Junior certificate HIGHER LEVEL Active Maths Old Sllabus Strand 5 πr m = - - πr Oliver Murph
Contents. Functions.... Functions and Graphs...5.3 Graphs...7.4 Linear Graphs...7.5 Quadratic Graphs...9.6 Real-Life Problems Solved b Graphs...3 Answers...9 Editor: Priscilla O Connor Designer: Liz White Laout: Compuscript Illustrations: Compuscript, Ror O Neill Oliver Murph, 0 Folens Publishers, Hibernian Industrial Estate, Greenhills Road, Tallaght, Dublin 4, Ireland All rights reserved. No part of this publication ma be reproduced or transmitted in an form or b an means, electronic, mechanical, photocoping, recording, or otherwise, without prior written permission from the publisher. The publisher reserves the right to change, without notice, at an time, the specification of this product, whether b change of materials, colours, bindings, format, tet revision or an other characteristic.
Old Sllabus Functions and Graphs Learning Outcomes In this chapter ou will learn: ÂÂ ÂÂ ÂÂ About functions, domain, range and codomain To solve problems based on linear and quadratic graphs To appl graphs to solve real-life problems
. Functions A function is like a machine. When a number is put in, it is transformed and a new number emerges. The letter is often used to denote the input. The letter is usuall used to denote the output. Another name for the output is f(). f : + 5 YOU SHOULD REMEMBER... Let A = {,, 3} and B = {3, 4, 5, 6, 7}. We can define a function f from A to B in the following wa: f: A B: + This means that f is a function which maps elements of A onto elements of B, where the output is twice-the-input-plus-one. Under this function, 3, 5 and 3 7, as shown in the mapping diagram: The set A is called the domain. The domain is the set of inputs. The set B is called the codomain. The codomain is the set of allowable outputs. 3 7 The range of f is the set of elements of the codomain which are actuall used up. In this case, the range = {3, 5, 7}. The range, therefore, is a subset of the codomain. A f B 3 4 5 6 KEY W ORDS Domain Range Codomain Aes and scales Old Sllabus Functions and Graphs Eercise.. f: 4. The domain of f is {,, 3, 4}. Find the range.. g: +. The domain of g is {0,, }. Find the range. 3. List the elements of: (i) The domain of f f (ii) The codomain of f (iii) The range of f 3
4. A = {,, 3} and B = {,, 3, 4, 5} are two sets. f: A B: is a function. (a) Cop and complete the mapping diagram of f. (b) Write down: A 3 (i) The domain of f (ii) The codomain of f (iii) The range of f f B 3 4 5 5. The domain of f: 3 is {, 0, }. Find the range. 6. A = {, 0, } is a set of numbers. f: A A: is a function. (i) List the couples of f. (ii) Write down the domain of f. (iii) Write down the codomain of f. (iv) Write down the range of f. 7. f: 3 is a function. Cop and complete the mapping diagram of the function f. A 0 8. g: 4 9 is a function.? (i) Find g(3) and g( 3). f B?? 7 (ii) If g(k) = 55, find two possible values of k. 9. f: 5. The range of f is {5, 4, 3, 9}. Find the domain of f. 0. g: ( )( + ) is defined for R. (i) Find g(5). (ii) Investigate if g() + g(3) = g(5). (iii) If g() = 45, find two possible values of. (iv) If g(k) = 3k, find two possible values of k.. f: + (i) Evaluate f(3) and f ( 3 ). (ii) If f(t) = 3, find the value of t. 4. g: is a function. + (i) Find g() and g ( ). (ii) If g(n) = 3, find the value of n. 5 (iii) If g() + g() = g(k), find the value of k. 3. f: ( ) (i) Evaluate f(3). (ii) Evaluate f(). (iii) Find two values of for which f() = 0. (iv) Find two values of for which f() = 8. 4. f() = a + b. If f() = 5 and f() =, find a and b. 5. f: (3 )( 3) (i) Evaluate f(). (ii) Find a value of (other than ) for which f() = f(). 6. The domain of f: + 3 is { 3,,, 0,,, 3}. What is the least element in the range? 7. N is the set of natural numbers = {,, 3, 4, 5 }. f: N N: Describe in words: (i) The codomain (ii) The range 8. f: is defined for all real numbers. What is the least number in the range of f? 9. f: 5. If f() = kf ( ), find the value of k. 3 Old Sllabus Functions and Graphs
