Preliminary Exam Preparation, Study Notes

Preliminary Exam Preparation, Study Notes

Linear Equations and Functions

Algebraic Terms

An algebraic expression is a general statement involving pronumerals. Pronumerals are letters of the alphabet that stand for numbers. An algebraic expression is made up of terms.

·  When adding or subtracting it is important to remember to only add or subtract ‘like terms’.

·  When multiplying terms with powers of the same base, add the powers.

·  When dividing terms with powers of the same base, subtract the powers.

Formulas

A formula is an algebraic rule describing a relationship between pronumerals. For example, the volume of a cylinder has the formula

where r represents the radius of the cylinder’s base and h is its height. This means that the volume of the cylinder is the product of ‘

, the radius squared and the height.

Expanding Expressions

Algebraic expressions involving grouping symbols (brackets) can be expanded and simplified. Expanding means rewriting the expression ‘the long way’ and removing the grouping symbols.

Solving Equations

An equation contains an algebraic expression and an equals sign. To solve an equation, we find the value of the pronumeral that makes the equation true. The process of solving an equation requires the use of inverse (opposite) operations. To solve an equation you must:

1.  Perform inverse operations on both sides of the equation

2.  Aim to have the pronumeral on one side and a number on the other. E.g. x=12

Equations Involving Algebraic Fractions

For equations involving algebraic fractions, both sides can be multiplied by a common multiple of the denominators, found by multiplying the denominators together. This way, we convert all fractions into whole numbers and then solve the equation the usual way.

Equations and Formulas

Sometimes when solving a problem involving a formula, the answer is not immediately found after substituting into a value. Instead an equation results, which must then be solved.

Linear Functions

To calculate the gradient of a line you use the formula m=

.

·  A linear function forms a graph that creates a straight line.

·  A linear function has the form y=mx+b, where m and b are constant numbers.

·  A linear function has 2 terms: mx is called the linear term as it contains the variable x (but not raised to a power), while b is called the constant term because it is just a number (no variable).

·  Any function containing higher powers of x such as are called non-linear functions.

·  When a linear function is written in the form of y=mx+b, its gradient and y-intercept are easily identified. The gradient is m and the y-intercept is b.

·  A positive gradient means that the graph is sloping upwards (left to right)

·  A negative gradient means the graph is sloping downwards (right to left)

·  A gradient of -2 means that as the x-values increase by 1, the y values decrease by 2. To draw a gradient of -2 on the number plane, move across 1 unit (as usual), but then go down 2 units.

The Gradient as a Rate of Change

The gradient of a function is the rate of change of y. The gradient is not only a measure of steepness of a line, but also a measurement of how quickly y values change. The higher the gradient, the steeper the line and the faster y changes relative to x. When a line is horizontal it is evident that there is a ‘run’, but no ‘rise’. With vertical lines this is opposite with no ‘run’ and a ‘run’.

Linear Modelling

The graph of a linear function (y=mx+b) is a straight line, demonstrating that the variable y is changing at a steady rate. In fact, as x increases by 1 unit, y increases by m units. Because the value of units. Because the value of y depends on the value of x, y is called the dependant variable and x is called the independent variable. For y=mx+b, the gradient (m) is the rate of change of y, and the vertical intercept (b) is the value of y when x=0. If a observed number pattern suggests a linear relationship, then we use the linear function y=mx+b to model the situation, this is called a linear model.

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Statistical Samples and Displays

Statistical Investigations

In statistical investigations there are three main stages including:
1.  Collect and organize information or data
2.  Summarise and display this data
3.  Analyse and interpret this data

Interpreting Graphs

See page 116 for graph and explanation. No notes!!!

Types of Data

Categorical Data –obtained form a categorical variable is information that can be put into categories that are distinct or arranged in some order (for example, ‘do you own a pet?’, the answer is a ‘yes’ or ‘no’)
Numerical data is obtained from a numeric variable and is information represented by numbers. The data can be discrete or continuous.
Discrete: obtained through a counting process. The possible values are clearly separated from each other.
Continuous: obtained through a measuring process. The possible values are on a continuous scale.

Sample Types

Samples are often conducted when a population s too large or difficult to survey. A population refers to the total amount of items under consideration. There are three types of random samples including:
1.  Simple random sample: each member of the population is equally likely to be chosen, so the sample has the attributes of the whole population. For example, names drawn out of a hat or winning balls in lotto are picked by a tumbler.
2.  Systematic sample: the first member of the survey is chosen at random, then the others chosen at regular intervals. For example, every 20th light bulb is taken from a conveyer belt for testing.
3.  Stratified sample: the population divided into equal strata or layers, and then a random sample is taken from each strata or layer.

Sampling techniques

Sampling techniques include random numbers which can be used to simulate a variety of situations. To use a table of random numbers, chose any starting point, them move systematically up, down or diagonally.
Another example of a sampling technique is Bias and non-random sampling. Bias usually occurs in a non-random sample favoring one section of the population.

Constructing graphs

Data represented graphically is visually appealing, and we are more likely to notice paterns if data is represented in a statistical graph. A statistical display should be simple and interesting and make an impact, but should not mislead the reader.

Misleading graphs

Statistical graphs are often used to display information in a way that may mislead the reader. Advertisers use graphs to entice us to buy products, and company directors often use graphs to display statistical information to their advantage when dealing with shareholders or prospective clients.

Frequency histograms and polygons

A histogram is used to represent quantitative data and is a column graph with no spaces between the columns. The height of each column is represents the frequency of the scores.
See page 143.

Dot plots

A dot plot is a simplified type of histogram. It is easy to see where clusters of scores occur and each score is represented by a symbol, usually a dot. See page 147.

