Topic Variation. Topic 5: Graphs and networks. Topic 6: Sequences. Topic Investigating and comparing data distributions. Topic Relationships between two numerical values. Topic 7: Shape and measurement.

Topic 9: Trigonometric functions 1. Topic Trigonometric functions 2. Topic 4: Cubic polynomials. Topic Exponential functions. Topic 5: Higher-degree polynomials. Topic Introduction to differential calculus. Topic Differentiation and applications. Topic 7: Matrices and applications to transformations. Topic Anti-differentiation and introduction to integral calculus. Topic 8: Probability. Topic 1: Number systems: real and complex. Topic 7: Coordinate geometry.

Topic 2: Algebra and logic. Topic 8: Vectors. Topic 3: Sequences and series. Topic 9: Kinematics. Topic 4: Geometry in the plane. Divide both sides by 8. Cancel if possible. Here, divide the numerator and denominator by 2. Other ways of representing the answer are shown opposite. Subtract 1 at 2 from both sides. Divide both sides by 2t.

Write the equation with the desired variable on the left. Divide both sides by 2. Square both sides. Form the reciprocal of both sides to make k the numerator.

Once a variable is isolated, we may substitute values of other variables to calculate various values of the isolated variable. The following worked example illustrates some practical applications. Worked example 5. Multiply both sides by 9.

Divide both sides by 5. Add 32 to both sides. Write the equation with F first. Sometimes it may be appropriate to use a common denominator. Write the equation. Solve for r make r the subject as follows.

Divide both sides by. Take the square root of both sides, and write r first. As r is the radius, we take the positive root only. Write the formula that has s as the subject see part b above. Substitute the values given in step 1. Simplify and evaluate. Each of the following is a real equation used in business, mathematics, physics or another area of science. Make the variable shown in parentheses the subject in each case.

Make the variable in parentheses the subject and find its value using the given information. Chapter 1 Linear functions 7. Find the value of b in the triangle above. Given that the volume of a cone is cm3 and its radius at the widest point is 12 cm, find the height of the h cone, expressing your answer in terms of. Here are two examples of where gradient can affect our everyday lives. Can you think of others?

Scientists calculate the required gradient of solar panels so that the maximum amount of energy is absorbed. Calculate the gradient of the line passing through the points 3, 6 and 1, 8. Match up the terms in the formula with the values given. If the angle a line makes with the positive direction of the x-axis is known, the gradient may be found using trigonometry applied to the triangle shown below. The angle given is not the one between the graph and the positive direction of the x-axis.

Calculate the required angle. Without drawing a graph, calculate the gradient of the line passing through: a 2, 4 and 10, 20 b 4, 4 and 6, 14 c 10, 4 and 3, 32 d 5, 31 and 7, What is the. What is the gradient of the sloping section? Give your answer as a. One regulation requires that the maximum gradient of a ramp exceeding mm in length is 1. Find the horizontal length of the ramp required to meet the specifications. To sketch a graph from a linear equation expressed in general form, follow these steps.

Step 1 Plot the y-intercept on a set of axes. Step 2 Find and plot a second point on the line. Do this by substituting any value of x into the equation and finding the corresponding y-value.

Step 3 Join the two points. Alternatively, you can use a CAS calculator or other graphing technology. Sketching linear graphs using intercepts To draw a graph, only two points are needed. A line may then be drawn through the two points, and will include all other points that follow the given rule. The two points can be chosen at random; however, it is often easier to sketch a graph using the points where the graph crosses the axes.

These points are called x- and y-intercepts. Mark the intercepts on a set of axes. Join the intercepts with a straight line. The graphs of some equations do not have two intercepts, as they pass through the origin 0, 0. We could choose any other non-zero value. Worked example Note that the graph passes through 0, 0. Substitute another x-value. Plot the points 0, 0 and 1, 4 on a set of 3 axes.

