Objectives C H A P T E R 5 A Galler of Graphs To recognise the rules of a number of common algebraic relationships: =, =, = / and + =. To be able to sketch the graphs and simple transformations of these relations. In the previous chapter quadratics written in turning point form were considered. The technique of sketching their graphs through emploing transformations of the graph of the basic quadratic = was introduced. In this chapter some other common algebraic relations will be considered. Methods similar to those developed in Chapter will be used to sketch the graphs of these relations. First, it is necessar to be able to recognise the basic rule of each tpe of relation and its corresponding graph. In this chapter use of a CAS calculator is recommended to obtain the initial graphs. 5. Rectangular hperbolas Consider the rule = =, for =. We can construct a table of values for = for values of between and as follows. Cambridge Universit Press Uncorrected Sample Pages 8 Evans, Lipson, Wallace TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard 8
We can plot these points and then connect the dots to produce a continuous curve. Agraph of this tpe is an eample of a rectangular hperbola. It should be noted that when =, is undefined and there is no -value that will produce the value =. As approaches infinit in either direction, the value of approaches zero. The following notation will be used to state this. As, +.As,. These are read: Chapter 5 AGaller of Graphs 9 = As approaches infinit, approaches from the positive side and as approaches negative infinit, approaches from the negative side. As approaches zero from either direction, the magnitude of becomes ver large. The following notation will be used to state this. As +, and as,. These are read: As approaches zero from the positive side, approaches infinit and as approaches zero from the negative side, approaches negative infinit. The graph approaches both the -ais (the line = ) and the -ais (the line = ) but does not cross either of these lines. We refer to these lines as asmptotes. Hence, for the graph of =, the equations of (, ) = (, ) the asmptotes are = and =., In the diagram on the right, the graphs of =, = and = are shown., The asmptotes are the -ais and the -ais, and the have equations = and = (, ) = respectivel. (, ) As can be seen from the diagram, the graphs of = and = have the same shape and asmptotes as = but have been stretched (or dilated from the -ais). It is said that the dilation from the -ais, which takes the graph of = to the graph of =,isb factor and the point with coordinates (, ) on the graph of = is taken to the point with coordinates (, ) on the graph of =. Cambridge Universit Press Uncorrected Sample Pages 8 Evans, Lipson, Wallace TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard
5 Essential Mathematical Methods &CAS Dilations will be considered formall in the Chapter 6. = Now let us consider the graph of = ( ) +. = + ( ) The basic graph of = has been translated 6 unit to the right and units up. The equation of the vertical asmptote is now = and the = equation of the horizontal asmptote is now =., The graph now has -ais and -ais intercepts. These can be calculated in the usual wa to give further detail to the graph. -ais intercept: let = -ais intercept: let = = + = ( ) + = = ( ) = = ( ) intercept is, intercept is (, ) Using the above technique we are therefore able to sketch graphs of all rectangular hperbolas of the form = a h + k. Eample Sketch the graph of = +. The graph of has been translated unit to the left and units down. Asmptotes have equations = and =. -ais intercept: (, ) = 6, -ais intercept: (,) = Cambridge Universit Press Uncorrected Sample Pages 8 Evans, Lipson, Wallace TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard
Chapter 5 AGaller of Graphs 5 Eample Sketch the graph of =. The graph of = is obtained from the graph of = b reflection in the -ais. This graph is then translated unit to the right to obtain the graph of =. = Eercise 5A = (, ) = = Sketch the graphs of the following, showing all important features of the graphs: a = b = c = d = e = + f = g = h = + 5 i = j = k = + + + l = Write down the equations of the asmptotes for each of the graphs in Question. 5. The truncus Consider the rule = =. Constructing a table of values for = for values of between and we have as follows: 6 9 9 6 Cambridge Universit Press Uncorrected Sample Pages 8 Evans, Lipson, Wallace TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard
5 Essential Mathematical Methods &CAS We can plot these points and then connect the dots to produce a continuous curve. Agraph of this shape is called a truncus. Note that when =, is undefined, and that there is no -value that will produce a negative value of. Note: As, +.As, +. As +, and, as,. The graph of =, has asmptotes = and =. a All graphs of the form = + k will have the same basic truncus shape. ( h) Asmptotes will be the lines with equations = k and = h. Eample Sketch the graph of = ( + ). The graph of = is translated units to the left and units down. Eercise 5B = Sketch the graphs of the following, showing all important features: = a = ( + ) b = c = ( ) d = ( ) + e = ( + ) f = ( ) + g = ( + ) 6 h = ( ) + Write down the equations of the asmptotes of each of the graphs in Question. 5. = = / The rule = = / for corresponds to the graph shown opposite. It is one arm of the parabola =. = = Cambridge Universit Press Uncorrected Sample Pages 8 Evans, Lipson, Wallace TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard
Chapter 5 AGaller of Graphs 5 Coordinates of points on the graph of = include (, ), (, ), (, ) and (9, ). = All graphs of the form = a h + k (, ) will have the same basic shape as the graph (, ) of =. The rule = for ields a graph which is the reflection of the graph of = in the -ais. All graphs of the form = a ( h) + k will have the same basic shape as the graph of =. Eample Sketch the graph of = +. The graph is formed b dilating the graph of = b factor from the -ais, reflecting this in the -ais and translating it unit to the right and units up. When =, + =. Therefore =. Square both sides to give ( ) = 9 Therefore = 9 + =. (, ), The rule is defined for and the values the rule can take (the range) is all numbers less than or equal to, i.e.. Eample 5 Sketch the graph of = ( ) +. When =, = ( ) + = + The rule is defined for and the values the rule can take (the range) is all numbers greater than or equal to, i.e.. (, +) (, ) Cambridge Universit Press Uncorrected Sample Pages 8 Evans, Lipson, Wallace TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard
5 Essential Mathematical Methods &CAS Eercise 5C For each of the following rules, sketch the corresponding graph, giving the aes intercepts when the eist, the set of -values for which the rule is defined and the set of -values which the rule takes: a = + b = + c = d = + + e = + + f = + g = + h = ( ) i = ( + ) 5. Circles The equation + = r defines a circle whose centre is the origin (, ), with radius r. To the right is the graph of the circle with equation + =. All circles can be considered as being transformations of this basic graph. Changing the value of r obviousl enlarges or reduces the circle. As has been seen with other graphs, the basic graph ma be translated horizontall and/or verticall. The general equation for a circle is ( h) + ( k) = r. The centre of the circle is the point (h, k) and the radius is r. If the radius and coordinates of the centre of the circle are given, the equation of the circle can be determined. Foreample, if the radius is and the centre is the point (, 5) then the equation will be ( ( )) + ( 5) = ( + ) + ( 5) = If the equation of the circle is given, the radius and the centre of the circle can be determined and the graph sketched. Eample 6 Sketch the graph of ( ) + ( ) =. Cambridge Universit Press Uncorrected Sample Pages 8 Evans, Lipson, Wallace TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard
The equation ( ) + ( ) = defines a circle of radius, with centre at the point (, ). When =, + ( ) =. Therefore ( ) = and = ±. Eample 7 Chapter 5 AGaller of Graphs 55 (, ) (, + ) Sketch the graph of the circle ( + ) + ( + ) = 9. The circle has a radius of, and its centre at (, ). -ais intercepts can be found in the usual wa. (, + ) When =, + ( + ) = 9 (, ) 5 i.e. ( + ) = 8 6 = ± 8 7 = ± The equation of a circle ma not alwas be written in the form ( h) + ( k) = r. Epanding the general equation of a circle gives ( h) + ( k) = r h + h + k + k = r + h k + h + k r = Let h + k r = c, then an alternative form of the circle equation is achieved: + h k + c = Notice that in the above form of the circle equation the coefficients of and are both and there is no term. In order to sketch a circle with equation epressed in this form, the equation can be converted to the centre radius form b completing the square for both and. Eample 8 Find the radius and the coordinates of the centre of the circle with equation + 6 + = and hence sketch the graph. Cambridge Universit Press Uncorrected Sample Pages 8 Evans, Lipson, Wallace TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard
56 Essential Mathematical Methods &CAS Eample 6 Eample 7 Eample 8 B completing the square for both and we have + 6 + = ( 6 + 9) 9 + ( + + ) = ( 6 + 9) + ( + + ) = 5 ( ) + ( + ) = 5 radius = 5, centre is at (, ). Semicircles 8 Transposing the general equation of the circle + = r to make the subject, we have = r =± r We can now consider two separate rules, =+ r and = r,which represent the top half and bottom half of the circle respectivel. Eample 9 Sketch the graphs of: a =+ b = a Eercise 5D Write down the equations of the following circles with centres at C(h, k) and radius r. a C(, ), r = b C(, ), r = c C(, ), r = 5 d C(, ), r = e C(, ), r = 5 f C( 5, 6), r =.6 Find the centre, C, and the radius, r, ofthe following circles: a ( ) + ( ) = b ( ) + ( + ) = 5 c ( + ) + ( ) = 9 d + 6 6 = e ( + ) + ( + ) = 6 f + 6 + + 9 = g + + 6 = h + 8 + + = Cambridge Universit Press Uncorrected Sample Pages 8 Evans, Lipson, Wallace TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard b 8
Chapter 5 AGaller of Graphs 57 Sketch the graphs of each of the following: a + = 6 b + ( ) = 9 c ( + ) + = 5 d ( + ) + ( ) 69 = e ( ) + ( 5) = 6 f + 6 6 = Eample 9 g + + 6 = h + 8 + + 7 = i =+ 9 j = 5 k = 6 ( ) l = ( + ) Find the equation of the circle with centre (, ) which touches the -ais. 5 Find the equation of the circle whose centre is at the point (, ) and which passes through the point (, ). 6 Find the equation of the circle whose centre lies on the line = and which passes through the points (, ) and (6, ). 7 The equation to a circle is + + 6 =. Find the centre and radius. 8 Find the length cut off on the -ais and -ais b the circle + =. 9 The graph of + 9isasshown. (, ) Note that (, ) satisfies + 9. The coordinates of ever point in the shaded region satisf the inequalit. Sketch the graphs of each of the following. Use a dotted line to indicate that the boundar is not included. a + b + > c + 5 d + > 9 e + 6 f + < 8 Cambridge Universit Press Uncorrected Sample Pages 8 Evans, Lipson, Wallace TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard
58 Essential Mathematical Methods &CAS Review Chapter summar The standard graphs: = Rectangular hperbola = Truncus + = Circle = O Dilations of these graphs: = (, ) = (, ) = = (, ), + = + = = = = Cambridge Universit Press Uncorrected Sample Pages 8 Evans, Lipson, Wallace TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard
Chapter 5 AGaller of Graphs 59 Reflections in the aes: = = General equation for a circle: ( h) + ( k) = r The centre is at (h, k) and the radius is r. Alternative form: + h k + c = where h + k r = c. Translations of graphs: Eamples = + = = Multiple-choice questions (, ) (, ) = ( ) + ( ) = = = The circle with equation ( a) + ( b) = 6 has its centre on the -ais and passes through the point with coordinates (6, 6). The values of a and b are A a = and b = 6 B a = and b = C a = and b = D a = 6 and b = E a = 6 and b = The equations of the asmptotes of the graph of = 5 5 are A = 5, = 5 B = 5, = 5 C = 5, = 5 D = 5, = 5 E = 5, = 5 Review Cambridge Universit Press Uncorrected Sample Pages 8 Evans, Lipson, Wallace TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard
6 Essential Mathematical Methods &CAS Review For the rule = 5 +, when = a, = A 5 a + B a + 5 C a + a a D a + 5 E a a + If the -ais is an ais of smmetr and the circle passes through the origin and (, ), the equation of the circle shown is A + ( ) = B ( ) + = C ( + ) + = D + ( ) = E + = 5 The equations of the asmptotes of the graph of = 5 + ( ) are A =, = 5 B =, = 5 C = 5, = D = 5, = E =, = 5 6 The coordinates of the centre and the radius of the circle with the equation ( 5) + ( + ) = 9 are A ( 5, ) and B ( 5, ) and 9 C (5, ) and 9 D (5, ) and E (, 5) and 7 For = + and, will take the following values A [, ) B (, ) C [, ) D (, ] E (, ) 8 The equation of the circle which has a diameter with end points at (, 8) and (6, 8) is A ( + ) + ( 8) = 6 B ( + ) + ( + 8) = 6 C ( ) + ( 8) = 6 D ( ) + ( + 8) = E ( ) + ( 8) = 6 9 Which of the following is the equation for the graph of a circle? A = 6 B = 6 + C + = 6 D = 6 E = 6 The equation of the semicircle shown is A + ( ) = 9 B = 9 + C = 9 + D = 9 E = 9 + 8 7 6 5 Cambridge Universit Press Uncorrected Sample Pages 8 Evans, Lipson, Wallace TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard
Chapter 5 AGaller of Graphs 6 Short-answer questions (technolog-free) Sketch the graphs of each of the following: a e = b = c = d = + + = f = g = + h = + i = + j = + k = + + B completing the square, write each of the following in the form ( a) + ( b) = r : a + 6 + = b + + 5 = c + + = d + + 6 = e + = 6( + ) f + = 6 Find the equation of the diameter of the circle + + 6 = which passes through the origin. Find the equation of the diameter of the circle + + = 6 which cuts the -ais at an angle of 5. 5 Find the equation to the circle with centre C and radius r for each of the following and sketch the graph: a C(, ), r = 5 b C(, ), r = ( c C(, ), r = d C, ), r = 6 6 Sketch the graphs of the following semicircles: a = 9 b = 6 ( + ) c = d + = ( + ) Etended-response questions The following questions also involve techniques developed in chapters and. The line with equation = m is tangent to the circle with centre (, ) and radius 5 at the point P(, ). a Find the equation of the circle. b Show that the -coordinate of the point P satisfies the equation ( + m ) + 75 =. c Use the discriminant for this equation to find the eact value of m. d Find the coordinates of P. (There are two such points.) e Find the distance of P from the origin. Review Cambridge Universit Press Uncorrected Sample Pages 8 Evans, Lipson, Wallace TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard
6 Essential Mathematical Methods &CAS Review A circle has its centre at the origin and radius. a Find the equation of this circle. b Two lines which pass through the point (8, ) are tangents to this circle. i Show that the equations of these tangents are of the form = m 8m. ii Use techniques similar to those used in Question to find the value of m and, hence, the equations of the tangents. A circle has centre at the origin and radius 5. The point P(, ) lies on the circle. a Find the gradient of the line segment OP. b Find the gradient of the tangent to the circle at P. c Find the equation of the tangent at P. d If the tangent crosses the -ais at A and the -ais at B, find the length of line segment AB. Let P(, )beapoint on the circle with equation + = a. a i Give the gradient of the line segment OP, where O is the origin. ii Give the gradient of the tangent to the circle at P. b Show that the equation of the tangent at P(, )is + = a. c If = and a =, find the equations of the possible tangents. 5 The line with equation = ais the equation of the side BC of an equilateral triangle ABC circumscribing the circle with equation + = a. a Find the equations of AB and AC. b Find the equation of the circle circumscribing triangle ABC. 6 Consider the curve with equation = b + c. a Show that if the curve meets the line with equation = at the point (a, a) then a satisfies the equation a (c + )a + c + b =. b i If the line is a tangent to the curve, show that c = b. ii Sketch the graph of = and find the coordinates of the point on the graph at which the line with equation = is a tangent. c Find the values of k for which = + k: i meets the curve with equation = twice ii meets the curve with equation = once iii does not meet the curve with equation =. 7 For the curve with equation = and the straight line with equation = k, find the values of k such that: a the line meets the curve twice b the line meets the curve once. Cambridge Universit Press Uncorrected Sample Pages 8 Evans, Lipson, Wallace TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard