METHODS TEXTBOOK LINK:Mathematical Methods Cambridge TextBook .. (0 children). MathsQuest Methods Unit 1&2 Solutions Manual. Chapter 17 - Essential Mathematical Methods Unit 1&2 - Download as PDF File . pdf), Text File .txt) or read online. Chapter 17 of Essential Mathematical. Mathematical Methods Units 1 2 AC VCE - Free ebook download as PDF File . pdf), Text An Essential Mathematical Methods Textbook from Cambridge. Cambridge Mathematical Methods Australian Curriculum/VCE Units 1&2 provides a.
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CD-ROM provides complete worked solutions to the exercises contained in Essential mathematical methods 1&2 CAS by Michael Evans, Kay Lipson and. Essential Mathematical Methods CAS 1&2 with Student CD-ROM TIN/CP Version . ISBN paperback. Reproduction and Communication for. Prepare students for VCE success with enhanced print and digital versions. Integrated CAS calculator explanations, examples and problems.
Simultaneous equations can be used to find a polynomial model when the data are not sequential. Complete the square: Give a rule for the volume, V litres, of water in the container at time t minutes after the pouring starts. Sometimes to simplify the situation we represent such functions by a continuous line. In the previous exercise, you may have noticed that: To sketch a graph from a linear equation expressed in general form, follow these steps. The graph of a function cannot be crossed more than once by any vertical line.
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. Attempt to factorise by inspection. Write 2 3x and try factors of A few possibilities are: 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. This is often called a substitution method.
Factorise the following using an appropriate method. Factorise the expression within the brackets using the difference of two squares method. Use written algebra to find the values of k and m, if it is known that the three polynomials are identical. Use a mental arithmetic substitution technique to verify your answers. Sometimes it is clear that it is impossible to find whole numbers to complete a factorisation. In such cases, completing the square may be used to factorise a quadratic.
The method of completing the square involves manufacturing a perfect square so that an expression may be factorised as a difference of squares. Because this method often produces surds, factorising this way is sometimes referred to as factorising over R, where R is the set of real numbers that includes surds. Worked example 7. Use the method of completing the square to factorise the following over R.
Halve and square the x-coefficient 6 , and then add and subtract it from the equation. Since the same value has been added and subtracted, the expression is equivalent to that in the question. Halve and square the x-coefficient 7 , and then add and subtract it from the equation. Combine the first three terms as a perfect square, 7 2. Answer the following questions.
Discuss the differences in the formats of your answers. Complete the square: Since the surd is the square root of a negative number, there are no possible linear factors.
Write your observations. There are no linear factors. As a quadratic equation is a degree 2 polynomial highest x power of 2 , it will have at most two solutions. For A B to equal zero, either A or B or both must be zero. This is known as the Null Factor Law. Rearrange transpose so all terms are on the side on which the x2 term is positive.
Write terms in order of decreasing powers of x. The height of a triangle is 5cm more than its base length. If the area of the triangle is 18cm2, find the base length and height. Consider the quadratic equations below. Equation 1: Fixed point iteration Fixed point or simple iteration is a way of solving equations numerically rather than algebraically.
Repeat the process until successive values for g x are equal within a tolerance of, say, 0. In this example, the equal successive values of g x are 0. The equation could be: If the area of the rectangle is 40cm , find the length and width.
How long does it take the temperature to reach 20 C? The diagram at right demonstrates the idea of rectangular numbers. What mass will produce a bend of 8. A window washer drops a squeegee from a scaffold m off the ground. Does this form converge using fixed point iteration? If so, state the solution found. State the solution. Recall that when you cannot factorise quadratics by the method of sensibly guessing whole numbers, the method of completing the square may be used.
Completing the square may also be used to solve quadratic equations that dont appear to easily factorise. Solve the following, giving answers in exact surd form. Take the constant term in this case the 23 to the other side of the equation, remembering to change the sign.
Use the method of completing the square to solve the following equations, or to explain why there are no solutions. Apply the process of completing the square. Notice here that the next step, finding the square root of both sides of the equation, is not possible as negative does not exist.
You will soon learn a way to predict when this will happen. Rearrange and solve, giving the solution in exact form. An equation that is being solved using completing the square is at the stage shown below. An alternative to the methods of factorising by inspection or completing the square is to use the quadratic formula.
The derivation of the formula follows, and is based on the method of completing the square, but all you have to remember is the formula in the last step. Use the quadratic formula to solve the following without a calculator. At this stage, it is tempting to cancel 2 out of the 8 and 6, but both terms of the numerator must possess this factor, and initially they dont. However, in simplifying into 4 43 , the factor of 2 emerges; then the cancelling is possible in step 6. In the original equation, the coefficient of x2 was 3; also, there are surds in the answer.
These facts suggest that neither completing the square nor standard factorisation would have been straightforward methods. Using the quadratic formula was the most appropriate method for solving this equation.