Note that 4 is 1 1 , which is a little less 3 3 than 1 1. Simultaneous equations are groups of equations containing two or more variables. In this section, we look at pairs of linear equations involving the variables x and y. Each equation, as we have learned in previous sections, may be represented by a linear graph that is true for many x- and y-values.

If the graphs intersect when wouldnt they? Graphical solution Finding the point of intersection of two straight lines can be done graphically; however, the accuracy of the graph determines the accuracy of the solution.

Consequently, using a calculator to solve the equations graphically is more reliable than reading the solution from a hand-drawn graph. Use a CAS calculator to solve the following simultaneous equations graphically.

Using a CAS calculator, make y the subject of the second equation. Write the answer. The methods of substitution and elimination taught in earlier years may be used. Use the substitution method to solve the following simultaneous equations.

Write down and label the equations. Substitute [1] into [2] and label the resulting equation [3]. Solve [3] for x and label the solution as [4].

Substitute [4] into [1]. State the complete answer. Optional check: substitute [4] and [5] into [2] to check that these values for x and y make [2] true. Use the elimination method to solve these simultaneous equations. Rearrange [2] so it is in a similar form to [1]. Call this [3]. Write down [1] again. Substitute [6] into [1].

Two shoppers buy the following at a fruit shop, paying the amounts given. What was the cost of each apple and each banana? Decide on variable names for the unknown quantities. Write equations involving these variables. Work in terms of cents.

Choose a variable to eliminate, in this case b. Find [3] [4] and solve for a. Solve for b. Determine the cost of each type of lolly. The sum of two whole numbers, x and y, is The difference between them is 3.

Write two equations involving x and y and solve them to find the numbers. A farmer counts emus and sheep in a paddock, and notes there are 57 animals and feet. Assuming no animal amputees, how many of each animal are there? If delivery is free, how much did the supplier charge for each type of ball? If the mass of a square panel is 13 kg and that of a circular panel is 22 kg, how many of each panel are there in a truck loaded with 65 panels of total mass kg?

Consider a general linear graph containing the particular points x1, y1 , x2, y2 and the general point x, y which could be any point. Find the equation of the line having gradient 3 that passes through 7, List the given information. Substitute for all variables except x and y. Find the equation of the straight line containing the points 2, 5 and 3, 1.

Write down the points so they match the variables in the formula. Leave x and y as they are. Simplify and express in the two forms required. The gradients of two perpendicular lines, when multiplied together, equal 1.

This type of relationship is known as a negative 2. Do you notice anything special about each pair of graphs? Find the equation of a straight line having the gradient given and passing through the point listed. Find the equation of the line containing each pair of points.

Find the equation of the line passing through 3, Find the equation of the line containing 7, 2 that makes an angle of Find the value of a.

Does the point 4, 8 also lie on this line? Find an equation connecting height with time in months after planting, using the information supplied in the diagram below. The distance, d, between any two points on the Cartesian plane may be found using Pythagoras theorem applied to a right-angled triangle as shown at right. Find the distance between the points 3, 7 and 5, 2 correct to 3 decimal places. Match up 3, 7 and 5, 2 with x1, y1 and x2, y2.

Substitute into the formula for d and simplify. Match 5, 9 and 3, 11 with x1, y1 and x2, y2. Substitute values into the formula for M and simplify. How far is it from A to B as the. If drink stations D1, D2 and D3 are to be placed at the middle of each straight section, give the map coordinates of each drink station.

Many real-life applications, such as fees charged for services, cost of manufacturing or running a business, patterns in nature, sporting records and so on, follow linear relationships. Here, F is the fee in dollars, and t the time in hours. The 50 represents an initial fee for simply turning up, while the 30t is the amount charged for the time spent on the job.

Rent-a-Chef provides food cooked and served by a qualified chef at parties. Under what conditions would it be cheapest to hire Greased Lightning? Define convenient variables.

Write an equation for the cost of hiring both organisations. Use simultaneous equations to find when the cost is the same with each group. Since Greased Lightning has the higher per hour cost, after 3. Notes 1. An alternative approach would be to use a CAS calculator and find the point at which the two graphs crossed.

