A Sketch graphs of = a m b n c where m = or and n = or B Reciprocal graphs C Graphs of circles and ellipses D Graphs of hperbolas E Partial fractions F Sketch graphs using partial fractions Coordinate geometr areas of STud Sketch graphs of functions defined b: f( ) = a m b n for m, n =, and f( ) =, their asmptotic behaviour, a b c and nature and location of stationar points Sketch graphs of ellipses from the general Cartesian relation: ( h) ( k ) = a b Sketch graphs of hperbolas (including asmptotic behaviour) from the general Cartesian relation: ( h ) ( k ) =. a b These do not involve consideration of focusdirectri properties. ebookplus Digital doc Quick Questions a ebookplus Interactivities int-8 Graphs of power functions : m = and n = int-8 Graphs of power functions : m = and n =. int-86 Graphs of power functions : m = and n =. int-87 Graphs of power functions : m = and n =. Sketch graphs of = a m b n c where m = or and n = or In this section we shall be sketching graphs of functions of the form = a m b n c where m = or and n = or. Eamples of such functions are:. =,, a =, b =, c =, m =, n =. =,, a =, b =, c =, m =, n =. =,, a =, b =, c =, m =, n =. =,, 6, a =, c =, m =. = 7 6,, a =, b = 7, c = 6, m =, n = Vertical asmptote The vertical asmptote occurs for the -value that makes the denominator of the fractional part of the function. When the denominator of a fraction is, the fraction, and hence the whole function, will be undefined. Chapter Coordinate geometr
General algebraic reasoning. For graphs of the form = a m b n c, as, n, and hence b n. Therefore, and thus the equation of the vertical asmptote is = the value that makes the denominator.. For graphs of the tpe = a m b n c, the vertical asmptote will alwas be = or the -ais, as the denominator of the fractional part is alwas. horizontal, oblique (sloping) or curved asmptote To determine the non-vertical asmptote, cover up onl the fractional part with our hand; the part of the function ou can still see is the equation of this asmptote. When finding horizontal, oblique or curved asmptotes we consider ver large values of. The reciprocal of ver large values is ver small, so the fractional part is close to and contributes ver little to the value of. The value of will become ver close to the value of the function without the fractional part hence the method for finding the asmptote. Note: An oblique line is an line that is neither horizontal nor vertical. General algebraic reasoning For graphs of the form = a m b n c, as, n, and hence b n and so becomes insignificant in size. Thus a m c, making = a m c an asmptote. Note: If a = then the asmptote will be horizontal; otherwise, it will be oblique or curved. Worked eample Find the asmptotes for the function with the equation =. Think WriTe Find the vertical asmptote b considering values of that make the denominator. Find the horizontal or oblique asmptote b considering large values of or covering the fractional part of the equation. = As, Vertical asmptote: = As, Oblique asmptote: =, and hence., and hence. Sketching the graphs of rational functions This process involves several steps. Step. Break the given function into two separate, simpler functions. Step. Sketch the graph of each of the separate functions on the one set of aes. Do this in pencil and use either colour or some distinguishing feature (for eample dotted and dashed lines) for each graph. Maths Quest Specialist Mathematics for the Casio ClassPad
Step. Determine the asmptotes of the original function and pencil in how the graph of the function approaches these asmptotes. You should remember to consider:. large positive values for. large negative values for. values ver close to and either side of the vertical asmptote. Step. Work out the -intercept(s), -intercept and turning points for the given function to give greater accurac. Worked eample Sketch the graph of the function =, (do not include the turning points). ebookplus Tutorial int- Worked eample Think Divide = into two functions: = and =. Consider =. (a) This is an upright parabola. We need to state the verte and find several points either side of the verte to give an idea of the shape. (b) Sketch =. WriTe Let = and =. The graph = is an upright parabola with verte at (, ). Let =, = () = The points (, ) and b smmetr (, ), are on the parabola. Consider =. (a) This graph is a hperbola. We need to state the asmptotes and points on the curve either side of the vertical asmptote. For = : Vertical asmptote is =. Horizontal asmptote is =. Let =, = =, so (, ) is on the hperbola. Let =, = =, so (, ) is on the hperbola. Chapter Coordinate geometr
(b) Sketch = on the same aes as =. Consider the graph of =. (a) Let = to calculate the -intercepts. (b) Multipl both sides b. (c) Solve for. (d) Take the cube root of both sides to obtain. (e) Let = to determine the -intercept. For = : At -intercept, =. = = = = = = 6. No -intercept as the -ais is a vertical asmptote. Write down the equations of the asmptotes. Asmptotes are = and =. 6 Consider behaviour near asmptotes. (a) For >, both and are positive and as, and. (b) Therefore, from above the graph of. (c) For <, > and < and as, and. (d) Therefore, from below the graph of. (e) Consider values of either side of the vertical asmptote. As, and. So from above the graph of. As, and. So from below the graph of (as is negative). As, and is positive and. So from above the graph. As, and is positive and. So from above the graph of. Maths Quest Specialist Mathematics for the Casio ClassPad
7 Sketch the graph of = on the same aes as and. Approach from below, Approach from above.6 6 Approach from above Approach from above Stationar points (turning points) When we are working with turning points, we need to remember the following.. The derivative of functions of the form = a n is d d = na n. To determine a local maimum or a local minimum we solve d = for. d. Distinguish between the maimum and minimum values b eamining the sign of d d around the -value(s) found in the step above. The table below will help ou. If d = when = a, the following applies. d Local minimum Local maimum < a = a > a < a = a > a d d < slope is negative d d = slope is zero d d > slope is positive d d > slope is positive d d = slope is zero d d < slope is negative Worked Eample Find the local maimum and minimum values of, for = Think To find the derivative, epress the rule in inde form. Differentiate. d d. Write = = 8 = Chapter Coordinate geometr
At local maimum and minimum points, d d =. For d =, d 8 = To solve, multipl both sides b. 8 ( ) ( ) = ( ) 8 = Rearrange. 8 = = 8 6 Take the cube root of both sides. = 7 Find the corresponding -value b substituting = into =. = = 8 Evaluate. = = 9 State the turning point., is the turning point. Determine if, is a maimum or minimum. < = > For < : as = is a vertical asmptote we need d d < d d = d d > to pick a value of between and, sa =, and ( ) (7) evaluate d d. For > : pick a value greater than, sa =, and evaluate d d. State the nature of the turning point., is a local minimum. Worked eample Sketch the graph of = Think On the Graph & Tab screen, complete the entr line as: = Tick the bo and tap:! including all asmptotes and intercepts. WriTe/diSPla 6 Maths Quest Specialist Mathematics for the Casio ClassPad
To determine the equation of the oblique asmptote, divide = into two functions. To do this, tap: Action Transformation propfrac Complete the entr line as: propfrac( f ()) Then press E. Write the equations of the asmptotes. The equations of the asmptotes are: = and =. To determine the turning points, tap: Action Command Define Complete the entr line as: Define f( )= and press E. Then tap: Action Advanced solve Complete the entr line as: d solve ( ( ( ), ) d f = and press E. To find the -coordinates of the stationar points b substitution, complete the entr lines as: f ( ) f ( ) Press E after each entr. 6 Sketch the graph of =. Solving d d = for gives = or =. The coordinates of the stationar points are: Local minimum, Local maimum, (Asmptote) (Asmptote) Chapter Coordinate geometr 7