In the original equation, the coefficient of x2 was 1; also, there are no surds in the answer. This suggests that an easier factorisation method i. This applies only if the question does not specify a particular method.
Using the quadratic formula is not the most suitable method for solving this equation. Use the quadratic formula to explain the solutions that occur with each b-value. The structure of the quadratic formula means the changing value of the expression under the square root sign, b2 , is critical. There was no solution as negative cannot be resolved. Identify a, b and c the coefficients of x2, x and the constant respectively in each of the following quadratic equations.
How high could the anthill be when there are ants in the colony? She has a total of 12 square metres of pavers. She then finds that the value of x found above will need to be rounded either up or down to a multiple of 0. Calculate the effect this will have on her existing supply of pavers if she rounds: If the surface area of a particular container is 60 m2, determine its radius. Both x and y are in metres.
How high is the Gateway Arch? You may have found on occasions that no solutions or roots can be found for a quadratic equation.
If you were using the quadratic formula at the time, you would have found the trouble started when you tried to evaluate the square root part of the formula. The expression under the square root sign is called the discriminant. The discriminant is used to determine how many roots of an equation exist and is denoted by the upper case Greek letter delta. The word real is used to describe numbers we can deal with at present.
The set of real numbers includes positives, negatives, fractions, decimals, surds, rationals numbers that may be expressed as a ratio for example 49 and irrationals. In further studies of maths, you will learn about a way of dealing with square roots of negative numbers using what are known as imaginary numbers. Consider case i, two distinct solutions. Consider case ii, one solution. That is, 3 or 3 are our only choices.
Consider case iii, no solutions. An alternative method is shown in the next example. Since is a more complicated expression than those in the previous example, a graph of versus k on the vertical axis, k on the horizontal axis is useful. Recall how you sketched quadratic graphs in previous work, or see the next section.
This method involving sketching a graph of may be used as an alternative to the method shown in the previous example. Use this result to: Pick a value in the range from step 4 of part a. Do not actually work out any solutions. Use substitution and your previous answers to question 5 to complete the following table. There is no need to actually solve the equations. Quadratic functions are also power functions. The value of the power, n, determines the type of function.
Other power functions will be discussed later. When a quadratic function is written in turning point form, it is written in power form. In previous years work, you will have discovered the following connections between a quadratic function in turning point TP form, and its graph. Write the equation. If a quadratic function is not in power form or turning point form, it must be manipulated in order to answer questions like those posed in the previous example.
To do this, we use the method of completing the square as demonstrated in the following example. State the coordinates of the turning point and the maximum or minimum value of y. Ensure the x2 coefficient is 1.
It is. If not, divide the equation by whatever will change the x2 coefficient to 1. Halve the x-coefficient and square it. Add and subtract this value after the x term.
Write the rule for the function. Begin the process of completing the square by first taking out 2 as the common factor.
Multiply the 2 through the square brackets, leaving it as a factor of the curved brackets. Graphs of quadratic functions as power functions turning point form exercise 2i. What equation should be programmed into the automatic glass cutting machine, using the grid system on the diagram? The graph is: When we talk about sketching a graph, we mean drawing a diagram showing the main features not a true scale graph showing every point plotted accurately using a computer package or other means.
To sketch a quadratic graph, the following features should generally be apparent or labelled. All quadratics have a y-intercept. In this case, the coordinates of the turning point are b2 b 2a , c 4 a Using xt and yt for the coordinates of the turning point, we have.
The y-coordinate of the turning point may be found using the general turning point form above, by completing the square from scratch or by substituting the x-coordinate into the original equation. Recall also the two main types of parabolas: Sketch the graphs of the following, showing all intercepts and the turning point in each case.
Substitute into the original rule to find the y-coordinate of the TP. Find the x-coordinate of the turning point. Alternatively, since there is only one x-intercept 5 , it must be the turning point x-coordinate. Dont be put off if asked to sketch a quadratic graph whose equation doesnt have 3 terms. Such cases are easier to sketch, as the following example shows.
Sketch the graphs of the following equations. Factorise before finding x-intercepts. In this case, recognise a difference of squares. Use written algebra to: For this equation to have one x-intercept, the discriminant, , must equal 0. The x-coordinate of the turning point here, 2.
The amount of vertical translation will be the difference between the original and final y-intercepts. Answer the question. Using the discriminant Since finding x-intercepts for a quadratic graph involves solving a quadratic equation, we can use the discriminant to decide the number of x-intercepts such a graph has. Find the y-intercepts for each of the following. State the x-intercepts for each of the following. Use the quadratic formula to find exact values if possible for the x-intercepts of: Factorise first.
Without sketching, determine how many x-intercepts each of the following graphs have. Also, state the function rule for this new parabola in both expanded and turning point forms. Quadratic expressions, equations and functions are linked closely, as you have previously seen. Study the following table carefully. Two variables, one equals sign in the rule Can be sketched, as it is a set of ordered pairs Sketch is a parabola, with 0, 1 or 2 x-intercepts.