The cost of hiring a floodlit tennis court consists of a booking fee and an hourly rate. TV connected. How much does the salesperson receive for a week in which he signs up 33 households? Would it be advisable to sign up for the service plan if you expected to need 3 hours of service assistance during the life of a computer purchased from SuperComputers Inc?

After how many rides does an excursion to Fun World become the cheaper option for the same number of rides? Determine the number of hospital visits in a year for which the cost of health services is the same whichever company insures you. A competitor, Savus,. You are planning a holiday, and would prefer to use Savus. Under what conditions days hired could you justify this choice? Aim to get a single variable by itself.

Solve inequations the same way as equations, keeping the original inequality sign at each step, unless multiplying or dividing by a negative number. Substitute means to replace a variable with a value. Lines with the same gradient m are parallel. Simultaneous equations can be solved with a calculator.

Express your answer: a in surd form b to 3 decimal places. If there are only one- and two-dollar coins in the piggybank,.

Find k and h. The value of a must be: a 2 B 1 C 5 d 7 e 11 9 The gradient of the line joining 1, 0 and 4, 10 is: a 4. B Multiply [1] by 7 and put it equal to [2]. C Multiply [2] by 2 and put it equal to [1]. At noon one day he begins cycling from home at.

Any deviation from a straight path, no matter how slight, means the arm must be programmed for more than one direction. Will the robotic arm be able to 12 move in one direction only to drill all three holes?

Find the coordinates for a point D so that the four points form a parallelogram. Her results are shown in the table below. Length of spring cm 4 7 12 What is the natural or unstretched length of the spring?

Plot a graph of the students results. Draw a straight line through the points that best describes the data. Select two points on the line and use them to fit a linear equation to the line.

A second student conducts the same experiment on a similar spring. His results are shown below. Length of spring cm 5 10 16 21 24 28 Force applied N 0 10 20 30 40 The gradients of graphs such as the ones you have drawn give an indication of the stiffness of a spring. The greater the gradient, the harder it is to stretch the spring. The lower the gradient, the easier it is to stretch the spring.

Explain your answer. Chapter review diGital doC Test Yourself doc Take the end-of-chapter test to test your progress page Positive values make the graph slope up when moving or tracing to the right; negative values make the graph slope down when moving to the right.

It is not possible for the springs to have a negative length, so this point is not achievable. Quadratic functions Chapter ContentS 2a 2B 2C 2d 2e 2F 2G 2h 2i 2J 2k 2l Polynomials Expanding quadratic expressions Factorising quadratic expressions Factorising by completing the square Solving quadratic equations Null Factor Law Solving quadratic equations completing the square The quadratic formula The discriminant Graphs of quadratic functions as power functions turning point form Graphs of quadratic functions intercepts method Using technology to solve quadratic equations Simultaneous quadratic and linear equations diGital doC doc 10 Quick Questions.

A polynomial in x, sometimes denoted by P x , is an expression containing only non-negative whole number powers of x. The degree of the polynomial is given by the highest power of the variable x. This chapter will deal with polynomials of degree 2, or quadratics. The constant term is 1. The degree of the polynomial is 2. The leading term is 13x2 as it is the term with the highest power of x.

Chapter 2 Quadratic functions A polynomial may be evaluated by substitution of a number for the variable. A CAS calculator is particularly useful for performing multiple substitutions simultaneously. Using a CAS calculator, define the polynomial 1. To evaluate t x for x-values of 3, 2 and 5, 2. State the degree of each of the following polynomials.

What is the degree of the polynomial? What is the variable? Evaluate v 0. How much faster is she swimming at 0. A quadratic expression is a polynomial of degree 2. It must contain a quadratic term; any others a linear term and a constant term are optional. When expanding brackets, multiply everything by everything else as shown on the diagram at right. That is, first term everything in the second brackets, then second term everything in the second brackets. The above method can be used on all types of binomial expansions, though a couple of shortcuts for special cases are shown in worked examples 1, 2 and 3.