REMEMBER To sketch graphs of functions of the form = a m b n c where m = or and n = or : Step. Break the given function into two separate simpler functions. Step. Sketch the graph of each of the separate functions created, on the one set of aes. Step. Determine the asmptotes of the original function and pencil in how the graph of the function approaches these asmptotes. Consider: (a) large positive values for (b) large negative values for (c) values ver close to, and either side of, the vertical asmptote. Step. Work out the -intercept(s), -intercept and turning point for the given function to give greater accurac. Eercise A Sketch graphs of = a m b n c where m = or and n = or WE Find the asmptotes for each of the following. a = b = c = d = e = f = g = 8 h = MC For each of the following equations, choose the alternative that gives the correct asmptotes. 6 a = A = 6 and = B = and = C = and = D = and = E = b = A = and = B = and = C = and = D = and = E = and = c = 6 A = and = B = and = 6 C = and = 6 D = and = E =, = and = 7 d = 9 A = 9 and = B = 9 and = 7 C = 9 and = D = and = E = 9 and = WE Sketch the graph of each function given in Question. (Do not include the turning points.) WE For each of the functions given in Question, find the local maimum and/or minimum values of. 8 Maths Quest Specialist Mathematics for the Casio ClassPad
ebookplus Digital doc Spreadsheet Single graph plotter MC For each of the following equations, choose the alternative that gives the correct graph. 6 a = A B C D E b = A 8 B 8 C 8 D E c = A B C D E A d = B C D E 6 We Sketch the graphs of each of the following, showing stationar points (ma. and min.), intercepts and asmptotes. a = 6 9 b = c = d = e = 8 f = g = h = (Hint: When finding -intercepts, use the factor theorem to find a factor.) Chapter Coordinate geometr 9
B ebookplus Interactivit int-8 Reciprocal graphs 7 The volume of a solid clinder is 8π cm. a Show that the total surface area, A cm, is A = πr 6π r where r >. b Sketch the graph of A against r. State the equation of an asmptotes and the coordinates of the stationar point. c Hence, find the eact minimum total surface area. 8 A bo with a volume of cm has the shape of a rectangular prism. It has a fied height of cm, a length of cm and a width of cm. If A cm cm is the total surface area: cm a Epress A in terms of. b Sketch the graph of A against. cm c Find the minimum total surface area of the bo and the dimensions in this case. reciprocal graphs This technique involves sketching the graph of = from the graph of = f (). f( ). When f (), = f( ), the graph of = approaches the vertical asmptote(s). f( ). Therefore, the graph of = will have vertical asmptotes at the -intercepts of = f (). f( ). When f (), f( ), the graph of = f( ) approaches the horizontal asmptote (the -ais in this case).. These graphs also have common points: (a) When f () = ±, f( ) = ±. The graphs are in the same quadrant. (b) When f () <, (c) When f () >, f( ) <. f( ) >.. The -intercepts of f () determine equations of the asmptotes for the reciprocal of the functions. 6. The minimum turning point of f () gives the maimum turning point of the reciprocal function. 7. The maimum turning point of f () gives the minimum turning point of the reciprocal function. Note: If = f( ) then:. for f () =, = and for f () =, =. and for f () <, > and for f () >, <. Worked eample Sketch the graph of the function =, ± from 9 the given graph of = 9. 6 = 9 6 8 Maths Quest Specialist Mathematics for the Casio ClassPad
Think WRITE Sketch the graph of the function = 9 as given. 6 9 Work out the asmptotes for = 6 8. The -intercepts of = 9 are = ±. 9 The vertical asmptotes for = are 9 = ±. The horizontal asmptote is =. For the -intercept, =. The -intercept is (, 9 ). Draw a diagram based on the information gathered so far. 9 As, approaches from the positive direction ( ) As approaches from the negative direction ( ), 9 6 As, As, 9 Chapter Coordinate geometr
7 As, As, 9 8 To determine the shape of the graph near the -intercept, evaluate the value of when is ± and ±. 9 Sketch the graph of =. 9 6 (, ) 6 8 9 9 The following eamples show a different approach to sketching reciprocal functions. Worked Eample 6 Sketch the graphs of f () and g() on the same set of aes where f () = - - and g() =,, -. Think WRITE The graph of f () is an upright parabola, as a =. Calculate the -intercept. -intercepts: = so ( )( ) = so = or = and = or = State the coordinates of the -intercepts. The -intercepts are (, ) and (, ). Calculate the -intercept. -intercept: f () = The -intercept is (, ). The verte or turning point -coordinate is halfwa between the -intercepts. Turning point: = = Maths Quest Specialist Mathematics for the Casio ClassPad
6 Substitute to find the -value of the turning point. f () = () () = = The turning point is (, ). 7 Sketch the graph of f () =. (See below.) 8 Use the above to determine important features for g() =. For g() = : ( ) = ( )( ) 9 Vertical asmptotes occur when f () has its -intercepts. Vertical asmptotes: = and =,, and so g() Find the horizontal asmptotes. The horizontal asmptote is g() =. The reciprocal of the turning point for f () is a turning point for g(). Wherever f () = or, g() = or. As g() = f( ), the graphs of f () and g() are in the same quadrants. Sketch the graph of g() on the same aes as f (). The reciprocal of the turning point (, ) is (, ). g() (, ) f() (, ) Worked eample 7 Sketch the graphs of f () and g() on the same set of aes where: f () = ( ) and g() =, ( ). ebookplus Tutorial int- Worked eample 7 Think WriTe Work out important features for f () = ( ). This is an inverted parabola, as a =. Calculate the -intercept(s). -intercepts: ( ) = ( ) = = = State the coordinates of the -intercepts. The -intercept is (, ). Chapter Coordinate geometr
Calculate the -intercept. The -intercept: f () = ( ) = (9) = 9 The -intercept is (, 9). As the graph touches the -ais at (, ), it must also turn at this point. Hence, (, ) is the turning point. Sketch the graph of f (). (See below.) Use the above to determine important features for g() =. ( ) The turning point is (, ). For g() = : ( ) 6 Vertical asmptotes occur when f () has its -intercepts. Vertical asmptote: = 7 Find the horizontal asmptotes. 8 We cannot take the reciprocal of the turning point for f () as the reciprocal of is not defined it was worked out in Step 7 above that this was the vertical asmptote. 9 The -intercept of g() is the reciprocal of the -intercept of f (). Since g() = f( ) then g() = or when f () = or. As g() =, the graphs of f () and g() are in f( ) the same quadrants.,, and so g() ( ) The horizontal asmptote is g() =. The -intercept is (, 9 ). Sketch the graph of g() on the same aes as f (). 8 7 6 g() ( ) f() = ( ) 6 8 9 (, ) 9 Worked Eample 8 Sketch the graphs of f () and g() on the same set of aes where: f () = and g() =. Think WRITE Work out important features for f () =. Maths Quest Specialist Mathematics for the Casio ClassPad
This is an upright parabola, as a =. Find the -value of the turning point b f () = solving d d = or f () =. For f () =, = = = Evaluate f () when =. f ( ) = ( ) ( ) = 8 = The turning point is (, ). As the parabola is upright and turns at (, ) it is completel above the -ais and hence there is no -intercept. There is no -intercept. Calculate the -intercept. -intercept: f () = The -intercept is (, ). B smmetr (, ) is also on the curve. 6 Sketch the graph of f (). (See below.) Use the above to determine important features for g() =. 7 Since there are no -intercepts for f (), g() has no vertical asmptotes. 8 Find the horizontal asmptote., 9 The verte of g() is the reciprocal of the verte of f (). The -intercept for g() is the reciprocal of the -intercept of f (). There are no vertical asmptotes., and so g(). The horizontal asmptote is f () =. The verte is (, ). The -intercept is (, ). Since g() = the graphs of f () and g() are in f( ) the same quadrants. Sketch the graph of g() on the same aes as f (). f() 6 g() (, ) (, ) Chapter Coordinate geometr
REMEMBER. To sketch the graph of = from the graph of = f (): f( ) (a) find the vertical asmptote at the -intercepts of f () (b) the horizontal asmptote is the -ais (c) find the common points when f () = ±.. The graphs are in the same quadrant, that is, f () <, f( ) < and f () >, f( ) >. Note: If = f( ) then:. f () =, = and f () =, =. f () <, > and f () >, <. Eercise B Reciprocal graphs WE Sketch the graph of each of the following functions from the given graph. a =, ± 6 6 c,, 8 6 (, ) b =, 8 6 d =,, 8 6 (, ) (, ) e =, ( (, ), ) f =,, (, ) 6 8 6 Maths Quest Specialist Mathematics for the Casio ClassPad
g =, ± 9 = 9 (, ) (, ) (, 9) h =,, (, ) 8 6 8 Sketch the graph of each of the following functions, f () and g(), on the same set of aes. Show all asmptotes with equations and turning points. a WE6 f () =, g() =, b f () =, g() = c f () =, g() = d f () =, g() = e f () =, g() =,,,,,,, f f () = 8, g() = 8,, g WE7 f () = ( ), g() =, ( ) h f () = ( ), g() =, ( ) i f () =, g() =, j f () =, g() =, k WE8 f () =, g() = l f () =, g() = MC Consider the function f () =. a f () has asmptotes with equations: A =, = and = B =, = and = C =, = and = D =, and = onl E =, and = b The -intercept and turning point are respectivel: A (, ) and (, ) B (, ) and (, ) C (, ) and (, ) D (, ) and (, ) E (, ) and (, ) Chapter Coordinate geometr 7
c The graph of f () is best represented b: A B C D E A bo in the shape of a rectangular prism has a base of length cm and width ( ) cm. a Epress the area of the base, A cm, in terms of. b If the volume of the bo is fied at cm, epress the height, h cm, in terms of. c Determine the height of the bo when the length of the base is.9 cm. d Sketch the graph of h against. e Find the minimum height of the bo and the dimensions in this case. a The graph shown has the form = b c. Find the values of a, b and c. (, ) C Graphs of circles and ellipses Graphs of circles All points P(, ) which satisf the relation = r lie on a circle with centre (, ) and radius r. ebookplus Interactivit int-8 Elliptical graph r P (, ) 8 Maths Quest Specialist Mathematics for the Casio ClassPad
If the points P(, ) are translated (h, k) units then the relation becomes ( h) ( k) = r This relation represents a circle with centre (h, k) and radius r. (h, k) r P ( h, k) r P (, ) Assumes h, k Worked Eample 9 Sketch the graph of the circle with centre ( -, ) and radius. Write the Cartesian equation of this circle. Think Write the equation in the form ( h) ( k) = r where h =, k = and r =. Sketch its graph. WRITE The equation is ( ) ( ) =, ) 6 Worked Eample Sketch the graph of 6 =. State the coordinates of the centre and the radius. Think WRITE Complete the square in and. 6 = ( ) 9 ( ) = Epress the equation in standard circle form. Recognise that this is a circle and state the centre and the radius. ( ) ( ) = 6 This represents a circle with centre (, ) and radius. Chapter Coordinate geometr 9
Find the - and -intercepts. -intercepts: = 6 = = 6± 8 Sketch its graph. = and = -intercepts: = - = (, ) = ± 8 = 7 and = 7 (, 7 ) (, ) 7 6, ) (, 7 ) 6 Parametric equations of circles The rule for a relation can sometimes be epressed in terms of a third variable called a parameter. For the Cartesian equation of a circle = r, the variables and can be epressed in terms of a parameter t, so that the parametric equations are: = r cos (t) and = r sin (t), where t [, π] so = r cos (t) r sin (t) = r (cos (t) sin (t)) = r (since cos (t) sin (t) = ) Note: If t [, π] a full circle is obtained; if t [, π], a semicircle is obtained. For the Cartesian equation of a circle ( h) ( k) = r, the parametric equations are: = h r cos (t) and = k r sin (t), These can also be written as (t) = h r cos (t) and (t) = k r sin (t). Maths Quest Specialist Mathematics for the Casio ClassPad
The domain () and range () of the Cartesian equation can be determined from the range of these respective parametric equations. That is, the domain of the Cartesian equation is the range of (t) and the range of the Cartesian equation is the range of (t). Worked Eample Find the Cartesian equation of the circle with parametric equations = cos (θ ) and = sin (θ ), θ [, π]. State the domain and range of the circle. Think Rewrite the parameters b isolating cos (θ ) and sin (θ ). Square both sides of each equation then add them. Write = cos (θ ) and Use the Pthagorean identit: sin (θ ) cos (θ) =. ( ) ( ) = sin (θ ) ( ) ( ) = cos (θ) sin (θ) Epress the relation in standard circle form. ( ) ( ) = The domain is the range of the parametric Domain is [, ] = [, ] equation = cos (θ ). 6 The range is the range of the parametric equation = sin (θ ). Range is [, ] = [, ] = Graphs of ellipses If a circle with Cartesian equation = is dilated b a factor a from the -ais and b a factor b from the -ais then all points P(, ) on the circle become the points P (a, b) as shown at right. The basic equation of an ellipse is: a = b Its graph is shaped like an elongated circle see graph at right. This ellipse:. is centred at (, ). has vertices at ( a, ), (a, ) (found b letting = and solving), and (, b) and (, b) (found b letting = and solving). If this curve were shifted h units to the right and k units up, then the centre would move to (h, k) and its equation would become: ( h) ( k) a b P(, ) P (a, b) b a a b b a a b = Note: If a = b then the equation becomes ( h) ( k ) a a = and can be rearranged to ( h) ( k) = a (b multipling both sides b a ). This is the equation of a circle. Chapter Coordinate geometr
For an ellipse in the form ( h) ( k ) = we can deduce the following, which will a b help us to sketch the ellipse:. (h, k) are the coordinates of the centre of the ellipse.. The vertices are ( - a h, k), (a h, k), (h, - b k), (h, b k). Notes. a is half the length of the major ais (ais of smmetr parallel to the -ais if a > b), (ais of smmetr parallel to the -ais if a < b).. b is half the length of the minor ais (ais of smmetr parallel to the -ais if a > b), (ais of smmetr parallel to the -ais if a < b).. a, b are lengths and so are positive values. Worked Eample Sketch the graph of the function ( ) ( ) 9 Think Compare ( ) ( ) 9 = with ( h) ( k) =. a b The major ais is parallel to the -ais as a > b. The etreme points (vertices) parallel to the -ais for the ellipse are: ( a h, k) (a h, k) The etreme points (vertices) parallel to the -ais for the ellipse are: (h, b k) (h, b k) Find the - and -intercepts. On the Main screen, complete the entr line as: solve ( ) ( ) =, = 9 solve ( ) ( ) =, = 9 Press E after each entr. =. WRITE/Displa h =, k = and so the centre is (, ). a = b = 9 a = b = Vertices are: (, ) (, ) = (, ) = (6, ) and (, ) (, ) = (, ) = (, ) The -intercepts are: = =, The -intercepts are: 6 6 6 6 =, = Maths Quest Specialist Mathematics for the Casio ClassPad
6 Sketch the graph of the ellipse., 6 6 6 (, ) ( ) ( ) 9 (, ) (, ) (6, ), 6 6 (, ), 6 6, Worked Eample Sketch the graph of the function ( ) ( ) 9 6 =. Think Compare ( ) ( ) 9 6 ( h) ( k) =. a b = with WRITE/Displa h =, k = So the centre is (, ). a = 9 b = 6 a = b = The major ais is parallel to the -ais as b > a. The etreme points (vertices) parallel to the -ais for the ellipse are: ( a h, k) (a h, k) The etreme points (vertices) parallel to the -ais for the ellipse are: (h, b k) (h, b k) Find the - and -intercepts. On the Main screen, complete the entr line as: solve ( ) ( ) =, = 9 6 solve ( ) ( ) =, = 9 6 Press E after each entr. Vertices are: (, ) (, ) = (, ) = (, ) and (, ) (, ) = (, 8) = (, ) The -intercept is =. The -intercepts are: =, = Chapter Coordinate geometr
6 Sketch the graph of the ellipse.,, (, ) 6 (, ) (, ) (, ) 6 8 (, 8) ( ) 9 ( ) 6 Worked Eample Sketch the graph of the function 9 ( - ) =. Think Rearrange and simplif b dividing both sides b to make the RHS =. WRITE/displa 9( ) = Simplif b cancelling. 9( ) ( ) 9 Compare ( ) = with 9 ( h) ( k) = a b. Major ais is parallel to the -ais as a > b. The etreme points (vertices) parallel to the -ais for the ellipse are: ( a h, k) (a h, k) 6 The etreme points (vertices) parallel to the -ais for the ellipse are: (h, b k) (h, b k) 7 Find the -intercepts. On the Main screen, complete the entr line as: solve( 9( - ) =, ) = Then press E. = = h =, k = and so the centre is (, ). a = 9 b = as a, b > a = b = Vertices are: (, ) (, ) = (, ) = (, ) and (, ) (, ) or (, ) (, ) (,.) (,.) =, = Maths Quest Specialist Mathematics for the Casio ClassPad
8 Sketch the graph of the ellipse. 6 (, ) 9( ) (, ) (, ) (, ) (, ),, Worked Eample Sketch the graph of the relation described b the rule: 8 9 =. Think WRITE/displa To find the centre and the value of a and b using the CAS calculator, on the Conics screen tap: q Select the form of the equation (the sith from the top). OK Complete the entr line as: - 8 9 Tap: w Select the form ou want the equation to be transformed into (i.e. the seventh from the top). OK Chapter Coordinate geometr
The equation is transformed into the desired format, so the centre and a and b values can be determined. To draw a sketch of the ellipse, tap: ^ Adjust to the required window size if needed. Epress the relation in the standard form of an ellipse. 6 State the coordinate of the centre and the values of a and b. ( ) ( ) = The ellipse has its centre at (, ), with a = and b =. 7 State the coordinates of the vertices. Vertices are (, ), (, ) and (, ), (, 6). 8 Find the -intercepts. On the calculator, tap: Analsis G-Solve Intercept To find the other intercept, use the right arrow to move the cursor. Alternativel on the Main screen, complete the entr line as: solve( - 8 9 =, ) = Then press E. 9 Write the -intercepts. = = 6 6 6 Maths Quest Specialist Mathematics for the Casio ClassPad
Sketch the graph of the ellipse. (, 6) 6 (, ) 6, (, ) (, ) 6, (, ) Parametric equations of ellipses The parametric equations of an ellipse with Cartesian equation a = b sin (t), where t [, π]: cos (t) sin (t) = but cos (t) = a and sin (t) = b so = a b The parametric equations of an ellipse with Cartesian equation ( h) ( k ) a b are = h a cos (t) and = k b sin (t). = b are = a cos (t) and = Worked Eample 6 Determine the Cartesian equation of the curve with parametric equations = sin (t) and = cos (t ) where t R. Describe the graph and state its domain and range. Think Use a CAS calculator to sketch the graph on the Graph & Tab screen, in parametric mode, b tapping: Tpe Param Tpe Complete the entr line as: t = sin(t) t = - cos(t)! WRITE/displa Rewrite the parameters b isolating cos (t) and sin (t). Square both sides of each equation then add. = sin (t) and = cos (t) ( ) ( ) = sin (t) cos (t) 9 = Describe the relation. This represents an ellipse with centre (, ). Chapter Coordinate geometr 7
The domain is the range of the parametric equation = sin (t). 6 The range is the range of the parametric equation = cos (t). Domain is [, ] = [, ] Range is [, ] = [, ] remember For a circle with equation ( h) ( k) = r :. The centre is (h, k) and the radius is r.. The parametric equations are = h r cos (t) and = k r sin (t ). For an ellipse with equation ( h) ( k ) = a b :. The centre of the ellipse is (h, k).. The vertices are ( a h, k), (a h, k), (h, b k), (h, b k).. The parametric equations are = h a cos (t) and = k b sin (t ). The domain () and range () of the Cartesian equation can be determined from the ranges, respectivel, of these parametric equations. eercise C Graphs of circles and ellipses We9 Write the equation of the circle with the following centres and radii. a Centre (, ) and radius b Centre (, ) and radius 6 c Centre (, ) and radius d Centre (, ) and radius We Find the coordinates of the centre and the radii of the circles with the following equations. Sketch the graph in each case. a 6 = b = c = d = e 8 6 = f 9 = We Find the Cartesian equations of the circles with the following parametric equations. State the domain and range of each. a = cos (t), = sin (t), t [, π] b = cos (t), = sin (t), t [, π] c = cos (t), = sin (t), t [, π] d = sin (t), = cos (t), t R MC For each of the equations below, choose the correct alternative for the values of a, b, h, and k. (This question is like Worked eample, but don t draw the graph.) a ( ) ( 9 ) = 8 6 A a = 9, b = 8, h =, k = 9 B a = 8, b = 6, h =, k = 9 C a = 8, b = 9, h =, k = 9 D a = 9, b = 8, h =, k = 9 E a = 9, b = 8, h =, k = 9 ebookplus Digital doc Spreadsheet 8 Ellipses 8 Maths Quest Specialist Mathematics for the Casio ClassPad
b ( ) = 6 A a = 6, b =, h =, k = B a = 6, b =, h =, k = C a = 6, b =, h =, k = D a = 6, b =, h =, k = E a =, b = 6, h =, k = c ( ) ( ) = 8 A a = 9, b =, h =, k = B a = 8, b =, h =, k = C a = 9, b =, h =, k = D a = 8, b =, h =, k = E a = 8, b =, h =, k = d ( ) = 6 A a =, b =, h =, k = B a =, b =, h =, k = C a = 6, b =, h =, k = D a = 6, b =, h =, k = E a =, b =, h =, k = WE Sketch the graph of the following functions. a ( ) ( ) 9 ( ) c = 9 e ( ) ( ) g ( ) ( ) 6 = = = b ( ) ( ) 9 d 9 f ( ) = ( ) ( ) 9 h ( ) ( ) 9 7 = = = 6 WE Sketch the graph of the following functions. a ( ) ( ) 9 c e = b 9 ( ) 6 = d ( ) 9 9 = f 6 g ( ) ( ) 9 6 = = = ( ) ( ) 8 9 h ( ) = 7 MC Consider the ellipse with the equation ( ) ( ) =. 6 a The centre of the ellipse is: A (, ) B (, ) C (, ) D (, ) E (, ) b The maimum and minimum points on the ellipse are respectivel: A (, ) and (, ) B (, 7) and (, ) C (, 6) and (, ) D (, ) and (, 8) E (, ) and (, 6) 8 WE Sketch the graphs of the following functions. a 9( ) 6( ) = b 6 = c 6( ) = d ( ) 6( ) = 6 e 6( ) 9( ) = f ( ) 9( ) = = Chapter Coordinate geometr 9
9 We Sketch the graphs of the following functions. a 9 7 = b 9 6 8 = c 9 7 8 = d 9 9 = We 6 Determine the Cartesian equation of each of the curves with the following parametric equations. Describe the graph and state its domain and range. a = sin (t), = cos (t), t [, π]. b = cos (t), = sin (t), t [, π]. c = cos (t), = sin (t), t [, π ]. Give a pair of parametric equations which correspond to the following Cartesian equations. a = 9 b = 9 c ( ) ( ) = 6 d ( ) ( ) = 9 d ebookplus Interactivities int-8 Conical hperbola int-8 Conical hperbola (conjugate) Graphs of hperbolas Hperbolas have the following important characteristics. b b a a. The basic equation of a hperbola centred at (, ) is = a b. ( a, ) (a, ). If this curve were shifted h units to the a a right and k units up, then the centre would (, ) move to (h, k) and its equation would become ( h) ( k ) =. a b. The basic form of a hperbola centred at (, ) is shown at right. The vertices for this curve are at ( a, ) and (a, ) and the two asmptotes are given b b = a b and = a. When the hperbola is not centred at (, ):. For the curve of the function ( h) ( k ) =, the points on = are a b a b moved h units to the right and k units up (or has been replaced with ( h) and replaced with ( k)).. Therefore, the vertices are ( a h, k), (a h, k) and the centre is at (h, k). b. The asmptotes are at k = ( a h b ) and k = ( a h ) or b = ( a h ) k b and = ( a h ) k. To draw sketches of hperbolic relations we simpl:. Rearrange the equation into the appropriate general form and determine the values of a and b.. Write down the coordinates of the centre.. State the coordinates of the vertices.. Write down the equations of the asmptotes.. Sketch a hperbolic graph which fits the above information. Maths Quest Specialist Mathematics for the Casio ClassPad
Worked Eample 7 Sketch the graph of the hperbola with equation =. 9 Think The equation is in the correct form, so read off the values of a, b, h and k. WRITE As h =, k =, there are no translations. a = 9 b = a = b = Write the coordinates of the centre. The centre is at (, ). Write the coordinates of the vertices. The vertices are (, ) and (, ). Write the equations of the asmptotes. The asmptotes are = and =. Draw the asmptotes, plot the vertices and centre, and then sketch the hperbola. (, ) (, ) (, ) 8 6 6 8 9 Worked Eample 8 Sketch the graph of the hperbola with the equation ( ) 6 Think The equation is in the correct form, so read off the values of a, b, h and k. WRITE ( ) 9 =. h =, k = a = 6 b = 9 a = b = Write the coordinates of the centre. The centre is (, ). Write the coordinates of the vertices. The vertices are (, ) and (, ) or ( -, ) and (7, ). Write the equations of the asmptotes. The asmptotes: For each asmptote find the - and -intercepts. = ( ) = ( ) ( ) = - ( ) ( ) = ( ) 8 = - 9 8 = 9 = 7 = - For = 7 For = =, = 7 =, = = 7 = (, 7 ) (, =, = 7 =, = Chapter Coordinate geometr
6 The - and -intercepts for = are too close to each other so use one of these points, sa (, ), and the centre to sketch this line as both asmptotes intersect here. = 7 7 Find the -intercepts of the function. = : 8 Plot the vertices and centre and then sketch the hperbola. = ( 7, ) (, ) ( ) ( ) = 6 9 = 9± 9 9,, (, ) (, ) (7, ) 7 ( ) 6 ( ) 9 Worked eample 9 Sketch the graph of the hperbola with equation 6 9( ) =. Think Rearrange the equation b dividing both sides b to make the RHS =. Simplif b cancelling. Read off the values of h and k Work out values of a and b. WriTe 6 9( ) = 6 9( ) = ( ) = 9 6 h =, k =, translation of units up a = 9 b = 6 a = b = 6 as a, b > Write the coordinates of the centre. The centre is at (, ). Write the coordinates of the vertices. The vertices are:(, ) and (, ) or (, ) and (, ) 6 Write the equations of the asmptotes. The asmptotes are: 6 6 = ( ) and = ( ) = 6 ( ) = 6 6= 6 6= 6 ebookplus Tutorial int-6 Worked eample 9 7 Write the - and -intercepts for the asmptotes. 6 = 6 6 = 6 Intercepts for 6 = 6 are ( 6, ) and (, ). Intercepts for 6 = 6 are ( 6, ) and (, ). Maths Quest Specialist Mathematics for the Casio ClassPad