The solutions also known as the roots of a quadratic equation say, 3x2 4. They are also known as the zeros of the related expression here, 3x2 4.
Now consider a general quadratic function variable x whose graph is an upright parabola with a dilation factor from the x-axis of 1. Its x-intercepts are j and k. Find the y-values of each turning point by substituting the x-values in the corresponding functions. The intercepts and turning points are moving to the right horizontally by 1 unit for each step in the progression. This means that only the x-coordinates change.
Also, the coefficient of the x term in the expanded expression is the negative sum of the zeros, and the constant term is the product of the zeros. The turning points and zeros have been translated 3 units to the left. The second parabola has been translated 3 units to the left from the first parabola.
The answers are verified. At what value of t will the comet reach this moon? How long after being fed could the colony survive without further food before none were left?
Both h and d are in metres. The coefficient of the linear term had to be 5. When they returned to class the next day, Harry announced his expression was 0. Who was closest, and by how much? In previous studies you have dealt with pairs of simultaneous linear equations and solved these using algebra. The solution could also be represented graphically. The same is true when we have one linear and one quadratic equation as a pair of simultaneous equations.
Consider the following pair of simultaneous equations: That is, the coordinate pair 3, 10 fits both equations, so it is a solution. That is, the coordinate pair 1, 2 is also a solution. To illustrate the situation graphically, the related linear and quadratic functions must be sketched on the same set of axes. As shown below, there are three possible relationships: When a linear equation and a quadratic equation are solved simultaneously, a new quadratic equation is formed, as you will see in the following examples.
The turning point of the quadratic graph is not. Substitute x-values into the linear equation  to find the corresponding y-values.
Find intercepts for a sketch of the quadratic graph. The turning point is not required here. Sketch the linear and quadratic graphs on the same axes, and include the points of intersection found in part a. The turning point of the quadratic graph.
Substitute x-values into the linear equation  to find the corresponding y-value. Try to factorise. No solution using the Null Factor Law is apparent. Check the value of the discriminant of the quadratic in step 4.
Sketch the linear and quadratic graphs on the same axes. Note that the graphs do not intersect, indicating no solution. Find the possible values of m. The turning point of the quadratic. You may use a CAS calculator to verify solutions. Find the value of c.
Find the values of b and c. At what heights above sea level will the entrance and exit to the tunnel be, given the equations of the mountain profile and road path as shown on the plan? V shape on a set of axes as shown at right. A pole 11 metres high is firmly secured to the ground. A parabolic arch is to be attached to the pole 3 metres above the ground as shown in the diagram below. A metal rod will run directly from the top of the pole straight to the ground so that it just touches the arch in one position only.
Let y be the height above the ground in metres and x be the distance along the ground from the foot of the pole in metres. Give all answers in exact form unless told otherwise. Find where the arch meets the ground in terms of a, that is, find the x-intercept. Consider the discriminant. A polynomial in x, sometimes denoted by P x , is an expression containing only non-negative whole powers of x.
The general form of a degree n polynomial is: To expand: First term everything in the second brackets, then 1 2. Second term everything in the second brackets. Look for a common factor first. General quadratics: Perfect squares: Use when whole number factors are not apparent.
Halve and square the x-coefficient, then add and subtract this new term. Form a perfect square from three of the terms. Continue to factorise using a difference of squares. Halve and square the x-coefficient. Add and subtract the new term. Form a perfect square and solve for x. Turning point form: The turning point is at b, c and the dilation factor is a.
To convert to turning point form, complete the square. Find the y-coordinate of the turning point by substituting xt into the equation for y, or by using b2 or by completing the square. Combine all the information and sketch it. Functions with a positive x2 coefficient are shaped, and those with negative x2 coefficients are shaped. If the information you have gathered doesnt seem to fit, check for calculation errors. Simultaneous quadratic and linear equations.
Solve to find any x-coordinates of intersection. Substitute any x-coordinates into the linear equation to find the corresponding y-coordinates.
State the solutions or state that there are none. Plot area on the vertical axis and w on the horizontal axis. On the graph, label all intercepts and the turning point. How can this be ensured? Warm up with ten quick questions on quadratic functions page Learn about the life of Galois, a 19th century mathematician page Practise expanding differences of squares page Watch how to apply quadratic algebra skills to determine the dimensions of a triangle page 51 diGital doCS doc Expanding brackets and factorising quadratic expressions page Practise simplifying surds page Watch how to find the possible values of an unknown constant in a quadratic equation with zero, one and two solutions page Observe the discriminant and the number of solutions to a quadratic equation page Solve quadratic equations, understanding the discriminant and sketching parabolas page Observe axial intercepts of the graph of a quadratic in general form page 78 doc Observe the turning point of the graph of a quadratic in turning point form page Calculate the solutions to linear and quadratic simultaneous equations page Consolidate your understanding of how to solve simultaneous quadratic and linear equations page Chapter review diGital doC Test Yourself doc Using technology to solve quadratic equations 1 a 3.