Old Sllabus Functions and Graphs 0. g: 4 3. If g(0) + g() = g(), find the value of.. f: + a + b is a function. If f() = 4 and f() = 0, find the values of a and b.. f: + a + b is a function. f(3) = and f( 3) = 8. Find the value of a and of b. 3. f: a + b is a function. 0 If f(3) = 3 and f(8) = 7, find the values of a and b. Find, also, the value of f(5). 4. The diagram represents a function f: p + q. 3 z (i) Find the values of p and q. (ii) Find the values of and z. f 5 9 5. f: N N: a + c is a function, as illustrated on the diagram. 3 n f 0 m 4 (i) Find the values of a, c, m and n. (ii) Write down three natural numbers which are elements of the codomain but not the range. 6. f is a function defined as f: R R:. (i) Find f(6). (ii) Find two values of for which f() =. 7. f and g are two functions such that f() = + and g() = 7. (i) Find f(4). (ii) Find g(4). (iii) Verif that f( 5) = g( 5). (iv) Find a value of (other than 5) for which f() = g(). 8. f() = 5 is a function. (i) Evaluate f() f( ). (ii) Find the value of k for which f(k) f( k) = 60. (iii) Show that f( + ) = 5 + 3. 9. f: N N: 5 + 3 is a function. (i) Investigate if f(4) + f(3) = f(7). (ii) Find f(n + ). (iii) Find the value of n if f(n + ) =. (iv) Write down one number which is an element of the codomain but not the range. 30. f: 8 + b is a function. (i) If f(5) = 0, find the value of b. (ii) Find a value of (other than 5) such that f() = 0. 4
. Functions and gr aphs f() If the point (p,q) appears on the graph of some function = f(), this means that p is mapped onto q b this function. (p,q) i.e. f(p) = q p f q Worked Eample. This graph represents the function f: R R: + a + b. (i) Write down two equations in a and b, given that (,0) and (3,7) are on the graph. (ii) Find the values of a and b. (iii) If (,) is on the graph, find the value of. (iv) If ( n,0) is on the graph, where n N, find the value of n. Solution (i) (,0) is on the graph f() = 0 \ () + a() + b = 0 4 + a + b = 0 \ a + b = 4 (3,7) is on the graph f(3) = 7 (3) + a(3) + b = 7 \ 9 + 3a + b = 7 \ 3a + b = (ii) Solve the simultaneous equations: I II a + b = 4 3a + b = I a b = 4 II 3a + b = Add a = a + b = 4 \ 4 + b = 4 \ b = 8 Answer: a =, b = 8 f() ( n,0) 0 (,0) 3 (,) (3,7) (iii) We know now that f() = + 8. If (,) is on the graph then f() = \ () + () 8 = + 8 = \ 5 = Answer (iv) ( n,0) is on the graph f( n) = 0 \ ( n) + ( n) 8 = 0 n n 8 = 0 (n 4)(n + ) = 0 n = 4 OR n = Since n N, n = is rejected. Answer: n = 4 Old Sllabus Functions and Graphs 5
Eercise.. f: a + b is a function whose graph is illustrated below. 5. The diagram shows part of the graph of the function f: + p + q where {p, q, } R. (3,7) f() (,7) (3,0) = f() Find the values of a and b. 3 4 (0, ). The diagram shows the graph of f: + p + q, where p, q R. f() (,0) Find the values of a and b. (3,0) (i) Find the value of p and of q. (ii) If (,0) is a point on the graph (where 3), find the value of. 6. The diagram shows part of the graph of = a + b. (,) Old Sllabus Functions and Graphs 6 3. The diagram shows part of a function = a + b. (, 8) Find the values of a and b. (,3) 4. The diagram shows part of the graph of the function = + p + q where p, q R. (,0) (i) Find the values of p and q. (ii) If (0,n) is on the graph, find n. (3,8) (,4) (3,q) 3 (i) Find the values of a and b. (ii) Write q as a fraction. 7. The diagram shows part of the function = a + b 0. (,k) (, 6) (i) Find the values of a and b. (ii) Find the value of k. (3,7) (iii) Find two values of for which = 0.