8 Draw the asmptotes, plot the vertices and centre, and then sketch the hperbola. 6 6 (, ) (, ) (, ) 6 6 6 9( ) Parametric equations of hperbolas The parametric equations of a hperbola with Cartesian equation a b = a sec (t) and = b tan (t) = a sec (t) The graph of the parametric equation = a sec (t) shows how it affects the domain of the hperbola. π π For t,, [a, ), which represents the a right branch of a hperbola. a π π For and t,, ( -, - a], which represents the left branch of a hperbola. = are: a sec(t) t = b tan (t) The graph of the parametric equation = b tan (t) shows how it affects the range of the hperbola. b tan(t) π π π π For t, or t,, R. Generall, the domain of the hperbola with centre (, ) is (, a] [a, ) and the range is R. b b t Verifing the parametric equations of a hperbola We can verif the parametric equations of a hperbola with equation Since: a tan (t) = sec (t) sec (t) tan (t) = b = as follows. but sec (t) = a and tan (t) = b (from the parametric equations) a = [the Cartesian equation of a b hperbola with centre (, )] Chapter Coordinate geometr
Worked eample Similarl, the parametric equations of a hperbola with Cartesian equation ( h) ( k) = are: a b = h a sec (t) and = k b tan (t). Determine the Cartesian equation of the curve with parametric equations = sec (t) and π π = tan (t), where t,. Describe the graph and state its domain and range. Think Rewrite the parameters b isolating sec (t) and tan (t). Square both sides of each equation then subtract. WriTe = sec (t), and = tan (t) = sec (t) tan (t) 9 = Describe the relation. This represents a hperbola with centre (, ). The domain is the right branch of the hperbola [a, ). Domain is [, ) The range is R. Range is R. remember For a hperbola with equation ( h) ( k ) = : a b. The vertices are ( a h, k), (a h, k).. The centre is at ( h, k).. The asmptotes are at b k = a ( h) and k = b ( a h ).. The parametric equations are = h a sec (t) and = k b tan (t). eercise d ebookplus Digital doc Spreadsheet Graphs of hperbolas Graphs of hperbolas MC For each of the equations below, choose the correct alternative for the equations of the asmptotes and the coordinates of the vertices. a = 8 6 A = 8, = 8 9 9, ( 9, ), (9, ) B = 9 8, = 9 8, ( 9, ), (9, ) b C = 8 9, = 8 9, ( 8, ), (8, ) D = 9 8, = 9 8, ( 8, ), (8, ) E = 8 9, = 8 9, ( 9, 8), (9, 8) ( ) = 6 A = 6, = 6, ( 6, ), (6, ) B =, = ( 6, ), (6, ) C =, = (, ), (, ) D = 6, = 6, (, ), (, ) E =, = (, ), (, ) Maths Quest Specialist Mathematics for the Casio ClassPad
c ( ) = 9 A = 9, = 9, ( 6, ), (, ) B 6 =, 6 =, (, ), (, ) C = 9, = 9, (, ), (, ) D = 9, = 9, (, ), (, ) E 6 =, - 6 =, ( 6, ), (, ) d ( ) ( ) = 6 A =, =, (, ), (, ) B =, = (, ), (6, ) C =, = (, ), (6, ) D =, =, (, ), (, ) E =, = (, ), (, ) Sketch the graphs of the hperbolas with the following equations. a WE7 = b = 9 9 c = d = 9 9 9 6 e = f g WE8 i ( ) ( ) 9 ( ) ( ) 9 k ( ) ( ) = = = = 6 h ( ) ( ) j l 9 ( ) = ( ) ( ) 9 = = MC The rule representing the graph shown at right is: A C ( ) = B = E = = D = MC The graph which best represents the relation ( ) ( ) 9 A B 6 (, ) (, ) (8, ) 6 8 ( 8, ) 8 6 = is: (, ) 6 8 (, ) Chapter Coordinate geometr
C D 8 6 (, ) (, ) (6, ) 6 (, ) (, ) (7, ) 6 8 6 8 E 6 (, ) (, ) 6 6 ebookplus Digital doc Worksheet. e We 9 Sketch the graph of the hperbola with the equation: a 6 = b 9 6 = c = d 9 = e 6( ) 9( ) = f ( ) 9( ) = g 6( ) ( ) = h 9 6( ) = i 9( ) 7 = 6 j ( ) 9( ) = 6. 6 We Determine the Cartesian equation of each of the curves with the following parametric equations. Describe the graph and state its domain and range. a = sec (t) and = tan (t), where t R. b = sec (t) and = tan (t), where t R. c = sec (t) and = tan (t), where t ( π, π ) d = sec (t) and = tan (t), where t ( π, π ) 7 Give a pair of parametric equations which correspond to the following Cartesian equations. a ( ) ( ) ( ) ( ) = b = 9 6 8 Find the asmptotes for the hperbola =. Partial fractions Adding fractions to obtain a single fraction is a familiar and basic process. For eample, = or = 7 = ( ) ( ) ( )( ) = = 6 Maths Quest Specialist Mathematics for the Casio ClassPad
The reverse of this process is to split a fraction into the sum of simpler fractions. These simpler fractions are referred to as partial fractions. For eample, using the above eamples:. 7 Fractions = Partial fractions. = Splitting a rational epression into partial fractions is useful when antidifferentiating rational epressions, as we will see in Chapter 6. It is also useful for sketching graphs of some rational epressions, as will be seen in the net section. For now, we concentrate on the methods for splitting a fraction with a quadratic denominator into partial fractions. Equating polnomials We regularl use the smbol, which means is identicall equal to, to indicate that two epressions are equal for all allowable values of. Two polnomials of degree n are equal if the coincide for more than n values of. For eample, suppose a( ) b( ). If values of a and b are found so that the polnomials (degree ) on each side of this identit are equal for two substituted values of, then the must be equal for all values of. The most convenient values of to substitute into the above identit are = and = as the allow a and b to be solved b direct substitution. This is demonstrated in the following worked eample. Worked Eample Determine the values of a and b in the following identities: a a( ) b( ) b a( ) b 9. Think Write a Substitute = to eliminate a from the identit. a a( ) b( ) Let = 7b = 7b = 7 Solve for b. b = Substitute = to eliminate b from the identit. Let =. 7a = 7a = Solve for a. a = State the solution. The solution is a = and b =. b Substitute = to eliminate a from the identit. Solve for b. b a( ) b 9 Let =. b = ( ) 9 = 6 9 b = Chapter Coordinate geometr 7
Substitute b = and =, sa (or an value of other than ), since b cannot be eliminated from the identit. Let =. a = 9 Solve for a. a = 6 a = State the solution. The solution is a = and b =. Creating partial fractions The general procedure for splitting a rational epression P ( ), where Q() is a quadratic Q ( ) epression, into partial fractions is outlined in the following steps. Step. If the degree of P() is greater than or equal to the degree of Q() then divide Q() into P() and split the rational part into partial fractions using the following steps. Step. Factorise the denominator Q(). Step. Equate P ( ) a b where R() and S() are factors of Q() and Q ( ) R ( ) S ( ) are usuall linear. Step. Epress the right-hand side of the identit in the same form as the left-hand side, with the same denominators: P ( ) as( ) br ( ) Q ( ) Q ( ) Step. Equate the numerators: P() as() br(). Step 6. Solve for a and b b substitution. Step 7. Substitute a and b into the partial fractions. Notes. If Q() is a perfect square then Steps to will be similar to but not eactl like those prescribed above. This will be demonstrated later in Case.. The solution can be quite easil checked b adding the partial fractions which should equal the original rational epression. Case : Factorised denominator If f (), g() and h() are linear functions, then: f( ) a b gh ( ) ( ) g ( ) h ( ) Worked Eample Epress as partial fractions. ( )( ) Think Write a b Equate the rational epression to. a b ( )( ) Epress the right-hand side of the identit into the same form as the left-hand side. a ( ) b ( ) ( )( ) 8 Maths Quest Specialist Mathematics for the Casio ClassPad