First term everything in the second brackets gives 18x2 21x. Second term everything in the second brackets gives 30x Combine the middle x terms. Remember the shortcut: Square the first term, double the product of the two terms and square the last term. Square the first term to get 4x2. Square the last term to get Rewrite the question so x is the first term in both brackets.

This is not essential as long as all combinations of terms are multiplied in the next step. Expand the brackets first. Multiply the brackets contents by 2. Write the expression. Expand the first pair of brackets. Expand the second pair of brackets. Subtract the two expanded groups in the order given.

Use new brackets for clarity as shown. Apply the negative sign to the contents of the second brackets. Collect like terms and simplify.

Expand the second and third bracketed terms. Subtract the second result from the first result to obtain P x. Simplify your answers to questions in this exercise as fully as possible. Expand the following. Use a calculator to verify the answers. Factorising is the reverse process to expanding. It involves writing an expression as a product of two or more factors. Four methods of factorising will be considered. Take out the highest common factor from every term in the expression and place it in front of the expression.

This makes the factorisation process simpler if further factorisation is required. We would then use inspection see below on the bracketed quadratic. This method involves finding factors of a c that add up to b, i. Arrange the expression in order of decreasing powers of x. Coefficients are numbers or variables in front of x2 and x terms.

If yes, you have a perfect square. Factorise the following. Take out a common factor of 3xy. Make the common factor negative so the leading term inside the brackets will be positive.

Look for a common factor. There isnt one. Is there a common factor? Yes 2. Attempt to factorise by inspection. Write 2 3x and try factors of There are no common factors. There seem to be a few square numbers in the expression, which looks suspiciously like a perfect square. The square root of the first term is 3x, and the square root of the last term is 5 or 5.

Since we need a negative middle term, take 5. Note: This is often called a substitution method. Factorise the new version of the expression. Replace X with x 6. Factorise the following using an appropriate method. Try the questions below. Converting units of time 1 Convert each of the following to the units shown in brackets. Thursday 6. C h a p t e r 1 E a r n i n g m o n e y 3 Calculating salary payments Methods of payment A payment received by an employee for doing a job is called income.

There are many different ways people are paid for performing a job. In this section we are going to look at some of these methods of payment: salaries, wages, commission, royalties, piecework and overtime. Salaries Many people employed in professional occupations are paid a salary. Such employees include teachers, lawyers, accountants and some doctors. The amount paid does not change, regardless of the number of hours worked. Salaries are usually calculated on an annual basis. A salary is therefore usually stated as an amount per annum, which means per year.

Salaries are paid in weekly, fortnightly or monthly amounts. To make calcu- lations about salaries, you will need to remember the following information. Calculate the amount that Dimitri is paid each fortnight. To do this, we need to know the number of days or hours worked per week.

If Charlotte works an average of 42 hours per week, calculate her equivalent hourly rate of pay. A salary is usually calculated on an annual basis and can be paid in weekly, fortnightly or monthly instalments. To calculate information about equivalent daily or hourly rates of pay, we need information about the number of days and hours worked by the employee.

C h a p t e r 1 E a r n i n g m o n e y 5 6 Copy and complete the table below for food production employees. If Fiona works an average of 40 hours per week, calculate the equivalent hourly rate of pay. Give your answer correct to the nearest cent.

Calculate the number of hours that Henry will need to work each week to earn more money than Garry does. A wage is paid at an hourly rate. The hourly rate at which a person is usually paid is called an ordinary rate. The wage for each week is calculated by multi- plying the ordinary rate by the number of hours worked during that week.

We must also consider the number of hours each has worked. Wages are compared by looking at the hourly rate. To calculate the hourly rate of an employee we need to divide the wage by the number of hours worked. Using a similar method we are able to calculate the number of hours worked by an employee, given their wage and hourly rate of pay. The number of hours worked is found by dividing the wage by the hourly rate.