8. The diagram shows part of the function = a + b. (,0) (5,0) 9. The diagram shows part of the graph of the function f() =, where a, b R. a + b (n, 8) (i) Find the values of a and b. (ii) Find the value of n, where n < 0. (, ) (, ) (i) Find the values of a and b. (ii) Find the value of t if f(t) = 3. 4 4 0. The diagram shows part of the graph of = a + b. (i) Find the values of a and b. (ii) Verif that f(3 + ) = f(3 ). (iii) Find two values of for which f(3 + ) = 0. 3 4 5 6.3 GraphS There is a ver good case for saing that Sir Isaac Newton (64 77) was the greatest mathematician and phsicist who ever lived. In phsics, he discovered the laws of motion, the laws of gravitation, the laws of light and the laws governing the collisions of spheres. In mathematics, his greatest invention was the calculus : a method of determining the slope of graphs at an point. Despite his remarkable discoveries, Newton remained modest about his achievements. He wrote: I do not know what I ma appear to the world, but to mself I seem to have been onl a bo plaing on the sea-shore, and diverting mself in now and then finding a smoother pebble or a prettier shell than ordinar, while the great ocean of truth la all undiscovered before me..4 Linear graphs The graph of an function of the form = a + b (where a and b are constants) is called a linear graph because the graph forms a line. Old Sllabus Functions and Graphs 7
Worked Eample. A car is accelerating uniforml. At time t seconds after it passes a point (p), its velocit (v) in metres per second is given b the formula v = + t. (i) Cop and complete the table. t 0 4 6 8 0 v (ii) Draw the graph of v for 0 t 0. Estimate from the graph: Solution (iii) The velocit at t = 3.6 seconds (iv) The time when the velocit is 9 m/s (i) t 0 4 6 8 0 v 6 0 4 8 3 (ii) v 30 9 0 9 Old Sllabus Functions and Graphs 0 0 4 3.6 6 8 0 8.5 (iii) Draw a line from t = 3.6, to the graph and then to the v-ais. The reading is approimatel v = 9 m/s. (iv) Draw a line from v = 9 to the graph and then to the t-ais. The reading is t = 8.5 seconds. Eercise.3. Draw the graph of = f() = 3 in the domain 3 4, R. Use our graph to estimate: (i) f(.3) (ii) The value of for which f() = 6 t. Draw the graph of = f() = 3 in the domain 3, R. Use our graph to estimate: (i) The value of f() when = 0.8 (ii) The value of for which f() = 0 3. Using the same scales and aes in the domain 4, R, draw the two graphs = and = 8 4. Find the point of intersection of the two graphs. 8
4. A car is travelling at a constant speed of 5 m/s until it passes a point p. The driver then decelerates. The speed (v) in metres per second thereafter is given b v = 5 3t, where t is the time (in seconds) after the car passes through p. Draw the graph of v for 0 t 5, t R. Use our graph to estimate: (i) The speed at t =.3 (ii) The time when the speed = 0 m/s (iii) The time when the car stops 5. The time (in minutes) for which a whole salmon should be cooked is given b the formula t = 0(m + ), where m = the mass (in kg) of the salmon. (i) Cop and complete the following table and hence draw the graph. 6. The conversion formula for changing from gas mark oven temperature to degrees Celsius is C = 0 + 4G, where G = the gas mark and C = degrees Celsius. Draw a conversion graph for 0 G 6, G R. (i) Estimate the temperature (in degrees Celsius) corresponding to gas mark _. (ii) Estimate the gas mark corresponding to 83 Celsius. 7. C = 5 (F 3) is the formula which relates 9 the temperature in degrees Fahrenheit (F) to the temperature in degrees Celsius (C). (i) Cop and complete the following table, giving the values to the nearest integer (whole number). F 0 0 40 60 80 00 C 8 Mass (kg) 0 3 4 5 6 Time (mins) 0 60 (ii) Estimate the time taken to cook a 3.4 kg salmon. (iii) A salmon is cooked for hours. What is its mass?.5 Quadratic graphs (ii) Draw a linear graph to illustrate this data. (iii) Estimate the temperature in degrees Celsius when it is 77 Fahrenheit. (iv) Estimate the temperature in degrees Fahrenheit when it is 5 Celsius. 8. Draw the graphs of the three linear functions: f: ( 3) g: 4 h: 3 in the domain 4, R. Show that the three lines are concurrent (i.e. that the all pass through the same point). An graph of the form = a + b + c, where a, b, c R, a 0 is called a quadratic graph. If a is a positive number, the graph looks like this: If a is a negative number, the graph looks like this: Old Sllabus Functions and Graphs 9