Equate the numerators. a( ) b( ) Substitute = to eliminate a. Let =. Solve for b. = b b = 6 Substitute = to eliminate b. Let =. 8 = a 7 Solve for a. a = 8 Epress the original rational epression as partial fractions. = ( )( ) Worked Eample Case : Perfect square denominator If f () and g() are linear, then: f( ) a b [ g ( )] g ( ) [ g ( )] Write the rational epression as partial fractions. ( 7) Think a b Equate the rational epression to. ( 7) ( 7) Epress the right-hand side of the identit into the same form as the left-hand side. Write a b ( 7) 7 ( 7) a( 7) b ( 7) Equate the numerators. a( 7) b Substitute = 7 to eliminate a. Let = 7. - = b Solve for b. b = 6 Substitute another value for, sa =, and b =. Let = 7 - = - 7a 7 Solve for a. - 7a = 7 8 Epress the rational epression as partial fractions. a = = ( 7) 7 ( 7) or = ( 7) ( 7) Case : Denominator is not factorised If the denominator of a rational epression is not factorised, then factorise it first before splitting it into partial fractions. Chapter Coordinate geometr 9
Worked eample B first factorising the denominator, epress as partial fractions. 9 Think The denominator factorises as a difference of perfect squares. Equate the rational epression to Epress the right-hand side of the identit into the same form as the left. WriTe = 9 ( )( ) a b. a b a ( ) b ( ) 9 Equate the numerators. a( )( ) Substitute = to eliminate a. Let =. = 6b 6 Solve for b. 6b = 8 b = 7 Substitute = to eliminate b. Let =. = 6a 8 Solve for a. 6a = a = 9 Epress the rational epression as partial fractions. = 9 Case : The degree of the numerator the degree of the denominator If the degree of the numerator of a rational epression is greater than or equal to the degree of the denominator, then divide the denominator into the numerator first before splitting the fractional part into partial fractions. Worked eample Epress each of the following as partial fractions. a b 7 ( )( ) c Use a calculator to determine the answers to parts a and b. Think WriTe/diSPla ebookplus Tutorial int-7 Worked eample a The degrees of the numerator and denominator are both. Epand the denominator so that it can be divided into the numerator. a Degrees are both. = ( )( ) 8 Maths Quest Specialist Mathematics for the Casio ClassPad
Use long division to divide. 8 Epress the original rational epression in terms of the quotient and remainder. Equate the fractional part to a b. 6 Epress the right-hand side of the identit in the same form as the left-hand side. = ( )( ) ( )( ) a b ( )( ) a ( ) b ( ) ( )( ) 7 Equate the numerators. a( ) b( ) 8 Substitute = to eliminate a. Let =. = 6b 9 Solve for b. b = Substitute = to eliminate b. Let =. = 6a Solve for a. a = Epress the fraction as partial So = ( )( ) fractions. Rewrite the original epression as partial fractions. b The degree of the numerator () is greater than the degree of the denominator (). Use long division to divide the denominator into the numerator. = ( )( ) b 7 7 7 6 Epress the original rational epression in terms of the quotient and remainder. Factorise the denominator of the fraction part. Equate the fractional part to a b. 7 = = ( )( ) a b Chapter Coordinate geometr
6 Epress the right-hand side of the identit in the same form as the left-hand side. a ( ) b ( ) ( )( ) 7 Equate the numerators. a( ) b( ) 8 Substitute = to eliminate a. Let =. 8 = b 9 Solve for b. b = 9 b = Substitute = to eliminate b. Let =. = a Solve for a. a = a = Epress the fraction as partial fractions. So = ( )( ) Rewrite the original epression as partial fractions. 7 = c On the Main screen, tap: c Action Transformation epand Complete the entr line as: epand ( )( ) epand 7, Press E after each entr. Write the answers. = ( ) ( ) 7 = REMEMBER. If f (), g() and h() are linear functions, then: f( ) a b (a) gh ( ) ( ) g ( ) h ( ) f( ) a b (b) [ g ( )] g ( ) [ g ( )] Maths Quest Specialist Mathematics for the Casio ClassPad
. If the denominator of a rational epression is not factorised, then factorise it first before splitting it into partial fractions.. If the degree of the numerator of a rational epression is greater than or equal to the degree of the denominator, then divide the denominator into the numerator first before splitting the fractional part into partial fractions. Eercise e Partial fractions WE Evaluate the values of a and b in the following identities. a a( - ) b( ) b a( ) b( ) 9 c a( ) b d a( ) b( ) e a( ) b( ) 9 f a( ) b( ) 6 WE Epress the following as partial fractions. a d ( )( ) 7 ( )( ) b e 7 ( )( ) ( )( ) c f ( )( ) ( )( ) MC The respective values of a and b in the identit a b( ) are: A, B, C, D, E, a b MC The respective values of a and b in the identit are: ( )( ) A, B, C, D, E, WE Write each of the following rational epressions as partial fractions. 9 7 9 a ( ) b ( ) c ( ) d ( ) e ( ) 6 WE B first factorising the denominator, epress each of the following functions as partial fractions. 7 7 a b 6 c 67 8 d 9 e f 8 7 WE Epress each of the following functions as partial fractions. a 6 ( )( ) d b ( )( ) e 8 7 8 c f 8 ( )( ) 6 9 7 8 MC Consider the rational epression 7. a After long division the epression simplifies to: 7 A B C 6 7 D E 9 Chapter Coordinate geometr
f b When epressed as a partial fraction, the epression simplifies to: A B C 6 D E Sketch graphs using partial fractions In this section we investigate how partial fractions can be used to assist in the graphing of rational functions. Emphasis is placed on locating asmptotes and the addition-of-ordinates method of graphing. Addition of ordinates The graph of a function that involves the addition of two (or more) simpler, familiar functions can be obtained b graphing the two simpler functions on the same set of aes and then adding the ordinates (-values). For eample, consider the graph of the function. Sketch the graphs of = and =, with broken lines, on the same set of aes.. For several values of add the -values, and, to obtain. Some eamples are: (a) When =, = and =, so = =, giving the point (, ). (b) When =, = and =, so = =, giving the point (, ). (c) When =, = and =, so = =, giving the point =. (, ) (, ) (, ).. Repeat until the shape of the function is deduced. The graph of the function = f () g() can be obtained b graphing = f () and = g() on the same set of aes and then adding the ordinates. (, ) (, ) (, ) (, ) Graphs of the tpe = a b c g ( ) a As we saw earlier in this chapter, graphs of the form = g b c ( ) have asmptotes c = and = g(). b Maths Quest Specialist Mathematics for the Casio ClassPad
a The graph of the function = g() can be obtained b: b c a. sketching the graphs of = b c and = g() (an asmptote) on the same aes. adding the two b the addition-of-ordinates method. Notes. If g() = d, a constant, then the graph of a = d can be sketched b recognising b c that it is a hperbola. Assuming that a, b, c and d are greater than, the sketch is:. Wherever possible, verif graphs using a graphics calculator. c b d c b a, b, c, d a b c d d Worked Eample 6 Sketch the graphs of each of the following, showing an asmptotes and aial intercepts. a = b = Think a The graph is a hperbola with asmptotes = and = (-ais). Write a Asmptotes: = = There is no -intercept ( = is an asmptote). No -intercept Substitute = into the equation to find the -intercept. Sketch the graph. When =, = (, ) b Divide the denominator into the numerator. b Epress the rational function as partial fractions. The graph is a hperbola with asmptotes = and =. 9 = Asmptotes: = = Chapter Coordinate geometr
Subsitute = into the original equation and solve for to determine the -intercept. Substitute = and solve for to determine the -intercept. =, = = =, = = 6 Sketch the graph.,, Worked eample 7 Sketch the graph of the function = 6. ebookplus Tutorial int-8 Worked eample 7 Think WriTe/diSPla Use a CAS calculator to epress the rational function as partial fractions b tapping: Action Transformation propfrac Complete the entr line as: propfrac 6 Then press E. Epress the function as partial fractions. = Sketch the graphs of = (asmptote) and = on the same aes. 6 Maths Quest Specialist Mathematics for the Casio ClassPad
Determine an -intercepts. =, 6 = ( )( ) = = and = 6 Determine the -intercept. =, = = 6 Add the two graphs b addition of ordinates 6 to obtain the graph of =. (, ) (, ) (, ) Graphs of the tpe = a b c f( ) g( ) a b Graphs of the form = c, where a, b and c R and f () and g() are linear f( ) g ( ) epressions, have two vertical asmptotes and the line = c behaves as a horizontal asmptote a b everwhere ecept for the point where f( ) g ( ) =. a b The graph of = c is obtained b: f ( ) g ( ) a. sketching the graphs of = f( ) and b = c on the same aes g ( ) a b. solving f( ) g ( ) = to find where = c. adding the two graphs b the addition-of-ordinates method. Worked Eample 8 Sketch the graph of the function = b first epressing it as partial fractions. State the equations of all vertical asmptotes and determine an aial intercepts. Think Write Factorise the denominator of the rational epression. ( )( ) Epress the rational epression as partial fractions in a b a b the form. a ( ) b ( ) ( )( ) Chapter Coordinate geometr 7
