VCE mATHEmATiCS. 2ND EDiTiON. UNiTS 1 & 2. mATHS QUEST 11 mathematical methods CAS. CASiO CLASSPAD EDiTiON. TEA. CHER EDiTiON . Maths Quest Maths B Year 11 for Queensland Solutions Manual contains the 1 Each of the following is a real equation used in business, mathematics, 1 Use a graphics calculator or other method to sketch graphs of the following on the. "This calculator companion is also available as a PDF file on Maths Quest 11 Mathematical Methods CAS Third Edition eBookPLUS"--Back cover. For secondary.
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Mathematical Methods CAS - Maths Quest 11 - Ebook download as PDF File . pdf), Text File .txt) or read book online. Text-book for Math Methods Units 1 & 2 for. Maths Quest 11 Mathematical Methods CAS Prelims - Download as PDF File . pdf), Text File .txt) or read online. Maths Quest 11 Mathematical Methods CAS. John Wiley &. Sons Australia. John Wiley & Sons Australia / Photo taken by Peter Merriner. caite.info Shutterstock alice-photo / caite.info 1p
Introduction ix. C Multiply  by 2 and put it equal to . Continue to factorise using a difference of squares. Subtract the second result from the first result to obtain P x. Find the x-coordinate of the turning point.
Robyn Robyn Ellen — Notes: Target Audience: Dewey Number: Reproduction and communication for other purposes Except as permitted under the Act for example. Ltd Mathematics — Textbooks. All inquiries should be made to the publisher. Ltd 42 McDougall Street. Mathematics — Problems exercises. This resource contains: Icons appear for the eBookPLUS to indicate that interactivities and eLessons are available online to help with the teaching and learning of particular concepts.
Student textbook Full colour is used throughout to produce clearer graphs and headings. Worked examples also contain CAS calculator instructions and screens to exemplify judicious use of the calculator. Introduction ix. Exercises also contain questions from past VCE examination papers as well as Exam tips.
Exercises contain many carefully graded skills and application problems. Each chapter concludes with a summary and chapter review exercise containing examinationstyle questions multiple choice. Also included are questions from past VCE exams along with relevant Exam tips. Technology is fully integrated in line with VCE recommendations. Exam practice sections contain exam style questions. Many worked examples have eBookPLUS icons to indicate that a Tutorial is available to elucidate the concepts being explained.
A selection of questions are tagged as technology-free to indicate to students that they should avoid using their calculators or other technologies to assist them in finding a solution. Career profiles and History of mathematics place mathematical concepts in context. Tutorial icons link to one-way engagement activities which explain the worked examples in detail for students to view at home or in the classroom. Two tests per chapter. Interactivity icons link to dynamic animations which help students to understand difficult concepts.
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To include a comma in your tag, surround the tag with double quotes. Please enable cookies in your browser to get the full Trove experience. Skip to content Skip to search. Home This edition , English, Book, Illustrated edition: Maths quest 11 mathematical methods CAS: Scoble, Patrick, author. Physical Description iv, 84 pages: Published Milton, Qld.
Other Authors Williams, Robyn, author. Karanikolas, Nicolaos, author. Boucher, Kylie, author. Roberts, Gayle, author.
Nolan, Jennifer, author. Phillips, Geoff, author. Edition 3rd edition. Subjects Mathematics -- Problems, exercises, etc. Mathematics -- Study and teaching Secondary -- Australia -- Victoria. Mathematics -- Textbooks. The first term in the brackets must be x2. Imagine the expansion of the expression in step 3. Rewrite P x. Write the original polynomial in factorised form.
Consider the x2 term from step 6. Consider the x term from step 6. This must equal —9x2 from the original cubic. This must equal ax3. Factorise the quadratic term. This must equal —2c. This confirms step 7. Consider the constant term Consider the x3 term 2x3. This must equal —2x from the original cubic. Write P x. Use the formula to factorise.
Worked example 15 Factorise the following using the sum or difference of cubes formula.
