200 9 Sequences and functions In this section you will… • use a term-to-term rule to make a sequence of numbers. 9.1 Generating sequences This is a sequence of numbers. 1 2 1 2 4 5 1 2 7 … … Each term is 11 2 more than the term before, so the term-to-term rule is ‘add 11 2 ’. You can generate a sequence when you are given the first term and the term-to-term rule. For example, when the first term is 3 and the term-to-term rule is ‘multiply by 2 and add 5’ you get the sequence 3, 11, 27, 59, … Key word generate Tip 2nd term is 3×2+5=11 3rd term is 11×2+5=27 4th term is 27×2+5=59 Worked example 9.1 a Write the term-to-term rule and the next two terms of this sequence. 6, 8 1 4 , 10 1 2 , 12 3 4 , … b The first term of a sequence is 4. The term-to-term rule of the sequence is: multiply by 3 and then add 2. Write the first three terms of the sequence. Answer a Term-to-term rule is: add 2 1 4 You can see that the terms go up by 2 1 4 every time as 6 2 8 1 4 1 4 + = , 8 2 10 1 4 1 4 1 2 + = , etc. Next two terms are 15 and 17 1 4 You keep adding 2 1 4 to find the next two terms: 12 2 15 3 4 1 4 + = and 15 2 17 1 4 1 4 + = b First three terms are 4, 14, 44 Write the first term, which is 4. Then use the term-to-term rule to work out the second and third terms: second term=3×4+2=14 third term=3×14+2=44 We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
201 9.1 Generating sequences Exercise 9.1 1 Complete the workings to find the term-to-term rule and the next two terms of each sequence. a 7, 10 1 2 , 14, 17 1 2 , … 7+3 1 2 =10 1 2, 10 1 2 +...=14, 14+...=17 1 2 The term-to-term rule is: add The next two terms are: 17 1 2 + = + = b 10, 9.8, 9.6, 9.4, … 10−0.2=9.8, 9.8− =9.6, 9.6− =9.4 The term-to-term rule is: subtract The next two terms are: 9.4− = − = Tip For Question 3, part a, work out 1+1.4=2.4, then 2.4+1.4=…, then …+1.4=… 3 Write the first three terms of each sequence. Show your working. First term Term-to-term rule First term Term-to-term rule a 1 Add 1.4 b 6 Add 4 1 2 c 20 Subtract 2.5 d 40 Subtract 5 1 3 e 0.4 Multiply by 2 f 9 Divide by 2 2 For each of these sequences, write i the term-to-term rule ii the next two terms. a 5, 5 1 4 , 5 1 2 , 5 3 4 , … b 7 1 3 , 8 2 3 , 10, 111 3 , … c 5.4, 5.8, 6.2, 6.6, … d 9, 8 1 2 , 8, 7 1 2 , … e 10, 9 3 5 , 9 1 5 , 8 4 5 , … f 17, 16.2, 15.4, 14.6, … We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
202 9 Sequences and functions 4 Copy these sequences and fill in the missing terms. a 2, 4 1 5 , , 83 5 , ,13, 15 1 5 b 5, 83 7 , 116 7 , , , 22 1 7 , c 25, 24 3 4 , , , 24, , 23 1 2 d 100, 89 1 2 , , 68 1 2 , , 47 1 2 , e 8, , , 8.9, 9.2, , f , , 24, 23.6, , , 22.4 Think like a mathematician 5 How can you answer these questions without working out more of the terms in the sequences? a In the sequence 0.4, 0.8, 1.2, 1.6, 2, 2.4, …, what is the first term greater than 10? b Is 45 a term in the sequence 5, 7 1 2, 10, 12 1 2 , 15, ...? c Is 5 1 3 a term in the sequence 30, 26 2 3 , 23 1 3 , 20, ...? Discuss your answers. 6 Write the first three terms of each of these sequences. The first one has been started for you. a first term is 8 term-to-term rule is: multiply by 2 then subtract 5 first term=8 second term=8 × 2−5=16−5=11 third term=11 × 2−5= −5= b first term is 15, term-to-term rule is: subtract 9 then multiply by 3 c first term is 12, term-to-term rule is: divide by 2 then add 5 7 The first three terms of a sequence are 8, 10, 14, … a Which of these cards, A, B or C, shows the correct term-to-term rule? A multiply by 3 then subtract 14 B divide by 2 then add 6 C subtract 3 then multiply by 2 b Which is the first term in this sequence greater than 50? We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
203 9.1 Generating sequences 8 Arun works out the terms in this sequence: First term is 10, term-to-term rule is subtract 6 then multiply by 2. Read what Arun says. Is Arun correct? Show your working. 9 Work out the first three terms in each sequence. a first term is 4, term-to-term rule is: multiply by 3 then subtract 10 b first term is 10, term-to-term rule is: subtract 2 1 2 then multiply by 2 c first term is −6, term-to-term rule is: divide by 2 then add 5 10 Zara describes a sequence. a Work out the first three terms of the sequence. What do you notice? b Describe two different sequences that are like Zara’s. Compare your answers with a partner’s. The first term in this sequence which is a negative number is −4. My sequence has a first term of 5, and the term-to-term rule is ‘Multiply by 3 then subtract 10’ Think like a mathematician 11 Sofia works out the terms in this sequence: First term is 8, term-to-term rule is add 10 then divide by 2. Read what Sofia says. Is Sofia correct? Discuss your answers. I will never have a term in my sequence which is greater than 10. 12 This is part of Tania’s homework. Question The 10th term of a sequence is 50 2 5 . The term-to-term rule is add 4 3 5 What is the 20th term of the sequence? Answer 20th term=2 × 10th term=2 × 50 2 5 =100 4 5 a Explain why Tania’s method is wrong. b Work out the correct answer. Show all your working. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
204 9 Sequences and functions 13 The 7th term of a sequence is 442. The term-to-term rule is add 3 then multiply by 2. What is the 4th term of the sequence? Show all your working. Activity 9.1 On a piece of paper, write three questions similar to those in Question 2, and three questions similar to those in Question 6. Write the answers on a separate piece of paper. Make sure the questions can be answered without using a calculator. Exchange questions with a partner. Work out the answers to your partner’s questions. Exchange back and mark each other’s work. If you think your partner has made a mistake, discuss with them where they have gone wrong. Summary checklist I can find the term-to-term rule for a number sequence. I can use the term-to-term rule for a number sequence. Look back at your answers to questions 12 and 13. Write a short explanation of the method you used to solve these problems. Discuss your method with a partner. Did they use the same method? Can you think of a better method? We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
205 9.2 Finding rules for sequences In this section you will … • make a sequence of numbers from patterns. 9.2 Finding rules for sequences This sequence of patterns is made from dots. Pattern 1 5 dots 7 dots 9 dots Pattern 2 Pattern 3 The numbers of dots used to make the patterns form the sequence 5, 7, 9, …, … As you go from one pattern to the next, two more dots are added each time. The term-to-term rule is ‘add 2’. You can use the term-to-term rule to work out the position-to-term rule. The term-to-term rule for this sequence is ‘add 2’, so start by listing the first three multiples of 2 and comparing them with the patterns of dots. Position 1 Position 2 Position 3 Multiples of 2: 1×2=2 2×2=4 3×2=6 Number of dots: 2+3=5 4+3=7 6+3=9 The pattern is formed by adding multiples of 2, shown as red dots, to the three blue dots at the start of each pattern. Pattern 1 5 dots 7 dots 9 dots Pattern 2 Pattern 3 The position-to-term rule for this sequence is: term=2×position number+3 Draw the next pattern, to check. Pattern 4: term=2×4+3=11 ✓ Key words position number position-to-term rule sequence of patterns 11 dots Pattern 4 We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