Sketch the graphs of same ais. = and = on the a( ) b( ) Let =. 6 = 6b b = Let =. = 6a a = So = or Determine the - and -intercepts. =, = = =, = = Solve the equation = to determine where the horizontal asmptote, =, is crossed. When =, = ( ) ( ) = = = = So horizontal asmptote is crossed at (, ). 6 Add the two graphs b the addition-of-ordinates method to obtain the graph of the rational function., (, ) 7 State the equations of the vertical asmptotes. Vertical asmptotes are = and =. 8 Maths Quest Specialist Mathematics for the Casio ClassPad
Worked eample 9 6 Sketch the graph of the function f( ) =, clearl 6 indicating all asmptotes. Think Divide the denominator of the rational epression into the numerator. WriTe 6 6 8 8 Epress f () in terms of the quotient and remainder. 8 f( )= 6 8 = ( )( ) ebookplus Tutorial int-9 Worked eample 9 Epress the fraction as partial fractions. 8 a b ( )( ) a ( ) b ( ) ( )( ) 8 a( ) b( ) Let =. = b b = Let =. = a a = Rewrite f () in partial fraction form. So f ( )= State the equations of the vertical asmptotes. Vertical asmptotes are = and =. 6 Sketch the graphs same ais. = and = on the 7 Solve = to find where the horizontal asmptote, =, is crossed. When =, = ( ) ( ) = 6 = 8 = = 8 So horizontal asmptote is crossed at ( 8, ). Chapter Coordinate geometr 9
8 Add the two graphs b the addition-of-ordinates method to obtain the graph of f (). ( 8, ) 8 remember a. The graph of = g ( ) can be obtained b: b c a (a) sketching the graphs of = b and c = g() (an asmptote) on the same aes (b) adding the two b the addition-of-ordinates method. a b. The graph of = c is obtained b: f( ) g ( ) a (a) sketching the graphs of = f( ) and b = g ( ) c on the same aes (b) adding the two graphs b the addition-of-ordinates method.. Use our graphics calculator to check our graphs. eercise F Sketch graphs using partial fractions ebookplus Use a graphics or CAS calculator wherever possible to verif the graphs obtained in the following eercise. We6a Sketch the graphs of each of the following functions. a = d = f We6b = b = e = g = 7 We7 Sketch the graph of the functions. c Digital doc Spreadsheet Single function grapher = a = d 7 = b = e = c = Maths Quest Specialist Mathematics for the Casio ClassPad
A WE8 Sketch the graph of the following functions b first epressing them as partial fractions. State the equations of all vertical asmptotes in each case. a = b = c = ( )( ) ( )( ) d = e = 7 MC Consider the function f( )= a As a partial fraction, f () is equal to: A D b The graph that best represents f () is: B B E. C C D E 7 MC The graph of the function f( )= has asmptotes with equations: A =, =, = B = 6, =, = C =, =, = D = and = E =, =, = 9 6 MC If g( )= then its graph has asmptotes described b the equations: A =, =, = B =, = C =, = and = D =, =, = E = and = 7 WE9 Sketch the graph of each of the following, clearl indicating all asmptotes. a f( )= b f( )= d g ( )= e f( )= 7 6 c g ( )= 6 6 Chapter Coordinate geometr
8 Sketch the graph for each of the following. For questions a and b, find the turning points. a = b = c = 8 6 8 6 8 9 8 d = e = 6 f = A 9 Given the function f( ) = ( α β ) αβ, state the equations of all the asmptotes and find the turning point, stating the domain. State the conditions on k such that the graph of = has: k 6 a two vertical asmptotes b onl one vertical asmptote c no vertical asmptotes. State the conditions on k such that the graph of = has: k a two vertical asmptotes b onl one vertical asmptote c no vertical asmptotes. Find the equation of the graph shown. = 9 = A If the curve = B b c has a vertical asmptote at = and has a horizontal asmptote at = and a turning point at ( -, ) then find the values of A, B, b and c. Maths Quest Specialist Mathematics for the Casio ClassPad
Summar Sketch graphs of = a m b n c where m = or and n = or Step. Break the given function into separate simpler functions. Step. Sketch the graph of each of the separate functions created, on the one set of aes. Step. Determine the asmptotes of the original function and pencil in how the graph of the function approaches these asmptotes. Consider:. large positive values for. large negative values for. values ver close to and either side of the vertical asmptote. Step. Work out the -intercept(s), -intercept and turning point for the given function to give greater accurac. Reciprocal graphs To sketch the graph of = from the graph of = f (): f( ). find the vertical asmptote at the -intercepts of f (). the horizontal asmptote is the -ais. find the common points when f () = ±. The graphs are in the same quadrant:. f( ) <, < f( ). f( ) >, >. f( ) Note: If = f( ), then:. f () =, = and f () =, =. f () <, > and f () >, <. Graphs of circles For a circle with equation ( h) ( k) = r :. the centre is ( h, k) and radius r. the parametric equations are = h r cos (t) and = k r sin (t). Graphs of ellipses ( h) ( k) For = : a b. the centre of the ellipse is ( h, k). the vertices are ( a h, k), (a h, k), (h, b k), (h, b k). the parametric equations are = h a cos (t) and = k b sin (t). Graphs of hperbolas ( h) ( k) For = : a b. vertices are ( a h, k), (a h, k). centre is at ( h, k). Asmptotes are at b k = a ( h) and k = b ( a h ). The parametric equations are = h a sec (t) and = k b tan (t). Chapter Coordinate geometr
Partial fractions If f (), g() and h() are linear functions, then: f( ) a b. gh ( ) ( ) g ( ) h ( ) f( ) a b.. [ g ( )] g ( ) [ g ( )] If the denominator of a rational epression is not factorised, then factorise it first before splitting it into partial fractions. If the degree of the numerator of a rational epression is greater than or equal to the degree of the denominator, then divide the denominator into the numerator first before splitting the fractional part into partial fractions. Sketch graphs using partial fractions a The graph of = g ( ) can be obtained b: b c a. sketching the graphs of = b c and = g() (an asmptote) on the same aes. adding the two graphs b the addition-of-ordinates method. a b The graph of = c is obtained b: f( ) g ( ) a. sketching the graphs of = f( ) and b = g ( ) c on the same aes. adding the two graphs b the addition-of-ordinates method. Use our graphics or CAS calculator to check graphs. Maths Quest Specialist Mathematics for the Casio ClassPad
chapter review Short answer Write down the equation of the asmptotes for =. The function = is broken into the functions = and = -, which appear on the graph shown. Describe the behaviour of the function near the asmptotes. Without an further calculations, sketch the graph of the function. a Calculate the eact value of the turning point for the graph of the function =. b Calculate the -intercept of the graph of the function =. c Sketch the graph of the function =, showing intercepts and the turning point. Sketch the graph of the function = showing intercepts and the turning point., The graph of the function f () = is shown below. 6 6 (, ) 6 f() = 6 Sketch the graph of the function = f( ). 6 a Sketch the graph of the function = 7 6, showing the turning point and intercepts. b Hence, on the same set of aes, sketch the graph of the function =. 7 6 7 Sketch the graph of each of the following. a ( ) = 6 b ( ) ( 6 ) = 6 c 7( - ) ( ) = 8 d e ( ) = 9 ( ) = 9 6 f ( ) - 9 = 6 g ( - ) ( - ) = h - 8 = i 9-8 6 = j - 6 = 6 8 Determine the Cartesian equation from the following parametric equations. a = cos (θ), = sin (θ), θ [, π] b = cos (t), = sin (t), t R c = sec (t), = tan (t), t R 9 Epress each of the following rational functions in partial fraction form. 9 a = b = 7 c 7 = Sketch the graph of each rational function in question 9. Consider the hperbola with equation ( c) ( ) = where c is a real constant. 9 The equation of one of the asmptotes of this hperbola is =. a Show that c = -. Chapter Coordinate geometr