In some cases, wages are increased because an allowance is paid for working in unfavourable conditions. C h a p t e r 1 E a r n i n g m o n e y 7 For example, a road worker may be paid an allowance for working in the rain.

In these cases, the allowance must be multiplied by the number of hours worked in the unfavourable conditions and this amount added to the normal pay. This type of allowance is also paid to casual workers. When you are employed on a casual basis you do not receive any holiday pay and you do not get paid for days you have off because you are sick.

The casual rate is a higher rate of pay to compensate for this. For working on wet days he is paid a wet weather allowance of 86c per hour. A wage is money earned at an hourly rate. To calculate a wage we multiply the hourly rate by the number of hours worked during the week.

To calculate an hourly rate we divide the wage by the number of hours worked. To calculate the number of hours worked we divide the wage by the hourly rate. Allowances are paid for working under unfavourable conditions.

The total allowance should be calculated and then added to the normal pay. A casual rate is a higher rate of pay for casual workers to compensate them for having no holidays and receiving no sick leave. Calculate the casual rate earned by casual waitresses. Find the least time Zhong must work if he is to earn more money than Rema does. Name Wage Hours worked Hourly rate A. Work through the following steps.

Open a spreadsheet and enter the following information. Highlight cells E7 to E11 and select the Fill Down option. The wages for each employee should now be calculated and be formatted as currency. If you now change the hours worked by each employee, his or her gross pay should update automatically.

Calculate the hourly rate to which her salary is equivalent. Answer to the nearest cent. Commission and royalties Commission is a method of payment used mainly for salespeople. When paid com- mission, a person receives a percentage of the value of goods sold. A royalty is a payment made to a person who owns a copyright. For example, a musician who writes a piece of music is paid royalties on sales of CDs; an author who writes a book is paid according to the number of books sold.

Royalties are calculated in the same way as commission, being paid as a percentage of sales. This means that the com- mission rate changes with the value of sales. This type of commission is commonly used in real estate sales. In these examples, each portion of the commission is calcu- lated separately. This is to ensure that the person earns some money even if no sales are made. To calculate this type of pay, you will need to add the retainer to the commission.

In some cases, the commission does not begin to be paid until sales have reached a certain point. A commission is earned when a person is paid a percentage of the value of sales made.

Some commissions are paid on a sliding scale. In these cases, each portion of the commission is calculated separately and then totalled at the end. How much does Ursula receive in commission? What rate of commission does Asif receive? Who earns the most money? How much should his sales be? It is commonly paid for jobs such as car detailing and letterbox delivery.

Chapter 6: Right-angled triangles. Practice assessment 3 Problem solving and modelling task Unit 4. Practice assessment 4 Practice internal examination Unit 4. Chapter 7: Summarising and interpreting data. Chapter 8: Comparing data sets. Practice assessment 1 Problem solving and modelling task Unit 3. Practice assessment 2 Practice common internal assessment Unit 3. Chapter 1: Identifying and describing associations between two variables. Chapter 7: Compound interest loans and investments.

Chapter 2: Fitting a linear model to numerical data. Chapter 8: Reducing balance loans. Chapter 9: Annuities and perpetuities. Chapter 3: Time series analysis. Chapter Graphs and networks. Chapter 4: The arithmetic sequence. Chapter 5: The geometric sequence. Chapter Networks and decision mathematics. Practice assessment 3 Unit 4 examination. Practice assessment 4 Units 3 and 4 examination. Chapter 6: Earth geometry and time zones. Practice assessment 2 Unit 3 examination. Chapter 1: The logarithmic function 2.

Chapter 8: The second derivative and applications of differentiation. Chapter 2: Calculus of exponential functions.

Chapter 9: Cosine and sine rules. Chapter 3: Calculus of logarithmic functions. Chapter 4: Calculus of trigonometric functions. Chapter 5: Further differentiation and applications. Chapter Discrete random variables. Chapter 6: Antidifferentiation. Chapter General continuous random variables.

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