206 9 Sequences and functions Worked example 9.2 This pattern is made from blue squares. a Write the sequence of the numbers of squares. b Write the term-to-term rule. c Draw the next pattern in the sequence. d Explain how the sequence is formed. e Work out the position-to-term rule. Answer a 4, 7, 10, … There are 4 squares in the first pattern, 7 in the second and 10 in the third. b add 3 The term-to-term rule is ‘add 3’. c Pattern 4 Pattern 4 will have 10+3=13 squares. d The pattern is formed by adding multiples of 3, shown as red squares, to the one blue square at the start of each pattern. e Position number 1 2 3 4 Term 4 7 10 13 3×position number 3 6 9 12 3×position number+1 4 7 10 13 Position-to-term rule is: term=3×position number+1 Draw a table showing the first four position numbers and terms. The term-to-term rule is ‘add 3’, so add a row to the table which shows 3×position number. (3×1=3, 3×2=6, 3×3=9, 3×4=12) You can see that each number in this row is 1 less than the equivalent number in the sequence. So if you add 1, you will get the terms of the sequence. (3+1=4, 6+1=7, 9+1=10, 12+1=13) We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
207 9.2 Finding rules for sequences Exercise 9.2 1 This pattern is made from squares. a Write the sequence of the numbers of squares. b Write the term-to-term rule. c Draw the next pattern in the sequence. d Explain how the sequence is formed. e Copy and complete the table to find the position-to-term rule. Position number 1 2 3 4 Term 3 5 2×position number 2 4 2×position number+ The position-to-term rule is: term=2×position number+ 2 This pattern is made from dots. a Write the sequence of the numbers of dots. b Write the term-to-term rule. c Draw the next pattern in the sequence. d Explain how the sequence is formed. e Copy and complete the table to find the position-to-term rule. Position number 1 2 3 4 Term 6 10 ×position number ×position number+ The position-to-term rule is: term= ×position number+ We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
208 9 Sequences and functions 3 This pattern is made from rectangles. a Write the sequence of the numbers of rectangles. b Write the term-to-term rule. c Draw the next pattern in the sequence. d Copy and complete the table to find the position-to-term rule. Position number 1 2 3 4 Term 3 8 ×position number ×position number− The position-to-term rule is: term= ×position number− Think like a mathematician 4 This pattern is made from squares. Razi thinks that the position-to-term rule for the sequence of the numbers of green squares is: term=2×position number+3 Is Razi correct? Explain the method you used to work out your answer. Discuss the method you used with other learners. Did you use the same method or a different method? What do you think is the best method to use? We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
209 9.2 Finding rules for sequences 5 This is part of Harsha’s homework. Question Work out the position-to-term rule for this sequence of triangles. Answer The sequence starts with 4 and increases by 2 every time, so the position-to-term rule is: term=4 × position number+2 a Explain the mistake Harsha has made. b Work out the correct answer. Activity 9.2 a Design your own sequence of patterns made from a shape of your choice. Draw the first four patterns in your sequence. Draw a table to show the number of shapes in each of your patterns. Work out the position-to-term rule for your sequence. b Ask a partner to check that your work is correct. 6 Work out the position-to-term rule for each sequence. a 10, 15, 20, 25, … b 10, 30, 50, 70, … Tip Draw a table like the ones in questions 1 to 3 to help you. Think like a mathematician 7 This pattern is made from hexagons. How many hexagons will there be in Pattern 20? Show how you worked out your answer. Discuss the method you used with other learners. Did you use the same method or a different method? What do you think is the best method to use? We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
210 9 Sequences and functions 8 Mia is using trapezia to draw a sequence of patterns. There are marks over the first and third patterns in her sequence. Pattern 1 Pattern 2 Pattern 3 Pattern 4 How many trapezia will there be in Pattern 18? Show how you worked out your answer. Tip 2×n is usually written as 2n. Summary checklist I can find and use the term-to-term rules for number sequences drawn as patterns. I can find and use the position-to-term rules for number sequences drawn as patterns. You already know how to work out the position-to-term rule of a linear sequence. Example: The sequence 5, 7, 9, 11, …, … has position-to-term rule; term=2×position number+3 You can also write the position-to-term rule as an nth term expression. To do this, you replace the words ‘position number’ with the letter n. So, in the example above, instead of writing: term=2×position number+3 you would write: nth term=2×n+3 or, more simply: nth term=2n+3 9.3 Using the nth term In this section you will … • use algebra to describe the nth term of a sequence. Key word nth term We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
211 9.3 Using the nth term Worked example 9.3 a The nth term expression of a sequence is 2n−1 Work out the first three terms and the tenth term of the sequence. b Work out the nth term expression for the sequence 7, 10, 13, 16, … Answer a 1st term=2×1−1=1 To find the first term, substitute n=1 into the expression. 2nd term=2×2−1=3 To find the second term, substitute n=2 into the expression. 3rd term=2×3−1=5 To find the third term, substitute n=3 into the expression. 10th term=2×10−1=19 To find the tenth term, substitute n=10 into the expression. b Position number (n) 1 2 3 4 Term 7 10 13 16 3×n 3 6 9 12 3×n+4 7 10 13 16 nth term=3n+4 Draw a table showing the position numbers and terms. The term-to-term rule is ‘add 3’, so add a row to the table which shows 3×n. (3×1=3, 3×2=6, 3×3=9, 3×4=12) You can see that if you work out 3×n+4, you will get the terms of the sequence. (3+4=7, 6+4=10, 9+4=13, 12+4=16) Exercise 9.3 1 Copy and complete the workings to find the first four terms of each sequence. a nth term=2n+1 1st term=2×1+1=3 2nd term=2×2+1= 3rd term=2×3+1= 4th term=2×4+1= b nth term=3n−2 1st term=3×1−2=1 2nd term=3×2−2= 3rd term=3×3−2= 4th term=3×4−2= 2 Work out the first three terms and the 10th term of the sequences with the given nth term. a n+6 b n−3 c 9n d 6n e 2n+5 f 3n−1 g 5n+3 h 4n−3 We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