b Sketch the hperbola on the following set of aes, clearl showing the asmptotes. 8 6 8 6 6 8 6 8 The graph that represents the function f( )= is: a b Eam tip Some students had both asmptotes correct, but one or both of their hperbola branches clearl did not ehibit asmptotic behaviour. VCAA Assessment report [ VCAA ] Sketch the graph of = on the aes below. Give the eact coordinates of an turning points and intercepts, and state the equations of all straight line asmptotes. [ VCAA 8] c d Multiple choice The graph at right could be described b the rule: A = B = (, ) (, ) C = (, ) D = E = e 6 Maths Quest Specialist Mathematics for the Casio ClassPad
The graph representing the function = a (, ) 9 is: 9 The graph of the rational function = has asmptotes: A = and = onl B =, = and = C = and = onl D =, = and = E =, = and = b (, ) 9 Consider the ellipse with equation ( ) ( ) = for questions and 6. 9 The maimum and minimum points are respectivel: A (, ) and (, ) B (, ) and (, ) C (, ) and (, ) D (, ) and (, ) E (, 6) and (, ) 6 The graph representing this equation is: a c (, ) 9 (, ) d b (, ) 9 (, ) e c (, ) 9 (, ) Chapter Coordinate geometr 7
d (, ) The rule that describes the hperbola shown below is: e (, ) (, ) (, ) (, ) 7 The rule that describes the ellipse at right is: A ( ) = B ( ) = C ( ) D E ( ) = = ( ) = 6 For questions 8 and 9, consider the hperbola ( ) ( ) =. 9 8 The hperbola has vertices given b: A (, ) and (9, ) B (, ) and (, 7) C (, ) and (, ) D (, ) and (, ) E ( 6, ) and (, ) 9 The graph of the hperbola has asmptotes with equations: A =± ( ) B =± ( ) C =± ( ) D =± ( ) 9 E =± ( ) (, ) A ( ) ( ) 9 6 B ( ) ( ) 6 9 C ( ) ( ) 6 9 D ( ) ( ) = = = = ( ) ( ) E = When epressed as partial fractions, 8 6 is equal to: A ( ) B ( ) C ( ) D 8 E 8 The rational epression 7 8 6 is 6 equal to: A B C 6 D 6 E 8 Maths Quest Specialist Mathematics for the Casio ClassPad
The graph of the rational function = is: A The rule describing the function graphed below is: (, ) (, ) (, ) B C D E (, ) (, ) (, ) A = B = c = d = e = A pair of parametric equations which correspond to the Cartesian equation = 9 is: A = 7 sec (t) and = 7 tan (t) B = 7 cos (t) and = 7 sin (t) C = 7 cos (t) and = 9 sin (t) D = 7 cos (t) and = 7 sin (t) E = 7 sin (t) and = 7 cos (t) 6 The graph of f( )=, where m and n mn are real constants, has no vertical asmptotes if: A m < n B m > n C m = - n D m < - n E m > - n Eam tip This question was not answered correctl b a majorit of students. Each of the four distractors was chosen b a significant proportion of the students who gave an incorrect response. It is unlikel that these responses were based on an particular misconception; rather, it appears that man of the students guessed the answer. VCAA Assessment report 7 [ VCAA ] (, ) A possible equation for the graph of the curve shown above is: Chapter Coordinate geometr 9
a a a =, a > b =, a < a a c =, a > d =, a > a e =, a < [ VCAA 8] 8 The equation a =, where a is a real constant, will represent a circle if: A a < - onl B a > - onl C a = ± onl D - < a < E a < - or a > [ VCAA 8] 9 P is an point on the hperbola with equation =. If m is the gradient of the hperbola at P, then m could be: A an real number B an real number in the interval ( -, ) C an real number in the interval [ -, ] D an real number in the interval R\( -, ) R E an real number in the interval [, ] [ VCAA 8] Etended response A drinking trough with semicircular ends is to be made from pressed metal. The volume of the trough is to be litres. a If r is the radius (in cm) of the semicircular end, show that the surface area of the trough is Sr () = π r, r >. r b Sketch the graph of surface area [S(r)] versus r. (Use a table of values.) c Ignoring an etra metal required to make the joins, find the minimum surface area of the trough and the corresponding value of r. An open bo is to be made from a roll of steel m wide (see the diagram at right). a Write down the epression for the length and width of the bo. b If the bo is to enclose a volume of 8 m, write a rule linking and. c Write an epression for the total surface area of the bo and then use substitution to make this a function of onl. d Show that the minimum surface area occurs when the bo is. m high and then find the minimum surface area that will enclose the given volume. (Hint: Epress the fractional part of the surface area as partial fractions before differentiating.) e Calculate the length that needs to be cut off the roll of steel to make the bo. A famil goes to the beach, and one of the bos takes a lilo and goes paddling. He is m from the shore, measured at right angles to the shore, and m from where the rest of the famil are ling on the beach. His father calls him to come back to shore, and he needs to get back in the quickest possible time. He paddles at m/s and runs along the beach at m/s. A diagram of his trip is shown at right. a Find an epression for the distance paddled through the water as a function of. b Using the equation distance travelled speed =, time taken find the time he takes to paddle to the shore in terms of. c Write an epression for the distance travelled along the beach in terms of and also the time taken travelling along the beach. d Write an epression for the total time he takes to get back to his famil. Use this epression to find the value of that gives a minimum value for the time of travel. e What is the minimum time of travel? Bo m m Famil 6 Maths Quest Specialist Mathematics for the Casio ClassPad
a Sketch the graphs of = and = on the same set of aes. b Hence, sketch the graph of = on the same set of aes as in a. Find the coordinates of the stationar points, the - and -intercepts and the equations of an asmptotes. c On a separate set of aes sketch the graph of =. d Eplain algebraicall how the graph in part c is obtained from the graph in part b. Hence, describe the transformation required. ebookplus e On the same set of aes as the graph in part c, sketch the graph ( ) of =. (Use a graphics calculator to assist.) Digital doc Test Yourself Chapter Chapter Coordinate geometr 6
ebookplus activities Chapter opener Digital doc Quick Questions: Warm up with ten quick questions on coordinate geometr. (page ) A Sketch graphs of = a m b n c where m = or and n = or Interactivities Graphs of power functions int-8: Consolidate our knowledge of graphs of the form = a m b n c, where m = and n =. (page ) Graphs of power functions int-8: Consolidate our knowledge of graphs of the form = a m b n c, where m = and n =. (page ) Graphs of power function int-86: Consolidate our knowledge of graphs of the form = a m b n c, where m = and n =. (page ) Graphs of power functions int-87: Consolidate our knowledge of graphs of the form = a m b n c, where m = and n =. (page ) Tutorial We int-: Watch how to sketch the graph of a rational function using addition of ordinates. (page ) We7 int-: Watch how to sketch the graphs of a quadratic and its reciprocal. (page ) Digital docs Spreadsheet : Investigate graphs using the single graph plotter. (page 9) B Reciprocal graphs Interactivit Reciprocal graphs int-8: Consolidate our understanding of reciprocal graphs. (page ) C Graphs of circles and ellipses Interactivit Elliptical graph int-8: Consolidate our understanding of graphs of circles and ellipses. (page 8) Digital docs Spreadsheet 8: Investigate graphs of ellipses. (page 8) D Graphs of hperbolas Interactivities Conical hperbola int-8: Consolidate our understanding of conical hperbolas. (page ) Conical hperbola int-8: Consolidate our understanding of conjugate conical hperbolas. (page ) Tutorial We 9 int-6: Watch how to sketch the graph of a hperbola. (page ) Digital docs Spreadsheet : Investigate graphs of hperbolas. (page ) WorkSHEET.: Sketch graphs of power functions, hperbolas and ellipses. (page 6) E Partial fractions Tutorial We int-7: Watch how to rationalise epressions as partial fractions. (page ) F Sketch graphs using partial fractions Tutorial We7 int-8: Watch a tutorial on how to sketch the graph of a rational function. (page 6) We9 int-9: Watch a tutorial on sketching the graph of a rational function b using long division and addition of ordinates. (page 9) Digital doc Spreadsheet : Investigate functions using a single function grapher. (page ) Chapter review Digital doc Test Yourself: Take the end-of-chapter test to test our progress. (page 6) To access ebookplus activities, log on to www.jacplus.com.au 6 Maths Quest Specialist Mathematics for the Casio ClassPad