Worked Eample.3 Draw the graph of the function f: 3 7 in the domain 3, R. Find, from our graph: Solution (i) The value of f(.5) (ii) The values of for which f() = 3 (iii) The minimum value of f() and the value of at which it occurs (iv) The solution set of 3 7 < 0 0 3 3 3 0 3 7 4 0 4 6 7 7 7 7 7 7 7 9 7 6 4 Points: (,9)(, )(0, 7)(, 6)(,)(3,4) 5 f() (i) Draw a line from =.5 on the -ais, up to the graph and across to the -ais. The reading is approimatel 7. \ f(.5) = 7 Old Sllabus Functions and Graphs.5 0 7 5 5 0 0.3 = 3 7.5 minimum point.5. 3 Worked Eample.4 (ii) Draw lines east and west from = 3 on the -ais. The corresponding readings on the -ais are =.5 and =.. (iii) The minimum value of f() is approimatel 7.5 at = 0.3. (iv) 3 7 < 0 \ f() < 0 \. < <.9 (where the graph is below the -ais) Using the same scales and aes, draw the graphs of the functions f: 4 and g: in the domain 4 3, R. Use our graph to estimate: (i) The range of values of for which 4 > 0 (ii) The solutions of the equation + 4 5 = 0 (iii) The value of 3 0
Solution 4 3 0 3 4 4 4 4 4 4 4 4 4 8 6 4 0 4 6 6 9 4 0 4 9 f() 4 4 5 4 4 Points: ( 4, 4)( 3,)(,4)(,5)(0,4)(,)(, 4)(3, ) 4 3 0 3 8 6 4 0 4 6 g() 9 7 5 3 3 5 Points: ( 4,9)( 3,7)(,5)(,3)(0,)(, )(, 3)(3, 5) The two graphs are shown. (i) 4 > 0 \ f() > 0 \ 3. < <. (where the graph of f() is above the -ais) 0 = g() 8 6 4.5.9.7 0.9.7 4 3 3 = f() 4 6 8 0 (ii) Change both sides until ou end up with f() [i.e. 4 ] on one side. (iii) + 4 5 = 0 +.5 = 0 (Dividing both sides b ).5 = 0 (Multipling both sides b ) 4 =.5 (Adding.5 to both sides) \ f() =.5 \ =.9 or = 0.9 = 3 \ = 3 (Squaring both sides) \ 0 = 3 (Subtracting from both sides) \ = 4 (Adding ( ) to both sides in order to get f() on the right) \ g() = f() \ We want the -value of the points of intersection of the two graphs. These are =.7 and.7 Of course, 3 is a positive number. \ 3 =.7 Old Sllabus Functions and Graphs
Old Sllabus Functions and Graphs Eercise.4. Draw the graph of the function f: 5 in the domain 4, R. Estimate from our graph: (i) The value of f(.) (ii) The values of for which 5 = 0 (iii) The range of values of for which 5 < 0 (iv) The minimum value of f(). Draw the graph of the function f: 3 7 in the domain 3, R. Estimate from our graph: (i) The values of for which 3 7 = 0 (ii) The values of for which 3 = 0 (iii) The range of values of for which 3 7 0 (iv) The minimum value of f() and the value of at which it occurs 3. Draw the graph of the function f: 4 + in the domain 3, R. Use our graph to estimate: (i) The solution set of 4 + = 0 (ii) The values of for which f() > 0 (iii) The solution set of + = 0 (iv) The maimum value of f() 4. Draw the graph of the function f: 6 in the domain 3 3, R. Estimate from our graph the values of for which: (i) 6 = + (iii) 6 ( + ) (ii) (6 ) = 5. Draw the graph of the function f: 3 3 4 in the domain 3, R. Use our graph to estimate: (i) The values of for which 3 = 3 + 4 (ii) The values of for which 5 = 0 (iii) The range of values of for which < 0 (iv) The minimum value of f() and the value of at which it occurs 6. Draw the graph of the function f: 4 + 6 7 in the domain 3, R. Use our graph to find: (i) The values of f(.8) (ii) The values of for which + 3 3 = 0 (iii) The range of values of for which ( + 3) < 7 (iv) A negative value of for which f() = f() 7. Cop and complete the following table for the function f: 4. 3 0.5 f() 4 Draw a graph of = f() in the domain 3.5, R. Use our graph to find: (i) The solutions of the equation + = 4 (ii) The range of values of for which + < 4 (iii) The maimum value of f() (iv) The values of which satisf the equation 4 + 5 = 0 8. Using the same scales and aes, draw the graphs of the functions f: + 6 and g: 4 in the domain 4, R. Use our graphs to estimate: (i) The values of for which f() = 0 (ii) The values of for which f() = g() (iii) The range of values of for which f() g()