Parabola 1: The turning point of the first graph has whole number integer coordinates. ChapTer ConTenTS 3a Expanding 3B Long division of polynomials 3C Polynomial values 3d The remainder and factor theorems 3e Factorising polynomials 3F Sum and difference of two cubes 3G Solving polynomial equations 3h Cubic graphs intercepts method 3i Quartic graphs intercepts method 3J Graphs of cubic functions in power function form 3k Domain, range, maximums and minimums 3l Modelling using technology 3m Finite differences.
Degree 4 polynomials quartics will also be considered. Worked example 1. Just as there is a shortcut for expanding perfect squares, there is also a shortcut for expanding cubes. Expand each of the following. The reverse of expanding is factorising expressing a polynomial as a product of its linear factors. Before learning how to factorise cubics, you must be familiar with long division of polynomials.
You may remember in earlier levels doing long division questions. Consider 3, or 3 The process used is as follows. Write 2 at the top. Write down the 6. Subtract to get 1. Bring down the 4 to form Write 4 at the top. Write down the Subtract to get 2. Bring down the 5 to form Write 8 at the top. The same process can be used to divide polynomials by polynomial factors. Write x2 at the top. Write down the 5x2 15x. Subtract to get Perform the following long divisions and state the quotient and remainder.
Note that there is no term in this equation.
Include 0x2 as a place holder. Think 1. Divide the first polynomial by the second, and state the quotient and remainder. Expand the right-hand side and collect like terms. Use the rules for expanding cubics and quadratics.
Copy the following table. Complete columns 2 to 5 of the table for each of the following polynomials. Copy and complete the following sentences, using your answers to questions 3 to 7 to find the pattern.
In the previous exercise, you may have noticed that: The remainder when P x is divided by x a is equal to P a.
This is called the remainder theorem. We could have derived this result as follows. If 13 is divided by 4, the quotient is 3, and the remainder is 1. ChapTer 3 Cubic and quartic functions. Worked example 9. Find the value of k. The factor theorem The remainder when 12 is divided by 4 is zero, since 4 is a factor of Similarly, if the remainder R when P x is divided by x a is zero, then x a must be a factor of P x.
This is called the factor theorem. Imagine P x could be factorised as follows: Find the value of m. Find the value of n. Find the value of b. Find a possible whole number value of c.
Find the possible values of d. Find a and b. Once one factor of a polynomial has been found using the factor theorem as in the previous section , long division may be used to find other factors. Look at the last term in P x , which is This suggests it is worth trying P 5 or P 5. Try P 5. Using short division The process of long division can take a lot of time and space. One short division method is shown here; it may take a little longer to understand, but it is quicker than long division once mastered.
Using the factor theorem, we can find that x 1 is a factor of P x. Actually, we know more than this: Factorise the following, using short division where possible. Try P 2. The first term in the brackets must be x2, and the last term must be Imagine the expansion of the expression in step 3. We have 2x2, and require 5x2. We need an extra 7x2. It is difficult to factorise a quartic using short division, so we will use long division here. Consider the x term from step 6.
This must equal 2x from the original cubic. This confirms step 7. Factorise the following as fully as possible. Two special cases of cubic polynomials, called sum of cubesand difference of cubes, are discussed in this section. There are shortcuts for factorising such cubic expressions.
Examples of each are shown in the table below. Consider the following expansions. That is, we have two formulas that may be used to factorise sums and differences of cubes. Factorise the following using the sum or difference of cubes formula. What are the values of m and n? Unlike a square root, a cube root can be only positive or negative, not both; for example, 3 8. Factorising to solve polynomial equations The Null Factor Law applies to cubic and quartic equations just as it does for quadratics.
Solve each of the following equations. Consider factors of the constant term that is, factors of 9 such as 1, 3. The simplest value to try is 1. Use the graph, or otherwise, to determine the charges for: B b Assuming that both the rabbit and the fox 5 were running along straight lines, calculate D whether the foxs path would cross the A rabbits track before the irrigation channel.
Explain your answer. Coordinate geometry and linear relations Review 9 A new light beacon is proposed at P 4, 75 for air y trac ying into an airport located at O 0, 0. It is O intended that the aircraft should follow a course over x beacons at P and Q 36, 4 , turning at Q towards the 20 Q 40 runway at O. It is thought that the main runway of the 40 N E W E airport will have one end of its centre line 30 S D at A 48, 10 , but the position of the other 20 B end of this line, B, has not been decided.
What is the equation of the line CD? Method 1 involves payment annually and method 2 involves payment each quarter that is, every three months. The charges for each method are as follows: Calculate which is the cheaper method of payment.