There are shortcuts for factorising such cubic expressions. Examples of each are shown in the table below. Apply the sum of cubes formula. What are the values of m and n? Use the Null Factor Law to solve. Consider the factorised equation to solve. Take out a common factor of x. The simplest value to try is 1. Subtract from both sides. Subtract 2 from both sides and simplify. Consider factors of the constant term that is. Take the cube root of both sides.
Think 1 2 3 4 5 WriTe Write the equation. Factorise the brackets using a difference of squares. Use long or short division to find the other factors of P x. Polynomial equations can also be solved using a CAS calculator.
The solutions can be found by using the solve function on a calculator page. How many other distinct solutions are there? When this occurs. For negative cubic graphs. The graphs below show the two main types of cubic graph. Combine information from the above steps to sketch the graph. Note the skimming of the x-axis indicative of a repeated factor.
Make each bracket equal to 0 and solve a mini-equation. Combine all information and sketch the graph. Worked example 19 Sketch graphs of the following. Factorise P x to find x-intercepts. The cubed factor.
Note the graph is a positive cubic. Write down the x-intercepts determined by making each bracket equal to 0 and solving for x.
All terms involving x are equal to zero. We can use the factor theorem and division of polynomials to achieve this. In this case there is only one x-intercept.
A y 2 3 Use the CAS calculator to help sketch the graph. The local minimum B is 0. Analyse the function with the CAS calculator to find the local maximum. Find all intercepts and stationary points. Find the intercepts and the coordinates a b c d of all turning points. Analyse the function with the CAS calculator to find the local minimum. Combine information from steps 1 to 3 to sketch the graph. The graphs shown below are the main types of quartic graphs.
As for the cubic functions. The graph has a positive x4 coefficient. Consider the general factorised quartic. Large positive values for x result in large positive values for y. Combine all the information above to sketch the graph.
Combine all the information from above to sketch the graph. Note that it touches the x-axis where there are repeated squared factors. A cubic power function has a stationary point of inflection at b. Cubic functions can be power functions. Although all linear and quadratic polynomials are also linear and quadratic power functions.
The same understanding of transformations can be used to sketch cubic functions. State the stationary point of inflection b. Stationary point of inflection 2. Find the y-intercept. The effect of a is illustrated below. Stationary point of inflection 0. Worked example 23 Sketch the graph of each of the following. Find the x-intercept. Stationary point of inflection 1.
Use the result from step 1. Use a CAS calculator to verify answers. The equation for this graph is: Intercepts are not required. The range is within this set. Use the Null Factor Law. The range is the set of y-coordinates of points on the graph.
The x-intercepts are 1. The range is not necessarily equal to R. We show this with a small coloured-in circle on the graph.
The domain and range of a restricted cubic function may be a smaller set of numbers. For the function f: If an end value is not included. Sometimes an extreme value is simply the y-coordinate of an end point of a graph. Note the right end point 7.
Use a CAS calculator to sketch the graph over the restricted domain. Sketch the graph over the restricted domain. Use the graph and the local maximum and minimum to determine the range. Use the CAS calculator to determine the minimum at point B. Calculate the value of the end points of the restricted domain. A method of finding maximums and minimums without a calculator will be covered in the study of calculus later in this book.
The local maximum is 1. Use the graph to determine the range. The range is [0. Note the use of a curved bracket to indicate that the end value is not included in the range.
The following example employs cubic regression. Worked example 26 Fit a cubic model to the following data using a CAS calculator.
Year of study x Wombat population W The graph below shows these data.
Write the equation. Write the equation and draw a rough sketch of the graph. This fits a best-fit cubic to the given data. The differences between successive y-values see table are called the first differences. Consider a difference table for a general polynomial of the form We begin the difference table by evaluating the polynomial for x values of 0.
We will call the first shaded cell nearest the top of the table stepped cell 1. The population at the start of each year is shown in the table below. The differences between successive first differences are called second differences. The differences between successive second differences called the third differences. If pairs of data values in a set obey a polynomial equation.
Year Population 0 1 2 3 4 5 6 7 8 9 10 Find and sketch a cubic model for the population. The second differences are constant. Substitute your values for a0. The curve is a quadratic.