9. Using the same scales and aes, draw the graphs of the functions f: + and g: 3 in the domain 4, R. Estimate from our graphs: (i) The values of f(3.5) and g(3.5) (ii) The values of for which = ( + ) (iii) The value of 5, eplaining how ou found this answer 0. Using the same scales and aes, draw the graphs of the functions f: 4 + 7 3 and g: + 5 in the domain 3, R. Using the graphs, estimate: (i) The value of for which g() = 0 (ii) The values of for which f() = 0 (iii) The values of for which f() = g() (iv) The range of values of for which g() > f(). Draw the graph of the function f: 5 3 in the domain 5, R. Using our graph: (ii) Draw the ais of smmetr of the graph and write down its equation in the form = k. (iii) Find the values of f(k + ) and f(k ). (iv) B drawing the graph of the line =, find the values of for which f() =.. The function f: 8 + is defined in the domain 3 4, R. (i) Cop and complete the table. 3 0 3 4 f() 4 6 4 (ii) Draw the graph = f(). (iii) Estimate the values of for which 4 = (3 + )(4 ), using our graph. (iv) Write down the equation of the ais of smmetr of the graph. (v) Find a positive value of for which f() = f( 0.5). (vi) Use the same scales and aes to draw the graph of g: + and hence estimate 7. (i) Estimate the maimum value of f()..6 Real-life problems solved b graphs Some of life s problems can be solved b drawing a graph. Here is an eample of such a problem. Worked Eample.5 A famil has a small garden. There is a straight wall along one side. Their daughter, Theresa, wants to make a flower-bed. Her parents sa that she can make a rectangular flower-bed using the wall as one side if the length of the other three sides is eactl 6 metres in total. Show that if is the width of the flower-bed, then the area (A) is given b the epression A = 6. Draw the graph of A = 6 for 0 3, R, and hence find the maimum possible area for the flower-bed. Wall Old Sllabus Functions and Graphs 3
Solution If = the width, then (6 ) = the length. A = length width 6 = (6 ) = 6 QED 0 3 6 0 6 8 0 8 8 A 0 4 4 0 6 4.5 4 A Maimum point Points: (0,0)(,4)(,4)(3,0) The maimum area is 4.5 m when =.5 m. 0.5 3 Eercise.5 Old Sllabus Functions and Graphs 4. Graph the function f: 5 3 in the domain 0 5, R. f() represents the height (in metres) of a football kicked from level ground, while represents the time (in seconds) after it is kicked until it hits the ground again. Use the graph to estimate: (i) The maimum height reached (ii) The height of the football after.3 seconds (iii) The time which the football spends in the air before it lands (iv) The number of seconds the football is more than 9 metres off the ground. The depth of water (d) in a harbour is given b the formula d() = + 5 + 7 where represents the time (in hours). = 0 is noon, = is p.m., = is p.m. etc. Draw the graph of d() in the domain 4, R. Use our graph to estimate: (i) The minimum depth in the harbour and the time it occurs (ii) The times when the water is 0 metres in depth (iii) The depth at.45 p.m. (iv) The length of time when the depth is less than metres 3. Using the same aes and the same scales, graph the two functions ( R): f: 5 in the domain 3.5 g: 6 in the domain 0 3 f() represents the height (in kilometres) of a foreign rocket which is launched at 4.30 p.m. ( = 3). g() is the height (in kilometres) of a missile which is launched at 5.00 p.m. ( = 0) to intercept the foreign rocket. Use the graphs to estimate: (i) The maimum height reached b the foreign rocket (ii) The time at which the missile intercepts the rocket (iii) The height at which the collision occurs 4. The perimeter of a rectangular room is metres. If = the length, show that the area (A) is given b A() = 6.