Clearly indicate the approximate number of units of electricity for which the cost is the same for both methods of payment. Use these formulas to calculate the exact number of units for which the cost is the same for both methods. Coordinate geometry and linear relations Review 14 In a metal fabricating yard which has been B ooded by overow from a local river, a A C large steel frame has been partly submerged. The coordinates of the ends relative to an overhead y m crane are A 10, 16 , B 16, 20 , C 24, 8 and B 20 D 18, 4.
The overhead crane moves eastwest 15 A along its rail, and the distance east from a point O 0, 0 is denoted by x. The cranes hook 10 C moves northsouth across the frame and the 5 D distance to the north of the south rail is denoted by y. Units are in metres. The steel frame is to 0 5 10 15 20 25 x m be raised out of the water by lifting it at the midpoint, M, of its middle section. Find the equation of the line along which the hook will be moved. The four points A, B, C and D stand away from the chip itself.
Units are in 25 mm. The unit S is a moveable micro-soldering unit, its tip being at P 0, It is desired to program the tip of the soldering iron, P, to solder wires to the points A, B, C and D, moving along the dashed lines as shown in the graph. The degree of a polynomial is given by the value of n, the highest power of x with a non-zero coecient.
This chapter deals with polynomials of degree 2. These are called quadratic polynomials. Polynomials of higher degree will be studied in Chapter 6. As an introduction to the methods of solving quadratic equations, the rst two sections of this chapter review the basic algebraic processes of expansion and factorisation. An algebraic expression is the sum of its terms. Example 3 Expand the following: You add the terms to complete the expansion.
Example 6 Expand: The graph of a quadratic function is called a parabola. Example 1 3 Simplify each of the following by expanding and collecting like terms: Quadratics 3A. Example 2 4 Simplify each of the following by expanding and collecting like terms: Example 3 5 Simplify each of the following by expanding and collecting like terms: Example 4 6 Simplify each of the following by expanding and collecting like terms: Example 5 7 Simplify each of the following by expanding and collecting like terms: Example 7 9 Simplify each of the following by expanding and collecting like terms: To nd the other factor, divide each term by the common factor.
The common factor is placed outside the brackets. This process is known as taking the common factor outside the brackets. The answers can be checked by expanding. Example 11 a Factorise 3x2 We have seen in the previous section that we can expand a product of two binomial factors to obtain a quadratic expression. That is, we want to start from the expanded expression and obtain the factorised form. We have already done this for expressions that are dierences of squares. We now turn our attention to the general case.
Example 13 Factorise x2 2x 8. A quadratic polynomial is called a monic quadratic polynomial if the coecient of x2 is 1. The quadratic polynomial x2 2x 8 factorised in the previous example is monic. Factorising non-monic quadratic polynomials involves a slightly dierent approach.
We need to consider all possible combinations of factors of the x2 term and the constant term. The next example and the following discussion give two methods. Example 14 Factorise 6x2 13x Solution There are several combinations of Factors of Factors of Cross-products add factors of 6x2 and 15 to consider. Here is a second method for factorising 6x2 13x 15 which still requires some trial and error but is more systematic.
We now apply this to factorising 6x2 13x First we look for two numbers that multiply together to give ac and add to give b. The two numbers are 18 and 5. We write: It is sometimes possible to take out a common factor rst to simplify the factorisation. Example 8 2 Factorise: Example 9 3 Factorise: Example 10 4 Factorise: Example 11 5 Factorise: Example 12 6 Factorise: Example 13 7 Factorise: Example 14, 15 8 Factorise: Example 16 9 Factorise: Example 17 10 Factorise: There are three steps to solving a quadratic equation by factorisation: Step 2 Factorise the quadratic expression.
We can check the answer for this example by substituting into the equation: Quadratics 3C. We will meet more such problems in Section 3L. Example 20 The perimeter of a rectangle is 20 cm and its area is 24 cm2.
Calculate the length and width of the rectangle. Solution Let x cm be the length of the rectangle and y cm the width. The width is 4 cm or 6 cm. Section summary To solve a quadratic equation by factorisation: Step 2 Factorise the quadratic polynomial. Give your answer correct to two decimal places. Example 18 3 Solve for x in each of the following: Example 19 4 Solve for x in each of the following: Calculate t if h is 76 metres. How many sides has a polygon with 2 65 diagonals?
Find v when the tractive resistance is Example 20 10 The perimeter of a rectangle is 16 cm and its area is 12 cm2. If the area of the triangle is 15 cm2 , calculate the altitude. How many members originally agreed to go on the bus? This is called polynomial form. The 2 line about which the graph is symmetrical is x called the axis of symmetry. When a is negative, the graph is reected in the x-axis. The vertex is now 0, 2 and the range is now all real numbers greater than or equal to 2. The vertex is now 0, 1 and the range is now all real numbers greater than or equal to 1.
