bx³ + cx² + dx + e where a, b, c, d and e are real numbers and a is not zero. Again, you need to know the roots of the function and the y-intercept to be able to sketch it. Reciprocal Graphs To sketch graphs of reciprocals, such as y = 1/x, y = 1/x², or -3/x, you need to know the asymptotes. These are lines that the graphs tend towards, but never touch or cross. Graphs in the form y = k/x or y = k/x² have asymptotes at x=0 and y=0 The greater the value of the numerator, the further the graph is from the coordinate axis. The asymptotes are still y=0 and x=0, however. Functions In maths, functions are relationships that map a value from a set of inputs to a single output. The set of inputs is known as the domain, and the set of possible outputs is the range. The roots of a function are the values of x for which f(x) = 0. There are two types of functions, one-to-one and many-to-one. Anything else is not a function: Composite Functions Two functions can be combined to form a composite function: fg(x) = f(g(x)) Apply g first, then apply f to this Piece-wise Defined Functions Often, functions will be split up into two or more parts, each of which applies for a certain range of values. Modulus The modulus of a number is its non-negative (or absolute) numerical value. For example |-3| = 3. The modulus of a function, therefore, is function where all input values give a positive output, regardless of whether or not the input (or x-value) is positive or negative: For a modulus functions y = |f(x)|: When f(x) ≥ 0, |f(x)| = f(x) When f(x) < 0, |f(x)| = -f(x) This is easiest shown on a graph of y=x: However, it is also possible to have the function of a modulus, rather than the modulus of a function. This is noted as y = f(|x|), not y = |f(x)|, and represents a reflection in the y-axis: It is important not to get confused between y = |f(x)| and y = f(|x|) Inverse Functions The inverse of a function, f‾¹(x) does the exact opposite to the original function, f(x) - it maps the range of the original function to its domain. Since functions cannot be one-to-many, inverse functions can only be one-to-one. f(x) and f‾¹(x) are inverses of each other ff‾¹(x) = x The domain of f(x) is the range of f‾¹(x) The range of f(x) is the domain of f‾¹(x) The graphs of f(x) and f‾¹(x) are reflections of each other in the line y=x Transformations There are a number of different types of graph transformations, that move every single point on a graph by a certain amount in a certain way. Transformations can be expressed as functions, or as vectors. When multiple transformations are combined, do one after the other. It is generally helpful to sketch out each individual transformation on a separate axis to avoid getting confused.
Often when trying to find the equation of a circle, you will be given a line that intersects with the circle twice (it may be the diameter). To work out the circle from this you need to know the midpoint: The perpendicular bisector of a cord will always pass through the centre of a circle. To find this, find the midpoint of the cord and find the equation of the line perpendicular to it (the negative reciprocal of the gradient) Equation of a Circle The points on a circle are all related through the equation of a circle. For a circle with centre (0,0) and radius r, the equation is: x² + y² = r² For a circle with centre (a, b) and radius r, the equation is: (x-a)² + (y-b)² = r² Expanded Form Sometimes, the equation is given or needed in expanded form: x² + y² + 2fx + 2gy + c = 0 (-f, -g) is the centre of the circle √(f² + g² -c) is the radius To go back from expanded form to the standard factorised form for an equation of a circle, it is quickest to complete the square: Intersecting Lines & Circles A straight line can intersect a circle once, twice, or not at all: To find out how many intersections there are between a circle and a straight line without sketching it, you can solve simultaneously: Circle Theorems
- The Binomial Expansion
Pascal's Triangle is used to expand binomial expressions like (a+b)ⁿ. It is created by summing adjacent pairs to find the number beneath this pair, starting from 1. Here are the first five rows: The (n + 1)th row of Pascal's triangle gives the coefficients in the expansion of (a+b)ⁿ Factorial Notation Parts of Pascal's triangle can be calculated quickly using factorial notation, ⁿCr (spoken "n choose r"): Expanding (a+b)ⁿ When n ∈ ℕ (when n is a positive integer) the binomial expansion is in its simplest form: The general term in an expansion of (a+b)ⁿ is given as: Expanding (1+x)ⁿ If n is a fraction or a negative number, you need to use this form of the binomial expansion: It is valid when |x| < 1 and when n ∈ ℝ The general term in this expansion is given as: x-term Coefficient When the x term has a coefficient, so the binomial is in the form (1+bx)ⁿ, treat (bx) as x, and follow the standard expansion for (1+x)ⁿ The expansion for (1+bx)ⁿ is valid for |bx| < 1, or |x| < 1/|b| Double Coefficients If the binomial is in the form (a+bx)ⁿ, you have to take a factor of aⁿ out of each term: The expansion for (a+bx)ⁿ, where n is negative or a fraction, is valid when |bx/a| < 1, or |x| < |a/b| Often, complex expressions can be simplified first by splitting them into partial fractions (see notes sheet on algebraic methods), then by using a binomial expansion.