212 9 Sequences and functions 3 Match each yellow sequence card with the correct blue nth term expression card. A 8, 9, 10, 11, … B 4, 8, 12, 16, … C −3, −2, −1, 0, … D 7, 14, 21, 28, … E 6, 8, 10, 12, … F 2, 6, 10, 14, … 4 The cards show one term from two different sequences. A 12th term in the sequence nth term is 8n−4 B 7th term in the sequence. nth term is 11n+16 Which card has the greater value, A or B? Show your working. i n−4 ii 2n+4 iii 4n−2 iv 4n v 7n vi n+7 5 Show that the first four terms of the sequence with nth term 1 4 n + 8 are 8 1 4 , 8 1 2 , 8 3 4 and 9. 6 Work out the first three terms and the 8th term of the sequences with the given nth term. a 1 2 n + 6 b 5 2 1 2 n − c 0.2n+1.5 d 4.5n−0.25 Think like a mathematician 7 a Work out the first four terms of the sequences with the given nth term. i 4n+12 ii 1 4 n+1 iii 12−4n iv 1−1 4 n b Discuss with a partner the answers to these questions. What is similar about the sequences in ai and aii? What is similar about the sequences in aiii and aiv? What is different about the sequences in ai and aiii? What is different about the sequences in aii and aiv? We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
213 9.3 Using the nth term 8 Look at this number sequence. 24, 18, 12, 6, 0, ... Simply by looking at the numbers in the sequence, explain why you can tell that the nth term expression for this sequence cannot be 2n+22. 9 Ian and Lin use different methods to work out the answer to this question. The nth term expression for a sequence is 4n+3. Is the number 51 a term in this sequence? Ian’s method Lin’s method Work out the terms in the sequence: n=1, 4n+3=7 n=2, 4n+3=11 n=3, 4n+3=15 Term-to-term rule is add 4, so sequence is 7, 11, 15, 19, 23, 27, 31, 35, 39, 43, 47, 51, … Yes, 51 is in the sequence. Make an equation and solve it to find n: 4n+3=51 4n=51−3 4n=48 n= 48 4 =12 Yes, 51 is the 12th term in the sequence. Continued c The cards show the nth terms of some sequences. Sort the cards into two groups. Give a reason for your choice of groups. 7 8 1 8 − n 15 2 3 − n 3n+7 1 2n +12 13−n 9−5n 1 4n −19 Discuss your choice of groups with other members of the class. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
214 9 Sequences and functions a Use Ian’s method and Lin’s method to work out the answer to this question. The nth term expression for a sequence is 3n+5. Is the number 48 a term in this sequence? b Write the advantages and disadvantages of Ian’s method and Lin’s method. c Which method do you prefer? Explain why. d Can you think of a better method? If you can, explain this method. e Use your preferred method to work out the answers to these questions. i The nth term expression for a sequence is 2n−3. Is the number 39 a term in this sequence? ii The nth term expression for a sequence is 6n+7. Is the number 60 a term in this sequence? 10 Copy and complete the workings to find the nth term expression for the sequence 8, 10, 12, 14, … Position number (n) 1 2 3 4 Term 8 10 12 14 2×n 2 4 2×n+ 8 10 12 14 nth term=2n+ 11 Work out an expression for the nth term for each sequence. Draw a table like the one in Question 10 to help you. a 6, 8, 10, 12, … b 5, 8, 11, 14, … c 6, 11, 16, 21, … d 3, 7, 11, 15, … e 2, 10, 18, 26, … f 2, 9, 16, 23, … 12 This pattern is made from rectangles. Pattern 1 Pattern 2 Pattern 3 Pattern 4 a Write the sequence of the numbers of rectangles. b Work out an expression for the nth term for the sequence. Draw a table like the one in Question 10 to help you. c Use your nth term expression to find the number of rectangles in the 20th pattern in the sequence. Tip What do you need to add to 2 to get 8? What do you need to add to 4 to get 10? etc. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
215 9.3 Using the nth term 13 Sofia and Marcus are looking at the number sequence: 4, 4 1 2 , 5, 5 1 2 , 6, ... Read what they say. I think the expression for the nth term of this sequence is 1 2 n+4 I think the expression for the nth term of this sequence is 4n+ 1 2 Is either of them correct? Explain your answer. 14 Work out an expression for the nth term for each sequence. a 9 9 9 10 1 4 1 2 3 4 , , , , ... b 4.6, 5.2, 5.8, 6.4, … c − − 1 1 − 0 1 2 1 2 , , , , ... d −0.6, 0.8, 2.2, 3.6, … Think like a mathematician 15 Zara and Arun are trying to work out the expression for the nth term of the sequence 8, 6, 4, 2, … They both start by drawing this table: Position number (n) 1 2 3 4 Term 8 6 4 2 …×n Read what they say next. As the term-to-term rule of the sequence is subtract 2, I think the next line in the table is 2×n As the term-to-term rule of the sequence is subtract 2, I think the next line in the table is −2×n a What do you think? Explain your answer. b Copy and complete the table. Use it to work out the expression for the nth term of the sequence. c Compare your answers with those of other learners in the class. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
216 9 Sequences and functions How well do you think you understand the nth term expressions? Give yourself a score from 1: Still need lots of practice, to 5: Feeling very confident. Summary checklist I can use the nth term expression for a number sequence. I can work out the nth term expression for a number sequence. 9.4 Representing simple functions In this section you will … • work out input and output numbers from function machines. Key words algebraically function function machine input inverse function map mapping diagram one-step function output two-step function Tip The numbers that go into the function machine are called the input. The numbers that come out of the function machine are called the output. 16 Work out an expression for the nth term for each sequence. a 18, 15, 12, 9, … b 11, 7, 3, −1, … c 7, 2, −3, −8, … A function is a relationship between two sets of numbers. You can draw a one-step function as a function machine, like this. 1 input output 6 2 1 2 + 5 4 1 4 7 1 2 9 1 4 We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
217 9.4 Representing simple functions You can also draw a function as a mapping diagram, like this. 0 1 2 3 4 5 6 7 8 9 10 0output 2 3 4 5 6 7 8 9 10 input 1 The input numbers map to the output numbers. You can also write a function algebraically as an equation. Use the letter x to represent the input numbers. Use the letter y to represent the output numbers. You can then show the previous function machine like this: x + 5 y You can write the input (x) and output (y) numbers in a table. x 1 2 1 2 4 1 4 y 6 7 1 2 9 1 4 You can also write the function as an equation like this: x+5=y but it is more common to write the equation like this: y=x+5 Tip You usually write a function equation starting with y=… Worked example 9.4 a Copy and complete the table of values for this two-step function machine. x × 2 + 1 y x 0 2 3 41 2 y b Draw a mapping diagram to show the function in part a. c Write the function in part a as an equation. Answer a x 0 2 3 4 1 2 y 1 5 7 10 To work out the y-values, multiply the x-values by 2 then add 1. 0×2+1=1, 2×2+1=5, 3×2+1=7, 4 1 2 ×2+1=10 We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
218 9 Sequences and functions Exercise 9.4 1 a Copy and complete the table of values for each one-step function machine. i x + 3 y ii x – 3 y x 1 2 1 2 4 5 1 2 y b Draw a mapping diagram for each function in part a. c Write each function in part a as an equation. 2 a Copy and complete the table of values for each two-step function machine. i x × 2 + 3 y ii x ÷ 2 – 3 y x 0 11 2 3 4 1 2 y b Write each function in part a as an equation. Continued b Draw a line connecting each x-value to its y-value. Draw an arrow on each line to show that 0 maps to 1, 2 maps to 5, 3 maps to 7 and 4 1 2 maps to 10. c y=2x+1 Write the equation with ‘y =’ on the left. Remember, you can write x×2+1 more simply as 2x+1. x 7 7 1 2 8 8 1 2 y x 8 10 15 19 y x 0 2 3 4 5 6 7 8 9 10 y 1 0 1 2 3 4 5 6 7 8 9 10 input output We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
219 9.4 Representing simple functions 4 Work out the missing values in the tables for these function machines. a i x × 3 – 1 y ii x ÷ 2 + 5 y x 5 1 2 12 1 2 y 8 26 b Write each function in part a as an equation. Think like a mathematician 3 Work out the missing values in the tables for these function machines. a i x × 2 y ii x + 4 ÷ 2 y x y 6 9 12 15 b Compare your answers with those of other learners in the class. Discuss the different methods you used. What do you think is the best method? x y 3 5 8 1 2 111 2 x 4 10 y 8 1 2 111 2 Activity 9.4 a On a piece of paper, draw two function machines of your own, similar to those in Question 4. Draw a table for each function machine and give two x-values and two y-values. On a different piece of paper, write the missing x-values and y-values. Exchange function machines with a partner and work out their missing x-values and y-values. b Exchange back and mark each other’s work. Discuss any mistakes. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
220 9 Sequences and functions Think like a mathematician 5 Hannah works out the answer to this question. Work out the missing values in the table for this function machine. x × 2 + 7 y This is what she writes. The function machine is: x × 2 + 7 y so the equation is y=2x+7 If you reverse the function machine, you get: x ÷ 2 – 7 y So the equation for the inverse function is y− 7 2 =x or x=y− 7 2 Use the inverse function to work out the missing values. When y=15, x= 15− 7 2 = 8 2 =4 When y=21, x= 21− 7 2 = 14 2 =7 When y=29, x= 29− 7 2 =22 2 =11 Answer is: x 4 7 11 y 15 21 29 a What do you think of Hannah’s method? What are the advantages and disadvantages of her method? Discuss your answers. b Use Hannah’s method to answer this question. Work out the missing values in the table for this function machine. x – 4 ÷ 3 y Compare your equations and answers with those of other learners in the class. x y 2 5 8 x y 15 21 29 We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
221 9.4 Representing simple functions 6 Copy and complete these inverse function machines and equations. a x + 2 y x – 2 y equation: y = x + 2 reverse equation: x = … b x ÷ 4 y x … y equation: y = … reverse equation: x = … c x – 3 y x … y × 8 … equation: y = 8(x – 3) reverse equation: x = … 7 Match each function equation with its inverse function equation. The first one is done for you: A and v You can draw function machines to help you if you want to. Tip In part c, remember you write (x−3)×8 as 8(x−3). A x × 6 y y = 6x v x ÷ 6 y x = y 6 i x=7y ii x=5y−2 iii x=7(y+3) iv x=y−8 v x= y 6 vi x= y − 4 2 A y=6x B y= x 7 C y=x+8 D y=2x+4 E y= x 7 − 3 F y= x + 2 5 We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
222 9 Sequences and functions 8 a Copy and complete the function machine for each table of values. i x y ii x y x −3 0 1.5 6.2 y 6 9 10.5 15.2 b Write each function in part a as an equation. 9 Sofia and Arun are looking at this function machine and table of values. x y I think the equation for this function is y=3x+1. I think the equation for this function is y=4x−3. Is either of them correct? Show all your working. 10 Work out the equation for this function machine and table of values. x y Explain how you worked out your answer. 11 Marcus is putting numbers into a two-step function machine. Read what Marcus says. Work out the equation for Marcus’s function. Show all your working. x 4 5.5 7 y 13 19 25 When x=4, y=11. As my x-values increase by 4, my y-values increase by 2. x −5 −1.5 2.5 4.25 y −15 −4.5 7.5 12.75 x 1 2 3 y 5 8 11 Summary checklist I can work out output values of a function machine. I can work out input values of a function machine. I can write a function as an equation. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
223 9 Sequences and functions Check your progress 1 For each of these sequences, write i the term-to-term rule ii the next two terms. a 2, 2 1 3 , 2 2 3 , 3, … b 6.7, 6.4, 6.1, 5.8, … 2 This pattern is made from dots. Pattern 1 Pattern 2 Pattern 3 a Write the sequence of the numbers of dots. b Write the term-to-term rule. c Draw the next pattern in the sequence. d Copy and complete the table to find the position-to-term rule. Position number 1 2 3 4 Term 3 5 ×position number ×position number+ The position-to-term rule is: term= ×position number+ 3 Work out the first three terms and the 10th term of a sequence with the given nth term. a 1 2 1 2 n + 8 b 5n−0.75 4 Work out an expression for the nth term for each sequence. a 9, 11, 13, 15, … b 15, 12, 9, 6, … 5 Work out the missing values in the tables for these function machines. a i x ÷ 4 – 1 y ii x + 9 × 2 y x 8 10 y 4 6 1 2 b Write each function in part a as an equation. x −5 − 1 2 y 22 29 We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
Getting started 1 There are 279 girls in a group of 450 children. What percentage of the group are a girls b boys? 2 Estimate, then work out a 40% of 600 b 140% of 600 c 0.5% of 600 3 Explain the difference between 25% and 0.25% 4 Xavier earns $20 per hour. Sasha earns $25 per hour. They are both given a pay increase of $2 per hour. a Write the increase as a percentage of Xavier’s pay. b Write the increase as a percentage of Sasha’s pay. 5 Copy and complete this table. 100% 10% 60% 120% 350% $850 $85 4.50 m 5.40 m 10 Percentages Percentages are often used, instead of actual values, in articles in newspapers, in magazines, on the internet and on television. 224 We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
225 10.1 Percentage increases and decreases Look at these two sentences: • The population has increased from 3.25 million to 3.77 million • The population has increased from 3.25 million by 16% The two sentences give the same information but in different ways. The absolute change is 3.77−3.25=0.52 million. The percentage change is an increase of 16%. Percentages are easier to interpret than actual values if you want to: • describe one number as a fraction of another • describe an increase or decrease • compare two different increases or decreases. In this unit, you will learn how to calculate percentage changes. You will understand how useful percentages can be. Tip 30 75 ÷ = 0 4. In this section you will … • learn to calculate percentage increases and decreases • learn to write a change in value as a percentage. 10.1 Percentage increases and decreases The price of a train journey increases from $75 to $105 The price increase is $105−$75=$30 To find the percentage increase, you must write the increase as a percentage of the original price. That is 30 75 × =× = 100% . 0 4 100 40 % % Suppose the price decreases from $75 to $60. The decrease is $15. You can write this as a percentage of the original price in a similar way: 15 75 × = 100% . 0 2 × = 100 20 % % The percentage decrease is 20%. For an increase or a decrease, 75 is the denominator of the fraction. Key words absolute change percentage decrease percentage increase We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