All other features of the graph are unchanged. The axis of symmetry is still the y-axis. In both cases, the range is unchanged and is still all non-negative real numbers. The vertex is now at 1, 0. The vertex has coordinates 1, 3. To add further detail to our graph, we can nd the axis intercepts: Therefore this graph has no x-axis intercepts. Quadratics 3D. The vertex or turning point is the point 0, 0 and the axis of symmetry is the y-axis.
The vertex or turning point is the point h, k. Similar results hold for dierent combinations of h and k positive and negative. Exercise 3D For each of the following, nd i the coordinates of the turning point ii the axis of symmetry iii the x-axis intercepts if any and use this information to help sketch the graph. This can be done by two dierent but related methods: Consider the expansion of a perfect square: However, by adding and subtracting a new term, we can form a perfect square as part of a new expression for the same polynomial.
In order to keep our original quadratic intact, we both add and subtract the correct new term. In the above example, the coecient of x2 was 1. If the coecient is not 1, this coecient must rst be factored out before proceeding to complete the square. The small rectangle to the right is moved to the base of the x by x square. The red square of area 1 unit is added.
Solving equations by completing the square The process of completing the square can also be used for the solution of equations. Example 25 Solve each of the following equations for x by rst completing the square: Solution Explanation a Completing the square: Sketching the graph of a quadratic polynomial after completing the square Completing the square enables the quadratic rule to be written in turning point form.
We have seen that this can be used to sketch the graphs of quadratic polynomials. Solution Take out 2 as a common factor and then complete the y square: Substitute this value into the 2a quadratic polynomial to nd the y-coordinate of the turning point.
Example 27 Use the axis of symmetry to nd the turning point of the graph and hence express in turning point form: Solution Explanation a The x-coordinate of the turning point is 2.
The method of completing the square allows us to deal with all quadratic equations, even though there may be no solution for some quadratic equations. First take out a as a factor and then complete the square inside the bracket. Hence state the coordinates of the turning point and sketch the graph in each case.
We can sometimes nd the x- and y-axis intercepts and the axis of symmetry from polynomial form by other methods and use these details to sketch the graph. Step 3 Find the equation of the axis of symmetry Once the x-axis intercepts have been found, the equation of the axis of symmetry can be found by using the symmetry properties of the parabola. The axis of symmetry is the perpendicular bisector of the line segment joining the x-axis intercepts.
Step 4 Find the coordinates of the turning point The axis of symmetry gives the x-coordinate of the turning point.
Substitute this into the quadratic polynomial to obtain the y-coordinate. Therefore the y-axis intercept is 0. The turning point has coordinates 2, 4. Therefore the y-axis intercept is 9. Therefore the y-axis intercept is You can also double click on the end values to change the window settings. Section summary Steps for sketching the graph of a quadratic function given in polynomial form: Step 1 Find the y-axis intercept. Step 2 Find the x-axis intercepts.
Step 3 Find the equation of the axis of symmetry. Step 4 Find the coordinates of the turning point. State the x-coordinate of the vertex. Find the other x-axis intercept. Example 28, 29 3 Sketch each of the following parabolas, clearly showing the axis intercepts and the turning point: Example 30 4 Sketch each of the following parabolas, clearly showing the axis intercepts and the turning point: The situation is a little more complex for quadratic inequalities.
We suggest one possible approach. Step 3 Use the graph to determine the set of x-values which satisfy the inequality. Using the TI-Nspire The calculator may be used to solve quadratic inequalities. The inequality symbol can be found in the Math3 keyboard. There is a general formula for nding the solutions of a quadratic equation in polynomial form.
This formula comes from completing the square for the general quadratic. The quadratic formula provides an alternative method for solving quadratic equations to completing the square, but it is probably not as useful for curve sketching as completing the square, which gives the turning point coordinates directly.
It should be noted that the equation of the axis of symmetry can be derived from this general formula: Also, from the formula it can be seen that: This will be further explored in the next section.
A CAS calculator gives the result shown opposite. Example 32 Solve each of the following equations for x by using the quadratic formula: You must use a multiplication sign between the k and x. Use the keyboard to enter the variables. Use the quadratic formula to calculate the x-axis intercepts.
The turning point coordinates are 2, 5. Give exact answers. Example 33 4 Sketch the graphs of the following parabolas. Use the quadratic formula to nd the x-axis intercepts if they exist and the axis of symmetry and, hence, the turning point. We sometimes say the equation has two coincident solutions. Example 34 Find the discriminant of each of the following quadratics and state whether the graph of each crosses the x-axis, touches the x-axis or does not intersect the x-axis.
From the graph it can be seen that 3 0 3 m. Quadratics 3I. For a, b and c rational numbers: Example 34 2 Without sketching the graphs of the following quadratics, determine whether they cross or touch the x-axis: If we wish to nd the point or points of intersection between a straight line and a parabola, we can solve the equations simultaneously.