Stem and Leaf plots

See page 149. A stem and leaf plot is represented by a column showing tens (stem) and ones (leaf) for example 4(stem) and 3(leaf) equates to a score of 43.

Radar Charts

Divide all outcomes by 360 degrees. If more then more, use separate color and indicate this. Label the graph
Ratios and Similar Figures

Simplifying Ratios

The parts of a ratio are called terms. A simplified ratio is one where the terms are whole numbers with not common factor. To simplify a ratio, we divide or multiply each of its term by the same number.

The Unitary Method

The unitary method can be used to solve problems involving ratios. The unitary method considers the value of one part and uses this to calculate other parts.

Dividing a Quantity into a Given Ratio

When dividing the value of the parts of a quantity divided into a given ratio, we can use the unitary method or consider each part as a fraction of the whole quantity.

Scale Factors and Centre of Enlargement

Similar figures are the same shape but not necessarily the same size.
·  All figures similar to a given figure will be an enlargement or reduction of that figure.
·  The scale factor shows by how much a figure is enlarged or reduced.
·  The original figure is called the object and the enlarged figure is called the image.
·  The matching angles in each figure are equal to preserve the same shape.
·  Two similar figures that have a scale factor of 1 are said to be congruent.
·  A centre of enlargement (page 172 for example) will help one to draw similar figure.

Properties of Similar Figures

·  Similar figures have all matching angles equal.
·  Similar figures have matching sides in the same ratio.

Using Shadows and Similar Triangles

On a sunny day, a sick of known length, such as a metre rule, and a long tape can be used to determine the height of trees, flagpoles or buildings in your neighbourhood with the aid of similar triangles. The stick is often referred to as a shadow stick. One way of finding the height of an object is by using the formula:

=

Scale Drawings

A scale drawing is usually a reduction of a real object, such as a building, but can be an enlargement of a very small object, such as a computer chip. The scale factor used in a scale drawing is called the scale. Some common scale drawings are house plans and maps. Examples of ways used to represent scales include; 10mm to 1m, 25:1, 1:100, 1cm = 2m and so on.

Floor Plans and Elevations

When an architect is drawing up two-dimensional plans for a new home, the view from the top looking down is called the floor plan or plan, and the views from the front, back and sides are called elevations. The plan is a scale drawing of the floor of the house. Yeah, (HELP?).

Symbols and Calculations from Plans and Elevations

No notes on this section, simply interpretations of graphs starting on page 188. Due to this and the fact that I’m special, I need HELP.

Earning and Taxation

Wages, salaries and overtime

A wage is a payment calculated on the number of hours worked in a given period of time, usually weekly. People who engage in manual or mechanical work earn a wage. The more hours they work the more pay they receive.
A salary is a fixed payment quoted as a yearly amount but paid weekly, fortnightly or monthly. People who engage in clerical and professional work earn a salary.
Overtime pay is paid to a wage earner who works beyond normal hours. Salary earners are not paid extra for overtime but may receive fringe benefits.

Commission, Piecework and Royalties

Not all workers are paid according to the amount of time they work.
Commission (earned by salespeople and agents) is calculated on the value of items sold.
Piecework (earned by dressmakers, fruit pickers and craftspeople) is calculated on the number of items made or processed.
Royalties (earned by writes, composers and inventors) are calculated on the number of copies sold or made of their creative piece of work.

Bonuses and Allowances

A bonus is paid to employees who produce work of a high quality or volume.
An allowance is paid to employees who either incur expenses in their line of work or work under dangerous conditions.
Annual leave loading is extra pay given during annual leave (usually 4 weeks at Christmas time). It is paid at a rate of 17.5% of 4 weeks normal pay.
A government allowance is paid by the Federal government to people who aren’t financially secure, such as the old, young, sick and unemployed.

Gross and Net Pay

Everyone who earns an income pays a percentage of it to the government as income tax. The government collects these taxes to fund public programs and services such as schools. Income tax is usually deducted from a persons everyday pay know as PAYE tax.
NET PAY=GROSS PAY – TAX – OTHER DEDUCTIONS

Household Bills

Features of a household bill include:
1.  Account number: Your customer ID
2.  Account period: The period of service covered by your bill
3.  Amount due, Dater Due: The amount to be paid and the due date
4.  Last bill: The amount charged on the last bill.
5.  Your payment: The amount paid on the last bill
6.  Fixed charges: Constant fees or charges for using the service
7.  Variable charges: charges based on the amount of the service used

Budgeting

A budget is a plan for managing your income wisely. A budget is divided into two sections: income and expenses. A balanced budget has its total expenses equal to its total income.

Income tax

The tax rate operates on a sliding scale. Not all income is taxed. For example, deductions can be made from work related expenses. Once this is subtracted one can work out their total taxable income.
TAXABLE INCOME = INCOME – ALLOWABLE (TAX) DEDUCTIONS

Goods and Services Tax

Since the year 2000 there has been GST added to most items. GST tax is on spending unlike income tax which is on earning.

Trigonometry

Pythagoras’ theorem

Pythagoras noted that the hypotenuse is the longest side of a right-angled triangle and is always opposite the right angle. He also noted that the square of the hypotenuse is equal to the sum of the squares of the other 2 sides. This is simplified through the formula

. When using the formulas you should:
1.  Draw a rough diagram if one is not provided.
2.  Decide whether the hypotenuse or shorter side needs to be found.
3.  Check whether the answer looks reasonable: compare it to the diagram.
4.  Make sure that the hypotenuse is the longest side, or a shorter side is not longer then the hypotenuse.

Investigating the Tangent Ratio