B drawing a graph of 6 in the domain 0 6, R, estimate: (i) The maimum area possible (ii) The length of the room if the area is 8.5 square metres 5. A straight wall runs through a farm. A farmer has 0 metres of fencing to make a rectangular pig-pen, using the wall as one of the sides. If = the width of the pen (in metres), show that the area (A) of the pen is given b the function A() = 0. (iii) The time taken for the stone to reach the sea (iv) The range of t for which 5 < h < 40 7. The sum of two numbers is 0. Let = the first number. (i) Write down an epression for the other number. (ii) Find an epression for the sum (S) of the squares of the two numbers in the form S() = a + b + c. (iii) Complete the table and hence draw the graph of = S() in the domain 0 0, R. 0 4 6 8 0 S() 00 68 (i) Cop and complete this table and draw the graph of = A() in the domain 0 0, R. 0 3 4 5 6 7 8 9 0 A() 0 8 3 3 8 0 (ii) Use our graph to estimate the width if the area is 5 m. (iii) What is the maimum area for the pen and what are its dimensions? 6. A bo throws a stone from the top of a cliff out to sea. The height h (in metres) of the stone above sea level is given b h(t) = 8 + 7t 5t, where t is the time (in seconds) after it is thrown. Draw the graph of h in the domain 0 t 6, t R. From the graph, calculate: (i) The height of the cliff (ii) The maimum height and the time when it is reached (iv) From our graph find the minimum value for the sum of the squares of the two numbers. (v) Estimate the values of for which S() = 80. 8. A woman bus apples for 0 cents each. She then sells them in the street. If she sells them for cents each, she can sell eight ever hour. For each cent b which she reduces the price, she sells two more apples per hour. (i) If the selling price is 0 cents, calculate the profit which the woman makes per hour. (ii) If she reduces the selling price b cents, show that the profit (p) per hour is given b the epression: p() = 96 + 6 (iii) Complete the table and hence draw a graph of = p() in the domain 0, R. 0 4 6 8 0 p() 0 96 0 (iv) Find the maimum profit per hour and the selling price which ields this profit. Old Sllabus Functions and Graphs 5
Revision Eercises. (a) f: 4 is a function. The domain of f is {,, 3, 4}. Find the range of f. (b) f: 6 is a function. (i) Evaluate f(3) and f(6). (ii) Find the value of k if f(6) = kf(3). (iii) Find two values of for which f() =. (c) The diagram shows part of the graph of the function = a b where a, b N. = f() 3. (a) A = {, 0, } f: A A: is a function. 0 f Cop and complete the graph of f. List the elements of: (i) The domain of f (ii) The codomain of f (iii) The range of f (b) f: R R: a + b is a function. 