It should be noted that depending on whether the straight line intersects, touches or does not intersect the parabola we may get two, one or zero points of intersection. Two points of intersection One point of intersection No point of intersection. If there is one point of intersection between the parabola and the straight line, then the line is a tangent to the parabola. As we usually have the quadratic equation written with y as the subject, it is necessary to have the linear equation written with y as the subject.
Then the linear expression for y can be substituted into the quadratic equation. Solution At the point of intersection: The result can be shown graphically. Quadratics 3J. Example 38 3 Prove that, for each of the following pairs of equations, the straight line meets the parabola only once: Consider the discriminant of the resulting quadratic.
Use the discriminant of the resulting quadratic. In this section these two ideas are extended for our study of quadratic polynomials. We recall from Chapter 2 that the letters a, b and h are called parameters. Varying the parameter produces dierent parabolas. For the parabola in this family that passes through the points 1, 7 and 2, 10 , nd the values of a and c.
Subtract 1 from 2: Substitute in 1: We now consider three important such families which can be used as a basis for nding a quadratic rule from given information.
These are certainly not the only useful forms. You will see others in the worked examples. Example 41 A parabola has x-axis intercepts 3 and 4 and it passes through the point 1, Find the rule for this parabola.
Example 42 The coordinates of the turning point of a parabola are 2, 6 and the parabola passes through the point 3, 3. Example 43 A parabola passes through the points 1, 4 , 0, 5 and 1, Example 44 Determine the quadratic rule for each of the following parabolas: These equations are then solved simultaneously to nd a and b.
You can either substitute the values for x, y prior to entering or substitute in the command line as shown. Remember to use Var to enter the variables a and b. Section summary To nd a quadratic rule to t given points, rst choose the best form of quadratic expression to work with. Then substitute in the coordinates of the known points to determine the unknown parameters.
Some possible forms are given here:. One point is needed to Two points are needed to determine a. Two points are needed to Three points are needed to determine a and b. For the parabola in this family that passes through the points 1, 2 and 0, 6 , nd the values of a and c. Example 41 3 a A parabola has x-axis intercepts 2 and 6 and it passes through the point 1, Example 42 b The coordinates of the turning point of a parabola are 2, 4 and the parabola passes through the point 3, Example 43 c A parabola passes through the points 1, 2 , 0, 3 and 1, 6.
The parabola passes through the point with coordinates 2, 8. Quadratics 3K. The parabola passes through the point with coordinates 1, 4 and one of its x-axis intercepts is 6. Find the values of a and b. The parabola has vertex 1, 6 and passes through the point with coordinates 2, 4.
Find the values of a, b and c. Example 44 7 Determine the equation of each of the following parabolas: Find the equation for the parabola. Find the equation of the parabola. The minimum height of the cable above the roadway is 30 m. Write its equation. Assuming that a quadratic model applies, nd an expression for the rate of rainfall, r mm per hour, in terms of t. Example 45 Jenny wishes to fence o a rectangular vegetable garden in her backyard. She has 20 m of fencing wire which she will use to fence three sides of the garden, with the existing timber fence forming the fourth side.
Calculate the maximum area she can enclose. Example 46 A cricket ball is thrown by a elder. It leaves his y hand at a height of 2 metres above the ground and 25, 15 the wicketkeeper takes the ball 60 metres away again at a height of 2 metres. It is known that after the ball has gone 25 metres it is 15 metres above the 2 x ground. Solution a The data can be used to obtain three equations: Quadratics 3L.
Example 45 1 A farmer has 60 m of fencing with which to construct three sides of a rectangular yard connected to an existing fence. Let x m be the length of one side.
Find a formula for the area A of the rectangle in terms of x. Hence nd the maximum area A. One piece is to be bent into a square and the other into a rectangle four times as long as it is wide.
Write a formula connecting y and x. Comment on the possible values of h. The y point A is 3. Use a CAS calculator. Quadratics Review. Nrich Taking out a common factor e. The value of c gives the y-axis intercept. Chapter 3 review For example:.
Quadratics Review 5 Sketch the graphs of each of the following: Find the equation corresponding to this parabola.
Find the maximum value of the product of such numbers. Review 15 Find the rule of the quadratic function which describes the y graph. What is the radius of a tank of height 6 m which has a surface area of h Extended-response questions 1 The diagram shows a masonry arch bridge of span 50 m.
The shape of the B curve, ABC, is a parabola. What is the greatest height of the deck above water level if the platform is to be towed under the bridge with at least 30 cm horizontal clearance on either side?
One piece is used to form a square shape and the other a rectangular shape in which the length is twice its width. After 1 hour the depth of water in the tank is 5 cm; after 5 hours the depth is 10 cm. For how long, from the beginning, can water be pumped into the tank at the same rate without overowing? Quadratics Review 4 The gure shows a section 90 m view of a freeway embankment to be built across a ood-prone xm river at.