226 10 Percentages Worked example 10.1 A library has 2800 books. Find the number of books if it a increases by 84% b decreases by 37% Answer a 84%=0.84 84% of 2800=0.84×2800=2352 There are 2352 more books so the total is 2800 + 2352=5152 b 37%=0.37 37% of 2800=0.37×2800=1036 There are 1036 fewer books so the total is 2800−1036=1764 Exercise 10.1 1 a Find 15% of $70 b Increase $70 by 15% c Decrease $70 by 15% 2 a Find 80% of 3200 people b Increase 3200 by 80% c Decrease 3200 by 80% 3 a Find 2% of 19.00kg b Increase 19.00kg by 2% c Decrease 19.00kg by 2% 4 How much will she have if she increases her savings by a 10% b 50% c 70% d 100% e 120%? 5 The population of a town is 45 000. The population is expected to rise by 85% in the next ten years. Estimate the population in ten years’ time. 6 Show that a 81 is 135% of 60 b 60.8 is 190% of 32 c 308 is 220% of 140 7 a What percentage of 950 is 380? b What percentage of 380 is 950? I have saved $240. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
227 10.1 Percentage increases and decreases 8 a What percentage of 40 years is 8 years? b What percentage of 8 years is 40 years? 9 A metal bar is 1.80metres long. It is heated and the length increases by 0.5%. a What is the absolute increase in length? b How long is the bar now? 10 Work out a 20% of 60km b 90% of 60km c 170% of 60km d 260% of 60km 11 Copy and complete this table. Amount 40% 140% 280% 420% $20 $8 $84 50kg 90m 126m 12 The mass of a child is 22kg. In the next 10 years, this mass increases by 150%. a Find 150% of 22kg. b Find the mass after 10 years. 13 A shop lists its prices in a table. a In a sale, all the prices are reduced by 30%. Calculate the sale prices. b How much would you save if you bought all three items in the sale? 14 Electricity costs are rising by 8%. The table shows the costs for one year for four customers. Copy the table and fill in the last column to show the costs for one year after the price rise. Customer Cost before the rise Absolute change ($) Cost after the rise A $415 B $629 C $1390 15 A garage is reducing the prices of cars. Calculate the new prices. Model Old price ($) Decrease (%) Absolute change ($) New price ($) Ace 15800 2.0 Beta 21300 12.0 Carro 24200 0.5 Item Price table $280 armchair $520 bed $1040 We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
228 10 Percentages 16 Mia sees this sign in a shop window: She says: ‘The original price of a coat was $120 so the price is now $84’ a Explain the calculation that Mia has done and why her statement is incorrect. b What is the price of the coat now? Prices reduced by 70% Think like a mathematician 17 There are 2000 people in a room. The number increases by P%. Then the number decreases by P%. a How many people are in the room if P=50? Show how you calculated your answer. b What happens if P=25? c Investigate other values of P. d Compare your answers with a partner’s. 18 A shop is selling a phone for $80. The shop increases the price by 10%. a Find the new price. After two weeks, the shop decreases the new price by 10%. Read what Arun and Sofia say. The price will go back down to $80. The price now will be less than $80. b Explain why Arun is wrong and Sofia is correct. c Find the price of the phone after the decrease. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
229 10.1 Percentage increases and decreases 19 The same shop is selling a television for $400. a The shop increases the price by 20%. Find the new price. b The shop increases the price by a further 20%. Here are three statements: • The new price is $560 • The new price is more than $560 • The new price is less than $560 Which statement is correct? Give a reason for your answer. c Show your answer to a partner. Is he or she convinced by your explanation? 20 a Sofia has savings of $500. She spends some money and says: My savings have decreased by 150%. Is it possible for her savings to decrease by more than 100%? b Arun has 500 g of rice. He says: I cooked some rice and the amount I have has decreased by 150%. What can you say about this statement? Summary checklist I can write one number as a percentage of another value. I can increase or decrease a value by a given percentage. I can calculate the percentage change from one value to another. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
230 10 Percentages Tip × 0.46 54% 100% 46% Worked example 10.2 The cost of a flight is $2300 Calculate the percentage change if a the cost is increased to $2850 b the cost is reduced to $1690 Answer a The multiplier for the increase is 2850 2300 = 1.239 to 3 d.p. 1.239=123.9% so the percentage increase is 23.9% b The multiplier for the decrease is 1690 2300 = 0.735 to 3 d.p. 0.735=73.5% so the percentage decrease is 100%−73.5%=26.5% The original value is always the denominator of the fraction. Tip × 1.65 65% 100% 100% In this section, you will learn a more efficient way to calculate percentage increases and decreases. Suppose you want to increase $275 by 65%. You start with $275=100% Then 65% of $275=$178.75 and the total is $453.75. 100%+65%=165% You can find 165% of $275 in a single calculation. 165%=1.65 and so 165% of $275=1.65×$275=$453.75 This is the value after the increase of 65%. To increase the value by 65% you used a multiplier of 1.65. Now suppose you want to decrease $275 by 54%. Again $275=100% So $275−54%=100%−54%=46% 46%=0.46 and so 46% of $275=0.46×$275=$126.50 This is the value after a decrease of 54%. To decrease the value by 54% you used a multiplier of 0.46. In general, original value×multiplier=new value You can also write this as multiplier= new value original value 10.2 Using a multiplier In this section you will … • learn to use a multiplier to calculate a percentage increase or decrease. Key word multiplier We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
231 10.2 Using a multiplier Exercise 10.2 In this exercise, always use a multiplier to calculate a percentage increase or decrease. 1 What multiplier would you use to a increase a value by 63% b decrease a value by 63% c increase a value by 103% d decrease a value by 88% 2 Match each percentage change to the correct multiplier. The first one is done for you: A and ii A 50% increase i × 0.2 B 80% increase ii × 1.5 C 80% decrease iii × 1.8 D 120% increase iv × 1.2 E 20% decrease v × 0.8 F 20% increase vi × 2.2 3 Write the multiplier for a an increase of 45% b an increase of 245% c a decrease of 45% 4 Here are some multipliers. Write the percentage change in each case. a × 0.75 b × 1.22 c × 3.33 d × 0.33 e × 0.03 5 Increase each of these numbers by 85% a 40 b 180 c 12 6 Find the value of 45kg after the following changes. a an increase of 20% b an increase of 170% c a decrease of 60% 7 a The mass of a girl is 26.5kg. Several years later her mass has increased by 62%. Calculate her new mass. Round your answer to 1 d.p. b A man has a mass of 172.4kg. He reduces his mass by 38%. Calculate his new mass. 8 a Increase 964 by 65% b Increase 357 by 195% c Decrease 560 by 84% 9 Change each length by the percentage shown. Length (mm) Change New length (mm) a 90 180% increase b 240 12% increase c 660 70% decrease d 320 7% decrease We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