Calculate these and place them in the next column. The first differences are not constant. Substitute this information into  and . Showing the third differences is optional. Calculate the first differences and place them in the next column. Note the first differences are constant. Often this is not the case. Simultaneous equations can be used to find a polynomial model when the data are not sequential. The relationship is linear. It may be necessary on occasions to adjust the table to achieve this.
Worked example 28 Complete a finite difference table based on these data and use it to determine the equation for y in terms of x. Substitute this information into . Calculate and fill in the first differences where possible. Each of the points are substituted into the general equation of the quadratic polynomial.
These can be solved using elimination or by using a CAS calculator. Worked example 29 Using simultaneous equations. Use a CAS calculator to solve the simultaneous equations. Write the rule. Think 1 2 WriTe Write down the general rule of a quadratic. Worked example 30 Using simultaneous equations. Think 1 2 WriTe Write the general rule for a cubic. Substitute each point into the general equation to get 4 simultaneous equations.
Substitute each point into the general equation to get three simultaneous equations. If x is the number of dots on the base of each diagram. Find the relationship between the number of dots x and the number of diagonals n.
Bring down the next term. Repeat until no variables remain to be divided. State the quotient and the remainder. Multiply and write the result underneath. How many? Enter data as lists. Circle or shade the stepped cells. Set up a table as shown and find differences by subtracting successive values value — previous value. Find the regression equation linear. Questions 15 and 16 refer to the following graph below. What is the equation of the translated graph and what are the coordinates of the point of inflection?
Sketch the translated graph. State one factor of f x. A mathematically able website designer has found the following equations for these features: Cubic River: Sketch the graph of f x. Find the distance between the proposed checkpoints. Using long or short division. Show at what times the height of this wave is exactly the same height as the pole. Part of the map involves two key features — the Cubic River and the Linear Highway.
Review the discriminant page 3k 3d domain. Factorising cubics and quartics using long division.
The last data point may be the beginning of a share price crash! L2 and L3 are parallel. B L1 and L3 are parallel. Label all key features. The coordinates of the second point of intersection would be closest to: Determine the values of a and b. The equation for this polynomial would best be described by which one of the following?
Its equation would be: Write your answer in exact form. Victor was delayed for 1 minute because he was tying up his shoelaces when he was m from the starting line. Express coefficients correct to 2 decimal places. Determine this cubic function. The distance in metres they each ran was recorded in 1-minute intervals. Write your answer in minutes and seconds.
It will travel from the gate to the seesaws. The individual distances are recorded in the table below. The longer side is 5 metres longer than the shorter side. Write your function in terms of Cs distance in metres and t minutes. Chapter 4 Relations. Anything contained in a set. A set is a collection of things.
These numbers are called irrational. N is the set of natural numbers. For example A capital letter is often used to refer to a particular set of things. Q is the set of rational numbers that is. R is the set of real numbers. It implies that there is nothing in the set. Z is the set of integers. The information in the table can be represented by a graph. The points are called x. The table below outlines this relation.
The cost of hiring a trailer depends on the number of hours for which it is hired. The relation appears to be linear. When graphing a relation. Q and R. Consider the following relation. In our example. It is a discrete relation as x can be only whole number values. Continuous variables include height. If a relationship exists between the variables. Some variables are referred to as continuous variables. It is a continuous relation as x can be any real number. Select values of x.
State the ordered pairs.
Estimate the value of R where this line touches the axis. From this point draw a horizontal line back to the vertical axis.
R beats per minute. In any defined domain. Worked exaMple 3 The pulse rate of an athlete. A particular relation is described by the following ordered pairs: Which graph below best represents this information? If n is the number of computers she sells per week and P dollars is the total amount she earns per week. C dollars. This is referred to as the implied domain of a relation. If the domain of a relation is not specifically stated. The set of all first elements of a set of ordered pairs is known as the domain.
If a relation is described by a rule. Worked exaMple 5 Illustrate the following number intervals on a number line. Worked exaMple 4 Describe each of the following subsets of the real numbers using interval notation. Place a closed circle on the point 1. The range is the set of values covered vertically by the graph.
The domain is the set of values covered horizontally by the graph. The range is the set of values covered by the graph vertically. Place a closed circle on the points 0.
State the coordinate points. Use a CAS calculator to view more of each graph if required. For any y-value there is only one x-value. Functions Relations that are one-to-one or many-to-one are called functions. A line through some x-values shows that two y-values are available: A line through any y-value shows that only one x-value is available: Worked exaMple 8 What type of relation does each graph represent?
Worked exaMple 9 State whether or not each of the following relations are functions. State the domain and range for each function. The shaded side indicates the region not required. Other power functions are: As x becomes very small approaches 0.
It dilates the graph from the x-axis. The horizontal asymptote and the range remain the same. Write a short statement about the effects of 1 a. The dilation factor does not affect the domain. The effect of this reflection cannot be seen in the basic graph. As a is positive. The equation of the graph is given by: Identify the values of a. If a is positive see graph below.
As can be seen from the graph. The domain is still [0. As you cannot get the square root of a negative number.
An inspection of the equation of the graph would also have revealed this. This translated graph has domain [1. There is no x-intercept. To help sketch the graph. There is no y-intercept. Sketch the graph by plotting the end point. Write the coordinates of the end point. Write a short statement about the effects each has on the basic graph of that function.
Inspection of the equation reveals that there is no y-intercept. State the shape of the graph. End point: Write a 1 Write the rule. The y-values are determined from the x-values. Simplify the expression if possible. The letters f. When using function notation the domain can be abbreviated as dom f and the range as ran f.
Domain co-domain. Fully defining functions To fully define a function: Worked exaMple 14 Express the following functions in function notation with maximal domain. It is usually R the set of real numbers. The actual values that y can be — the range — is determined by the rule.
If a function is referred to by its rule only. Domain Co-domain Rule Y is not necessarily the range but is a set that contains the range. The co-domain gives the set of possible values that contains y. Use a CAS calculator to obtain the graph of the function. To confirm the maximal domain and see the range.
State the maximal domain. Worked exaMple 15 State i the domain. Check whether both a vertical line and a horizontal line crosses only once. Worked exaMple 17 A one-to-one function. The graph of a relation is a function if any vertical line crosses the curve at most once. Worked exaMple 18 Which of the following graphs show a one-to-one function? Which of the following functions are one-to-one? A one-to-one function has. It is not a one-to-one function.
The functions are one-to-one for b and c. Only b is a one-to-one function. Note that the domains do not overlap. The graph of f x is shown at right.
If we have one relation. Worked exaMple 19 For each function graphed below state two restricted. For the function to be one-to-one. The inverse of a set of ordered pairs is obtained simply by interchanging the x and y elements.
The domain is restricted. The graph is a c y parabola with turning point 0. Only functions that are one-to-one have inverses. Worked exaMple 21 Sketch the graph of the following and then sketch the inverse. If these points are plotted on a set of axes. These two relations represent two semicircles that together make a complete circle: The rule that defines a circle with its centre at 0. The vertical-line test clearly verifies that the circle graph is not a function.
Assume each set of axes has the same scale for x and y. Draw a circle that passes through these four points. Draw a semicircle above the x-axis. State the domain and range of each. Worked exaMple 22 Sketch the graphs of the following relations. State the domain. Draw the circle. Draw a semicircle on the right-hand side of the y-axis. The range is [2. State the range. When using a CAS calculator to plot circle graphs. State your answers to 1 decimal place.
For a circle to be drawn the radius needs to be a positive number. State the cost function C h. Hence the domain is 0. Worked exaMple 24 The table describes hire rates for a removal van. B hours. The meter will accept a maximum of one-dollar coins.
P metres. P dollars. How many koalas per hectare are there 13 weeks after the virus struck? How long after the virus strikes are there 23 koalas per hectare? Will the virus kill off all the koalas? Explain why. That is, for any x-value there is only one y-value. The equations of the asymptotes are: If a is negative, the graph is reflected in the x-axis.