0 (,0) (7,0) If f() = 4 and f(4) = 7, find the values of a and b. Old Sllabus Functions and Graphs 6 (i) Find the values of a and b. (ii) If (, n) is also a point on the graph, find the value of n.. (a) f: N R: 5 + is a function. (i) Find the value of f(7). (ii) If f(n) = 9, find the value of n. (b) f: 3 is a function defined on R (real numbers). (i) Evaluate f() and f(). (ii) Investigate if f() + f() = f(3). (iii) If f(4) + f( 6) = f(n), find the two possible values of n. (c) f: is defined for all R \{}. (i) Calculate f(4) and f ( 4 ) as fractions. (ii) If f(k) =, find the value of k R. 3 (iii) Show that f() f ( ( )( + ) ) = ( )( ). (c) The diagram shows part of the graph of = p 3 + q. (,) (, 3) (3,n) (i) Find the value of p and q. (iii) Evaluate n. 4. Using the same scales and the same aes, draw the graphs of the two functions: f: 5 5 and g: in the domain 6, R. Use our graph to estimate: (i) The maimum value of f() and the values of at which it occurs (ii) The values of for which f() = 0 (iii) The values of for which f() = g()
5. Using the same scales and the same aes, draw the graphs of the two functions f: 8 + and g: + in the domain 3 5, R. Use our graph to estimate: (i) The range of values for for which 8 + > 0 (ii) The solutions of the equation 5 + = 0 (iii) The value of 7 6. Draw the graph of the function f: 0 + 7, in the domain 0 5, R. represents the time; each unit represents five minutes. Use the graphs to find: (i) The maimum height of the missile (ii) The length of time for which the missile is more than 0 km above the ground (iii) The time of the collision and the height at which it occurs 9. (a) The diagram shows part of the graph of = 5. 0 (i) Show the ais of smmetr of the graph and write down its equation in the form = k. (ii) Find a value of (other than.3) for which f() = f(.3). (iii) Estimate the range of values of for which > 0 7. (iv) Estimate the range of values of for which < f() <. 7. (a) Find the solutions of 3 = 0 correct to one decimal place, using the quadratic formula b ± b 4ac. a (b) Draw the graphs of f: 3 and g: on the same scales and aes in the domain 4 3, R. (c) Use our graphs to find the values of for which: (i) f() > 0 (ii) g() = f() (iii) g() f() 8. On the same scales and aes draw the graphs of f: 8 in the domain 4 3.5, R, and g: 7 4 in the domain 0 4.5, R. f() represents the height (in km) of a missile which is launched at 8.00 a.m. ( = 4). g() represents the height of a shell which is launched to intercept the missile at 8.0 a.m. ( = 0). 0 4 3 0 3 Use the diagram to estimate the values of for which 5 > 0. (b) abc is a triangle which is right angled at b, such that ab = bc = 7. c 7 A rectangle is to be inset, as shown, in the triangle. If is the length of the rectangle, show that the area (A) is given b A() = 7. B drawing the graph of = A() in the domain 0 7, R, find: (i) The values of for which the area is 8 square units (ii) The maimum area and the value of which gives rise to it a 7 b Old Sllabus Functions and Graphs 7