232 10 Percentages 12 a The population of a town increases from 63200 by 17%. Calculate the new population. b The population of a city increases from 7.35 million to 12.82 million. Calculate the percentage change. c The population of an island is 4120. The population decreases by 16.5%. Calculate the new population. Think like a mathematician 13 Work with a partner on this question. This table shows the changing population of China. a Calculate the percentage increase in the population from i 1950 to 1970 ii 1970 to 2000 iii 1960 to 1990 iv 1950 to 2010 Round your answers to 1 d.p. b In which decade was there the greatest percentage increase in population? c Use the data to predict the population of China in 2020. Justify your answer. Activity 10.2a a What do you notice about the answers to Question 11? b Write some more questions like this for a partner to answer. c Exchange questions with a partner and answer your partner’s questions. d Exchange back and check your partner’s answers. Discuss any mistakes. Year Population in millions 1950 554 1960 660 1970 828 1980 1000 1990 1177 2000 1291 2010 1369 10 An athlete has a resting pulse rate of 60 beats per minute. During a race, this increases to 160 beats per minute. a Calculate the percentage increase. b Calculate the percentage decrease after the race, when his pulse rate falls from 160 to 60 beats per minute. 11 a Increase 96 by 25% b Decrease 200 by 40% c Increase 60 by 100% d Decrease 240 by 50% We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
233 10.2 Using a multiplier Activity 10.2b Find out how the population of your country has changed from 1950 to 2010. Are the percentage changes similar to or different from the percentage changes in China? Compare your answers with the answers of other learners. Can you improve your answers? 14 Prices in a shop are reduced. Copy and complete this table. Original price Percentage reduction Reduced price $280 20% $420 45% $620 $217 $750 $705 15 The height of a tree is 3.65m. Find the new height if the height increases by a 15% b 132% c 260% 16 The depth of water in a well is decreasing. Calculate the percentage reduction from a Monday to Tuesday b Tuesday to Thursday c Monday to Friday Day Depth Monday 5.75m Tuesday 5.10m Wednesday 4.31m Thursday 3.58m Friday 2.46m Tip A decade is a period of 10 years. Summary checklist I can use a multiplier to calculate a percentage increase or decrease. I can identify a percentage increase by finding a multiplier. 17 Here are two sentences. • The population of India is 407% of the population of the USA. • The population of India is 307% more than the population of the USA. a Explain why both these sentences can be correct. b Compare your explanation with a partner’s. Can you improve your explanation or your partner’s? 18 Read what Marcus says: a Describe two different ways to check that Marcus is correct. b Which way do you think is better? Give a reason. When 650 is increased by 184% the answer is 1846. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
234 10 Percentages Check your progress 1 a Write 32 as a percentage of 80. b Write 80 as a percentage of 32. 2 Increase $240 by 35% a by first finding the increase in dollars b by using a multiplier. 3 In 1960 the population of Indonesia was 88 million. In 2010 the population of Indonesia was 242 million. a Calculate the percentage increase from 1960 to 2010. b Estimate the population in 2060 if the rate of increase does not change. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
235 Getting started 1 A pair of shoes costs $25 less than a coat. a If the coat costs $110, find the cost of the shoes. b The coat costs $x and the shoes cost $y Write a function to show y in terms of x. 2 One Singapore Dollar can be exchanged for 80 Japanese Yen. If d Singapore Dollars can be exchanged for y Japanese Yen, which of these equations is correct? y d = + 80 y d = − 80 y d = 80 y d = 80 3 Here is a function: y x = + 2 a Copy and complete this table of values. x −3 −2 −1 0 1 2 3 y 0 4 b Use the table in part a to draw a graph of y x = + 2 4 This graph shows the temperature of some water. 30 20 10 0 21 3 54 6 7 8 40 Temperature (°C) Time (minutes) y = −2x + 30 a How does the graph show that the water is cooling? b How long does it take for the water to cool by 10 °C? 11 Graphs We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
236 11 Graphs In the 17th century, the Frenchman René Descartes showed how to plot points on a grid and use this to draw lines and curves. In his honour, we still call this method ‘Cartesian coordinates’. (–2, 3) (1, –3) y = –2 x y (4, 2) y = x 1 0 –1 –1 –2 –2 –3 –3 2 3 1 2 3 4 René Descartes, 1596–1650 Cartesian grid You have used positive and negative numbers as coordinates to show points on a Cartesian grid. You know that equations involving x and y can correspond to lines and curves on such a coordinate grid. In this unit, you will concentrate on straight-line graphs. Two examples, y=−2 and y=x, are shown on the Cartesian grid above. Tip The charge per hour is multiplied by the number of hours, and this is added to the booking fee. In this section you will … • represent situations in words and using functions. 11.1 Functions The cost of hiring a hall is in two parts. There is • a booking fee of $15 • a charge of $40 per hour. The total cost of hiring the hall for 3 hours is $40×3+$15=$135 Suppose the hall is hired for n hours and the cost is $c. Then c=40n+15 This function shows how to work out the cost for any number of hours. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
237 11.1 Functions Worked example 11.1 The cost of hiring a digger is a fixed charge of $35 plus $10 per day. a Find the cost of hiring the digger for 7 days. b The cost of hiring the digger for n days is $y Write a function to find the cost for any number of days. Answer a For 7 days, the cost is $35+$10×7=$105 b y=10n+35 Exercise 11.1 1 Arun buys some books online. The cost is $6 for each book plus postage of $4. a Work out the total cost, including postage, of i 3 books ii 6 books iii 12 books. b Write a function to show the cost in dollars (c) of b books. 2 A plumber comes to a house to do a repair. He charges a fixed fee of $45 plus $30 per hour. a Work out the total cost for a job that lasts i one hour ii 3 hours iii 1.5 hours. b Write a function to show the cost in dollars (c) of a job that takes h hours. 3 Theatre tickets cost $12 each plus a booking fee of $3. a Work out the total cost, including the booking fee, of i 4 tickets ii 6 tickets iii 10 tickets. b If t tickets cost $d, write a function for d in terms of t. 4 The cost of scaffolding is $80 delivery plus $50 per week. a Work out the cost of hiring scaffolding for i 2 weeks ii 4 weeks iii 7 weeks. b The cost is $y for w weeks. Find an expression for y. If you want to hire the hall for 3 hours, then n=3 and c=40×3+15=135 The cost is $135. If you want to hire the hall for 6 hours, then n=6 and c=40×6+15=255 The cost is $255. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
238 11 Graphs 5 The cost of printing photos is $2 per photo plus a fixed charge of $3. a Work out the cost of printing 40 photos. b The cost of n photos is $c. Write a function for c. c Marcus pays $49. How many photos were printed for Marcus? 6 The cost of hiring a car is a fixed fee of $25 plus $45 per day. a Show that the cost of hiring a car for 7 days is $340. b Read what Sofia says: Explain why Sofia is not correct. c Write a function to show the cost in dollars (a) of hiring a car for n days. 7 A bamboo plant is 1.5m tall. It grows 0.2m every week. a Work out the height after i 2 weeks ii 4 weeks. b How long will it take until the bamboo is 3.5m tall? Justify your answer. c Write a function to show the height in metres (h) after t weeks. 8 Read what Zara says: a How old is Zara if her father is i 40 ii 52? b Zara is z years old and her father is f years old. Write a function for z in terms of f. c How old is Zara’s father if Zara is 30 years old? If the cost of hiring a car for 7 days is $340, then the cost of hiring it for 14 days is 2×$340=$680 I am 2 years less than half my father’s age. Think like a mathematician 9 Here is a function: r=18−3t a Work out the value of r when i t=2 ii t=5 iii t=0 iv t=6 b What happens when the value of t is more than 6? c A car is on a journey. The amount of fuel in the tank of the car after t hours is r litres, where r=18−3t What can you say about the possible values of t? We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
239 11.1 Functions 10 Here is a shape. All the lengths are in cm. The perimeter is p cm and the area is acm2 . a Show that p=2L+22 b Find a function for a in terms of L. 11 There are 108 litres of water in a tank. 9 litres flow out of the tank every hour. a How much water is in the tank after i 1 hour ii 3 hours iii 7 hours? b How long will it be until the tank is empty? c There are l litres of water in the tank after h hours. Complete this function: l=................... 12 If x=5 then y=30 Which of these functions could be correct? A y x = 6 B y x = + 4 10 C y x = + 30 D y x = − 40 2 E y=8 10 x − 13 When x=4, y=6 a Show that a possible function connecting y and x is y x = − 2 2 b Show that a possible function connecting y and x is y x = + 0 5 4. c Find three more possible functions if y=6 when x=4 Write them in the form y=................... 14 The cost of booking a room for a meeting is a fixed charge plus an amount for each person. The cost is $c for n people and c=8n+40 Explain what the numbers 40 and 8 show. 15 Here is a function: y x = + 20 15 a Describe a situation that this function could represent. You must explain what x and y stand for. You must explain what the numbers 20 and 15 tell you. b Look at a partner’s answer to part a. Is the answer clear? Can you improve it? 8 6 2 L L 3 3 Someone says to you: ‘Why do you need to describe situations like the ones in this exercise with a function when you can describe them in words?’ What would be your reply? We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
240 11 Graphs Here is a function: y x = − 2 1 You can substitute different values of x into the function to find the y-value, for example: • if x=3 then y=2×3−1=5 • if x=−2 then y=2×−2−1=−4−1=−5 • and so on. You can then complete a table of values like this. x −2 −1 0 1 2 3 4 y=2x−1 −5 −3 −1 1 3 5 7 The table gives you coordinates: (−2, −5), (−1, −3), (0, −1) and so on. You can use these coordinates to plot points on a grid. You can draw a straight line through all the points. 11.2 Plotting graphs In this section you will … • construct a table of values for a function • use the table to plot a graph. Key word plot y 4 3 2 1 0 –1 –1–2–3 54321 –2 –3 –4 5 6 7 8 x –5 –6 y = 2x − 1 Summary checklist I understand how a situation can be represented in words or as a function. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
241 11.2 Plotting graphs Worked example 11.2 Here is a function: y=8x+4 a Copy and complete this table of values. x −2 −1 0 1 2 3 y=8x+4 4 b Use the table to draw the graph of y=8x+4 Answer a x −2 −1 0 1 2 3 y −12 −4 4 12 20 28 b –1 1 –10 0 10 20 30 y x –20 –2 2 3 y = 8x + 4 For example, if x=−2 then y=8×−2+4= −16+4=−12 Choose a scale so that you can plot all the points. Exercise 11.2 1 a Copy and complete this table of values for the function y x = + 2 3 x −2 −1 0 1 2 3 y 1 7 b Use the table to plot a graph of y x = + 2 3 2 a Copy and complete this table of values for the function y x = + 3 2 x −2 −1 0 1 2 3 y −4 11 b Use the table to plot a graph of y x = + 3 2 We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
242 11 Graphs 3 a Copy and complete this table of values for the function y x = + 2 6 x −3 −2 −1 0 1 2 y 2 8 b Use the table to plot a graph of y x = + 2 6 c Extend the line on the graph to show that it goes through the point (3, 12). d Show that the coordinates (3, 12) are correct for the function y x = + 2 6 4 a Copy and complete this table of values for the function y x = −2 x −2 −1 0 1 2 3 y 4 −6 b Use the table to plot a graph of y x = −2 5 a Copy and complete this table of values for the function y x = − 4 x −2 −1 0 1 2 3 4 5 y 5 1 b Use the table to plot a graph of y x = − 4 c Where does the graph cross the x-axis? d Where does the graph cross the y-axis? e i Use the function to show that if x=10 then y=−6 ii Is the point (10, −6) on your line? 6 a Copy and complete this table of values for the function y x = + 10 30 x −2 −1 0 1 2 3 4 y 20 b Use the table to plot a graph of y x = + 10 30 Use a scale of 1 cm=1 unit on the x-axis and 1 cm=10 units on the y-axis. c Where does the graph cross the y-axis? d i Use the function to find the value of y when x=2.5 ii Does this correspond to a point on your line? e Read what Marcus says: Is Marcus correct? Justify your answer. f Compare your answers with a partner’s. Is your partner correct? Tip If x=−2 then y=−2×−2=4 If the graph is extended, (10, 130) and (20, 260) will be on the line. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
243 11.2 Plotting graphs 7 Here is a function: y x = + 2 40 a Complete this table of values. x −10 0 10 20 30 40 y 100 b Use the table of values to plot a graph of y x = + 2 40 c Where does the graph cross the y-axis? d Which of these points are on the line? P (15, 70) Q (50, 140) R (37, 114) S (−20, 0) T (100, 240) e Compare your answer to part d with a partner’s. Do you agree about which points are on the line? Who is correct? 8 Here is a function: y x = − 5 15 a Complete these coordinates of points on the graph of y x = − 5 15 i (4, ) ii (7, ) iii (0, ) iv (20, ) v (3, ) b Where does the graph of y x = − 5 15 cross the y-axis? c Where does the graph of y x = − 5 15 cross the x-axis? 9 The cost of hiring a drill is in two parts. There is a delivery charge of $5 plus $2 per day. a The cost of hire for n days is $c. Explain why c n = + 2 5 b Copy and complete this table of values. n 0 1 2 3 4 5 6 7 c 5 19 c Use the table to plot a graph of c n = + 2 5. You only need positive axes. d Why do you only need positive axes? 10 A motor runs on diesel. Initially there are 40 litres of diesel in the fuel tank. The motor uses 5 litres per hour. After h hours there are f litres of diesel remaining. a Explain why f=40−5h b Copy and complete this table of values. h 0 1 2 3 4 5 6 f 30 Tip c is the subject of the formula. Put c on the vertical axis. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