The height of the 45 45 embankment is x m and the width at the top is 90 m. This gure shows another section of the freeway which is to be constructed by xm cutting through a hillside. The depth of the 65 65 cutting is x m and the width of the cutting at the base is 50 m. If the area of this sheeting is A m2 , formulate a rule connecting m A and x.
A B P 1x x Many have thought that it has the perfect proportions for buildings. The rectangle is such that, if a square x is drawn on one of the longer sides, then the new rectangle is similar to the original. This value is the reciprocal of the golden ratio. A a Find distance PA in terms of x.
D 5m b i Find distance PC in terms of x. Answer correct to three decimal places. Another runner is moving along road CD. A O B iii Find the time s when the runners are 4 km apart. The rectangle has width x cm and length y cm.
Answers correct to two decimal places. This is called the locus of the midpoints. The path 30, 35 E 20, 45 starts at a point C 30, 15 and D nishes at a point D 60, Pat One boundary of the pond in x the park is parabolic in shape. In Chapter 2, we looked at linear graphs, sketching them and determining their rules given sucient information.
The features we concentrated on for linear graphs were the x-axis intercept, the y-axis intercept and the gradient. The features we concentrated on for graphs of quadratic polynomials were the x-axis intercepts, the y-axis intercept and the coordinates of the turning point vertex. In this chapter, we study some other common algebraic relations, and develop methods similar to those used in Chapter 3 to sketch the graphs of these relations.
The relations in this chapter have dierent types of key features. A gallery of graphs. We can plot these points and then connect y the dots to produce a continuous curve. A graph of this type is an example of a 2 1 rectangular hyperbola. Horizontal asymptote From the graph we see that, as x approaches innity in either direction, the value of y approaches zero.
The following notation will be used to state this: This is read: As x approaches innity, y approaches 0 from the positive side.
As x approaches negative innity, y approaches 0 from the negative side. Vertical asymptote As x approaches zero from either direction, the magnitude of y becomes very large. As x approaches zero from the positive side, y approaches innity.
As x approaches zero from the negative side, y approaches negative innity. Dilations will be considered formally in Chapter 7.
These can be calculated in the usual way to add further detail to the graph. Example 1, 2 1 Sketch the graphs of the following, showing all important features of the graphs: We can plot these points and then connect the dots to produce a continuous curve.
A graph of this shape is sometimes called a truncus. Example 3 1 Sketch the graphs of the following, showing all important features: The set of values the rule can take the range is 2, 3 all numbers greater than or equal to 3, i. The graph will have endpoint h, k. Example 4 1 For each of the following rules, sketch the corresponding graph, giving the axis intercepts when they exist, the set of x-values for which the rule is dened and the set of y-values which the rule takes: Example 5, 6 2 For each of the following rules, sketch the corresponding graph, giving the axis intercepts when they exist, the set of x-values for which the rule is dened and the set of y-values which the rule takes: All circles can be considered as being transformations x 1 0 1 of this basic graph.
As has been seen with other graphs, the basic graph may be translated horizontally and vertically. If the radius and the coordinates of the centre of the circle are given, the equation of the circle can be determined. Example 7 Write down the equation of the circle with centre 3, 5 and radius 2.
If the equation of the circle is given, the radius and the centre of the circle can be determined and the graph sketched. Solution Explanation The equation We can sketch the circle with a little extra work. Then we obtain an alternative form for the equation of a circle:. Notice that in the general form of the circle equation, the coecients of x2 and y2 are both 1 and there is no xy term.
In order to sketch a circle with equation expressed in this form, the equation can be converted to the centreradius form by completing the square for both x and y. The radius is 5 and the centre is at 3, 2. Similarly, solving for x will give you the semicircles to the left and right of the y-axis: Example 11 Sketch the graphs of: Example 7 1 Write down the equation of each of the following circles, with centre at C h, k and radius r: Example 9 3 Sketch the graphs of each of the following: Example 10 4 Find the centre, C, and the radius, r, of the following circles: Example 10 5 Sketch the graphs of each of the following: Example 11 6 Sketch the graphs of each of the following: The coordinates of every point in the shaded region 3 satisfy the inequality.
Use a dotted line to 3 0 3 indicate that the boundary is not included. For straight lines these included: For parabolas these included: In this section we are looking at some sucient conditions for determining the rules for the graphs of this chapter. Find the value x of a. Find x the values of a and k. Find the values of a and h. Note that Multiply both sides of equation 3 by 4 h: Example 15 Find the equation of the circle whose centre is at the point 1, 1 and which passes through the point 4, 3.
Solution Explanation Let r be the length of the radius. A gallery of graphs 4E. Find the values of a, h and k. Example 15 7 Find the equation of the circle whose centre is at the point 2, 1 and which passes through the point 4, 3. The general form of the rule is given for each graph. Search alert. Price Minimum Price. Maximum Price. Offer Type Offering Sponsored Links.
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