Summar of important points. A linear graph is the graph of an function of the form = a + b. The graph will be a straight line. = a + b. A quadratic graph is the graph of an function of the form = a + b + c, a 0. (i) If a is positive, the graph looks like this: a b a + b + c > 0 for < a or > b a + b + c < 0 for a < < b (ii) If a is negative, the graph looks like this: a b Old Sllabus Functions and Graphs a + b + c > 0 for a < < b a + b + c < 0 for < a or > b 8
Answers Eercise.. {3, 7,, 5}. {, 3, 9} 3. (i) {, } (ii) {,, 3} (iii) {, 3} 4. (b) (i) {,, 3} (ii) {,, 3, 4, 5} (iii) {, 3, 5} 5. {, } 6. (i) {(,0), (0,), (,0)} (ii) {, 0, } (iii) {, 0, } (iv) {0, } 8. (i) 7, 7 (ii) 4, 4 9. {0, _,, } 0. (i) (ii) No; 0 + ( 5) (iii) ±7 (iv), 4. (i) _ 5, 3_ 7 (ii) _ 3. (i) _ 3, _ 3 (ii) n = 3_ (iii) k = 7 3. (i) 3 (ii) (iii) 0, (iv) 4, 4. a = 7, b = 5. (i) (ii) _ 6. 3 (when = 0) 7. (i) Natural numbers (ii) Even natural numbers 8. (when = 0) 9. k = 6 0. 7_ 3. a = 3, b = 0. a =, b = 3 3. a = 8, b = 6; 4.6 4. (i) p = 7, q = 3 (ii) = 40, z = 5 5. (i) a = 3, c =, m = 5, n = 9 (ii) An number ecept, 0, 5, etc. (the range) 6. (i) 4 (ii) 4, 3 7. (i) 8 (ii) 9 (iv) 3 8. (i) 0 (ii) 6 9. (i) No (ii) 5n + 3 (iii) 3 (iv) An number ecept 3, 8, 3, 8,..., 30 30. (i) 5 (ii) 3 Eercise.. a = 5, b =. p =, q = 6 3. a = 7, b = 4. (i) p = 3, q = 0 (ii) 0 5. (i) p =, q = (ii) = 4 6. (i) a =, b = (ii) 5 7. (i) a =, b = 3 (ii) k = 5 (iii).5, 4 8. (i) 0, 7 (ii) 9. (i) a =, b = 3 (ii) 5_ 3 0. (i) a = 5, b = 6 (iii), Eercise.3. (i) 3 (ii).7. (i) 4.6 (ii).5 3. ( _, ) 4. (i) 8 m/s (ii).7 s (iii) 5 s 5. (ii) 56 mins (iii).75 kg 6. (i) 55 (ii) 4 _ 7. (iii) 5 C (iv) 59 F Eercise.4. (i) 4.6 (ii).4, 3.4 (iii).4 < < 3.4 (iv) 6. (i).3,.8 (ii) 0.5, (iii).3.8 (iv) 8. at = 0.75 3. (i).6,.6 (ii).6 < <.6 (iii) 0.6,.6 (iv) 4.5 4. (i),.5 (ii).7,. (iii).5 5. (i) 0.7,.7 (ii) =.8,.8 (iii) 0 < < (iv) 4.8 at = 0.5 6. (i) 7.5 (ii) =., 0.6 (iii).3 < < 0.7 (iv).6 7. (i).7,. (ii).7 < <. (iii) 4. (iv) =.4, 0.9 8. (i) 3.7,.7 (ii), (iii) 9. (i) 3.3, 4 (ii) 0.7,.7 (iii). 0. (i).5 (ii)., 0.3 (iii)., 0.9 (iv). < < 0.9. (i) 7.5 (ii) =.5 (iii) 3.5 and 3.5 (iv) 5,. (iii) =.3, 3.3 (iv) = _ (v).5 (vi) =.65 Eercise.5. (i) 8.75 (ii) 4.4 (iii) 5 seconds (iv) 3.6 seconds. (i) 3.75 metres at 0.45 a.m. (ii) 9 a.m. and.5 p.m. (iii) 0 metres (iv) 4 hours 3. (i) 5 km (ii) 5.7 p.m. (iii) 7.5 km 4. (i) 9 m (ii).3 or 3.7 5. (i).5 or 8.5 (ii) 50 m ; 5 m 0 m 6. (i) 8 m (ii) 54.5 at t =.7 (iii) 6 seconds (iv) 0.3 < t < ; 4.5 < t < 5. 7. (i) 0 (ii) 0 + 00 (iv) 50 (v). and 8.9 8. (i) 0 cent (ii) 96 + 6 (iv).8 at a price of 8 cent ( = 4 cent reduction) Revision Eercises. (a) { 3, 4, 0} (b) (i) 3, 30 (ii) 0 (iii) 3, (c) (i) a = 5, b = 4 (ii) n = 8. (a) (i) 6 (ii) 6 (b) (i), (ii) No; ( ) + 6 (iii) n = ±7 (c) (i) _, 4_ 7 (ii) _ 3. (a) (i) {, 0, } (ii) {, 0, } (iii) {, } (b) a =, b = 5 (c) (i) p =, q = 5 (ii) n = 49 4. (i).5 at = 5 (ii) 5.85, 0.85 (iii) 5, 5. (i) < < 4 (ii).4, 3.4 (iii).6 6. (i) =.5 (ii) 3.7 (iii) < 0.8, > 4. (iv) 0.5 < < or 4 < < 4.5 7. (a).3,.3 (c) (i).3 < <.3 (ii) 4, (iii) 4 8. (i) 9 km (ii) 0 mins (iii) 8.30 a.m. at 8 km 9. (a) 3 < <.5 (b) (i).4, 5.6 (ii).5 at = 3.5 9