244 11 Graphs c Use the table to draw a graph of f=40−5h d How does the graph show that there were 40 litres of diesel initially? e Use the graph to find the number of hours until the motor runs out of diesel. 11 The cost of hiring a car is a fixed charge of $35 plus $15 per day. a Write a function to show the cost in dollars (y) of hiring a car for d days. b Copy and complete this table of values. d 0 1 2 3 5 8 y 80 c Use the table to plot a graph of the function. d Use the graph to find the cost of hiring a car for 7 days. Use the function to check that your answer is correct. e How does the graph show the fixed charge? 12 The cost of hiring a van is $60 plus $20 per hour. a Work out a function to show the cost in dollars (c) of hiring the van for n hours. b Copy and complete this table of values. n 0 1 2 3 4 5 6 c 100 c Use your table to plot a graph to show the cost. d Read what Zara says: The cost for 12 hours is twice the cost for 6 hours. Is Zara correct? Give a reason for your answer. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
245 11.2 Plotting graphs 14 A function is y=2x+5 a Copy and complete this table of values. x −3 −2 −1 0 1 2 3 y −1 11 b Draw a graph of y=2x+5 c Where does the graph cross i the y-axis ii the x-axis? d Is there a way to use the function to predict where the line will cross the axes, before you make a table of values? Explain how you can do this. e Test your method from part d to see if it gives the correct answers for the function y = 2x+2 Summary checklist I can construct a table of values for a function of the form y=mx+c I can use the table of values to draw a graph of the form y=mx+c Think like a mathematician 13 A plant is initially 10cm high. It grows 2cm a week. After x weeks the height is ycm. a Write a function to show y in terms of x. b Copy and complete this table of values. x 0 1 2 3 4 5 6 y 10 There are three lines on this graph, A, B and C. c Which line shows the growth of the plant? d The other two lines show the growth of two other plants. Describe the growth of each of these plants. e Compare your answer to part d with a partner’s. Do you agree on the answer? x y 30 5 0 1 10 15 25 35 2 3 5 Height Weeks 20 40 4 6 A B C y = 10 + x y = 10 + 2x y = 10 + 5x We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
246 11 Graphs A function in the form y=mx+c where m and c are numbers is called a linear function. All the functions in Section 11.2 were linear functions. Here are some linear functions and the values of m and c. Function m c Function m c y=x 1 0 y=−5x+3 −5 3 y=5x+2 5 2 y=−2x−4 −2 −4 y=3x−4 3 −4 y=−x+10 −1 10 y=4x 4 0 y=12 0 12 Here is a table of values and a graph for the linear function y=6x−2 x −3 −2 −1 0 1 2 3 y −20 −14 −8 −2 4 10 16 y=6x−2 is the equation of the line. The graph crosses the y-axis at (0, −2) −2 is the y-intercept of this graph. You can see from the graph that if x increases by 1 then y increases by 6. x −3 −2 −1 0 1 2 3 y −20 −14 −8 −2 4 10 16 +1 +6 +1 +6 +1 +6 +1 +6 +1 +6 +1 +6 6 is the gradient of this graph. In the following exercise, you will investigate y-intercepts and gradients. 11.3 Gradient and intercept In this section you will … • learn to interpret the values of m and c for a function of the form y=mx+c Key words coefficient equation of a line gradient linear function x-intercept y-intercept Tip m is the coefficient of x. Tip y=−5x+3 can also be written as y=3−5x –5 0 5 10 15 20 y x –15 –20 –3 –2 –1 1 2 3 6 1 6 1 6 1 6 1 6 1 6 1 y = 6x − 2 We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
247 11.3 Gradient and intercept Exercise 11.3 1 a Complete this table of values. x −3 −2 −1 0 1 2 3 2x 2x+4 2x−3 b Use the table to draw, on the same axes, graphs of i y=2x ii y=2x+4 iii y=2x−3 c Find the gradient and the y-intercept of each line. The equations in part b are of the form y=2x+c where c is an integer. d i Write another equation of this type. ii Without drawing the graph of this equation, make a conjecture about what it looks like. iii Draw the graph to test your conjecture. 2 Here are three equations. y=3x y=3x+3 y=3x−1 a Draw a graph of each line. Use a table of values to help you. b Find the gradient and the y-intercept of each line. All the equations are of the form y=3x+c where c is an integer. c Draw the graph of another line of the same type. Think like a mathematician 3 Investigate the graphs of equations of the form y=4x+c where c is an integer. What do you notice about all your graphs? 4 a Copy and complete this table of values for y=−x+5 x −2 −1 0 1 2 3 4 5 y=−x+5 7 4 1 b Add another row to this table to show values of y=−x+2 c Draw graphs of y=−x+5 and y=−x+2 on the same axes. d Write the gradient of each line. e Write the equation of another line parallel to these lines. Tip Draw a number of graphs on the same set of axes. Tip y=−x+5 is the same as y=5−x We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
248 11 Graphs 5 a Copy and complete this table of values for y=−2x+9 x −2 −1 0 1 2 3 4 y=−2x+9 13 7 1 b Add another row to this table to show values for y=−2x+6 c Draw graphs of y=−2x+9 and y=−2x+6 on the same axes. d Write the gradient of each line. e Write the equation of another line parallel to these lines. 6 Here are six equations of lines. y=4x+6 y=6x+4 y=4x+2 y=2x+4 y=6x+2 y=2x+6 a Group together lines that are parallel. b Write the equation of one more line for each group. 7 These three lines are parallel. One of the lines is y=5x+5 x y –6 5 12 a Write the equations of the other two lines. b Write the equation of a parallel line that passes through the origin. 8 a Copy and complete this table of values. x −2 −1 0 1 2 3 x+3 4 2x+3 1 −x+3 5 1 b Use the table to plot graphs of y=x+3, y=2x+3 and y=−x+3 Plot all three graphs on the same axes. c On the same axes, draw the line y=3 d Write the gradient of each line. e Write the y-intercept of each line. f Write the equations of two more lines with the same y-intercept. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
249 11.3 Gradient and intercept 9 a Copy and complete this table of values. x −2 −1 0 1 2 3 x−2 −3 −1 3x−2 −5 1 −2x−2 0 −4 b Use the table to plot graphs of y=x−2, y=3x−2 and y=−2x−2 Plot all three graphs on the same axes. c Write the gradient of each line. d Write the y-intercept of each line. e Write the equations of two more lines with the same y-intercept. 10 The cost of a visit by an electrician is given by this function: y=25x+40 • y is the total cost in dollars • x is the number of hours • there is a fixed charge and a charge per hour a Copy and complete this table for the function. x 0 1 2 3 4 y b Use the table to plot a graph of y=25x+40 Use a scale of 2 cm to 1unit on the x-axis and 2 cm to 50units on the y-axis. c Find the y-intercept and the gradient of the graph. d What do the y-intercept and the gradient show about the electrician’s charges? 11 The cost of a holiday abroad is in two parts: • the cost of the airline flights • a charge for each night in the hotel The total cost is given by this function: y=100x+200, where • x is the number of nights • $y is the total cost. a Copy and complete this table of values. x 0 1 2 3 4 5 6 7 y 900 b Use your table to draw a graph of y=100x+200 c Explain how the graph shows the cost of the flights. d What does the gradient of the graph show? Tip You only need positive axes. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE