GCSE Maths: Exponential graphs, growth and decay (Higher)
Part of the GCSE Maths functions and advanced graphs chapter (Higher). Chapter overview · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Exponential graphs, growth and decay
Imagine a rumour. On day 0 one person knows it. Each day every person who knows it tells one more person, so the number doubles: 1, 2, 4, 8, 16, … The number after x days is 2x. A function with the unknown x in the power is called an exponential function. The general form is y = kx, where k is a positive number called the base.
To draw the graph, make a table of values. For y = 2x: 20 = 1 (any number to the power 0 is 1), 21 = 2, 22 = 4, 23 = 8, and for negative powers 2−1 = 1/2, 2−2 = 1/4. Every graph of y = kx has the same key features:
- it passes through (0, 1), because k0 = 1;
- it is always above the x-axis (y is never zero or negative), and the x-axis is an asymptote;
- if k > 1 the graph rises faster and faster: this is exponential growth;
- if 0 < k < 1 the graph falls, steeply at first and then more slowly towards the axis: this is exponential decay.
Growth and decay problems. Exponential functions model real situations where something changes by the same percentage in every period: compound interest, a growing population, a falling car value, a medicine leaving the body. You met the idea in Chapter 5. The method is:
- Find the multiplier for one period: for an increase of r% it is 1 + r/100; for a decrease it is 1 − r/100.
- The amount after n periods is starting amount × (multiplier)n.
- Use a calculator for the power. Keep all the digits until the final rounding.
To find when an amount reaches a target, build a table of values period by period (or use trial and improvement) until you pass the target. For a quantity that halves every T units of time (a half-life) the amount is start × (1/2)time ÷ T.
If a graph passes through (0, a), then y = a × kx: the a is the starting value. A second point gives k.
Key ideas
- y = kx passes through (0, 1), is always positive, and has the x-axis as an asymptote. k > 1: growth. 0 < k < 1: decay.
- k0 = 1 and k−n = 1/kn.
- Multiplier: increase r% = 1 + r/100; decrease r% = 1 − r/100. After n periods: start × multipliern.
- To find a time, use a table or trial and improvement.
- Through (0, a): y = a × kx. Use a second point to find k.
Key words. In 2x the number 2 is the base and x is the power (or index). A function with x in the power is an exponential function.
Method. Substitute each value of x into y = 2x and work out the power.
- Remember what a power means. 2x means 2 multiplied by itself x times. A power of 0 gives 1. A negative power means 1 divided by the positive power.
- Work out each value. Take the powers one at a time.x = −2: 2−2 = 1 ÷ 22 = 1 ÷ 4 = 0.25
x = −1: 2−1 = 1 ÷ 21 = 1 ÷ 2 = 0.5
x = 0: 20 = 1 (any number to the power 0 is 1)
x = 1: 21 = 2
x = 2: 22 = 2 × 2 = 4
x = 3: 23 = 2 × 2 × 2 = 8
x = 4: 24 = 2 × 2 × 2 × 2 = 16 - Write the values in the table.
x −2 −1 0 1 2 3 4 2x 0.25 0.5 1 2 4 8 16
Check. Each value in the table is 2 times the one before it (reading left to right). ✓
Watch out. A negative power does not make the answer negative: 2−3 is 1/8, not −8.
Key words. The multiplier is the number you multiply by to apply a percentage change in one step. Compound means each change is applied to the new amount, not the original.
Method. Growth and decay problems use a multiplier. Find the multiplier, raise it to the power of the number of years, and multiply by the starting amount.
- Find the growth multiplier. An increase of 4% means we add 4% of the amount. So we are left with 100% + 4% = 104% of the amount, and as a decimal that is the multiplier.104% = 104 ÷ 100 = 1.04
- Write the formula. The amount is multiplied by 1.04 once for each year. After n years we multiply n times, which is the multiplier to the power n.V = (starting amount) × (multiplier)n
- Put in the numbers: start = £3000, multiplier = 1.04, n = 5. Replace each letter by its value.V = £3000 × 1.045
- Work it out on a calculator. Do the power 1.045 first, then multiply by 3000. Keep all the digits until the final rounding.1.045 = 1.21665
V = £3000 × 1.21665 = 3649.96 - Round sensibly. Money is rounded to the nearest penny (2 decimal places).V = £3649.96
Check. Check the direction: the amount increases, so the answer should be bigger than £3000. It is £3649.96. ✓
Watch out. Do not work out the percentage once and multiply by the number of years. That gives simple growth. Compound growth multiplies by the multiplier every period, so use a power.
Key words. The multiplier is the number you multiply by to apply a percentage change in one step. Compound means each change is applied to the new amount, not the original.
Method. Growth and decay problems use a multiplier. Find the multiplier, raise it to the power of the number of years, and multiply by the starting amount.
- Find the decay multiplier. A decrease of 15% means we take away 15% of the amount. So we are left with 100% − 15% = 85% of the amount, and as a decimal that is the multiplier.85% = 85 ÷ 100 = 0.85
- Write the formula. The amount is multiplied by 0.85 once for each year. After n years we multiply n times, which is the multiplier to the power n.V = (starting amount) × (multiplier)n
- Put in the numbers: start = £14000, multiplier = 0.85, n = 3. Replace each letter by its value.V = £14000 × 0.853
- Work it out on a calculator. Do the power 0.853 first, then multiply by 14000. Keep all the digits until the final rounding.0.853 = 0.614125
V = £14000 × 0.614125 = 8597.75 - Round sensibly. Money is rounded to the nearest penny (2 decimal places).V = £8597.75
Check. Check the direction: the amount decreases, so the answer should be smaller than £14000. It is £8597.75. ✓
Watch out. Do not work out the percentage once and multiply by the number of years. That gives simple growth. Compound growth multiplies by the multiplier every period, so use a power.
Method. Use the multiplier repeatedly, building a table, until the amount reaches the target.
- Find the multiplier. Each year the amount is multiplied by the same number.103% = 1.03
- Work out the amount at the end of each year. Multiply by 1.03 each time. Stop as soon as the amount reaches or passes £1000.
After n years Amount 0 £800.00 1 £824.00 2 £848.72 3 £874.18 4 £900.41 5 £927.42 6 £955.24 7 £983.90 8 £1013.42 - Read off the answer. The first row that reaches or passes £1000 is after 8 years.n = 8
Check. Use the formula: £800 × 1.038 = £1013.42 (above £1000) but after 7 years it is £983.90 (not yet there). ✓
Watch out. Count the years carefully. The row marked 0 is the start, before any year has passed.
Method. Use the point on the y-axis to find a, then use the second point to find k.
- Write the general form. An exponential graph through the given points has the form y = a × kx, where a is the starting value and k is the multiplier.y = a × kx
- Use the point with x = 0 to find a. Any number to the power 0 is 1, so when x = 0 we get y = a × 1 = a.(0, 3) ⇒ a = 3
- Use the point (4, 48) to find k. Substitute x and y and then divide by a.48 = 3 × k4
k4 = 48 ÷ 3 = 16 - Find the number that gives this value when raised to the power. Ask: which number multiplied by itself 4 times gives 16?24 = 2 × 2 × 2 × 2 = 16, so k = 2
- Write the equation. y = 3 × 2x
Check. Test the second point: x = 4 gives 3 × 24 = 3 × 16 = 48. ✓
Watch out. Divide by a before looking for the power. Do not square-root or cube-root until you have kn alone.
Watch out
A 20% increase followed by a 20% decrease does not return you to the start. Do not add the percentage on every year: multiply by the multiplier every year. And remember a negative power does not make a negative answer: 2−3 = 1/8.Exercise 25F Exponential graphs, growth and decay
Calculator allowedKey words. In 2x the number 2 is the base and x is the power (or index). A function with x in the power is an exponential function.
Method. Substitute each value of x into y = 2x and work out the power.
- Remember what a power means. 2x means 2 multiplied by itself x times. A power of 0 gives 1. A negative power means 1 divided by the positive power.
- Work out each value. Take the powers one at a time.x = −2: 2−2 = 1 ÷ 22 = 1 ÷ 4 = 0.25
x = 0: 20 = 1 (any number to the power 0 is 1)
x = 2: 22 = 2 × 2 = 4
x = 4: 24 = 2 × 2 × 2 × 2 = 16 - Write the values in the table.
x −2 0 2 4 2x 0.25 1 4 16
Check. Each value in the table is 2 times the one before it (reading left to right). ✓
Watch out. A negative power does not make the answer negative: 2−3 is 1/8, not −8.
Key words. In 2x the number 2 is the base and x is the power (or index). A function with x in the power is an exponential function.
Method. Substitute each value of x into y = 3x and work out the power.
- Remember what a power means. 3x means 3 multiplied by itself x times. A power of 0 gives 1. A negative power means 1 divided by the positive power.
- Work out each value. Take the powers one at a time.x = −2: 3−2 = 1 ÷ 32 = 1 ÷ 9 = 19
x = −1: 3−1 = 1 ÷ 31 = 1 ÷ 3 = 13
x = 0: 30 = 1 (any number to the power 0 is 1)
x = 1: 31 = 3
x = 2: 32 = 3 × 3 = 9 - Write the values in the table.
x −2 −1 0 1 2 3x 19 13 1 3 9
Check. Each value in the table is 3 times the one before it (reading left to right). ✓
Watch out. A negative power does not make the answer negative: 2−3 is 1/8, not −8.
Key words. In 2x the number 2 is the base and x is the power (or index). A function with x in the power is an exponential function.
Method. Substitute each value of x into y = 10x and work out the power.
- Remember what a power means. 10x means 10 multiplied by itself x times. A power of 0 gives 1. A negative power means 1 divided by the positive power.
- Work out each value. Take the powers one at a time.x = −2: 10−2 = 1 ÷ 102 = 1 ÷ 100 = 0.01
x = −1: 10−1 = 1 ÷ 101 = 1 ÷ 10 = 0.1
x = 0: 100 = 1 (any number to the power 0 is 1)
x = 1: 101 = 10
x = 2: 102 = 10 × 10 = 100 - Write the values in the table.
x −2 −1 0 1 2 10x 0.01 0.1 1 10 100
Check. Each value in the table is 10 times the one before it (reading left to right). ✓
Watch out. A negative power does not make the answer negative: 2−3 is 1/8, not −8.
Key words. In 2x the number 2 is the base and x is the power (or index). A function with x in the power is an exponential function.
Method. Substitute each value of x into y = 0.5x and work out the power.
- Remember what a power means. 0.5x means 0.5 multiplied by itself x times. A power of 0 gives 1. A negative power means 1 divided by the positive power.
- Work out each value. Take the powers one at a time.x = −3: 0.5−3 = 1 ÷ 0.53 = 1 ÷ 0.125 = 8
x = −1: 0.5−1 = 1 ÷ 0.51 = 1 ÷ 0.5 = 2
x = 0: 0.50 = 1 (any number to the power 0 is 1)
x = 2: 0.52 = 0.5 × 0.5 = 0.25
x = 4: 0.54 = 0.5 × 0.5 × 0.5 × 0.5 = 0.0625 - Write the values in the table.
x −3 −1 0 2 4 0.5x 8 2 1 0.25 0.0625
Check. Moving one step to the right multiplies the value by 0.5. ✓
Watch out. A negative power does not make the answer negative: 2−3 is 1/8, not −8.
Method. Go up from x = 2.5 to the curve, then across to the y-axis.
- Find the known points either side. The graph shows (2, 4) and (3, 8).22 = 4 and 23 = 8
- Read the graph between them. At x = 2.5 the curve is between 4 and 8, and nearer to 6 than 5 because the curve is bending upwards.22.5 ≈ 5.7
Check. Calculator: 22.5 = 5.66. ✓
Watch out. The curve bends, so do not just take the midpoint (6) of 4 and 8; the true value is a little below.
Method. Go across from y = 10 to the curve, then down to the x-axis.
- Find the known points either side. The graph shows (3, 8); the next whole power is 24 = 16.23 = 8 and 24 = 16
- Read the graph. 10 is between 8 and 16, so x is between 3 and 4, a little more than 3.x ≈ 3.3
Check. Calculator: 23.3 = 9.85, close to 10. ✓
Watch out. 10 is closer to 8 than to 16, so x is closer to 3 than 4.
Method. Work out the value of y when x = 1 and compare it with the diagram.
- Substitute x = 1. The graphs cross the line x = 1 at different heights.y = 21 = 2
- Decide which graph. At x = 1 the blue graph is at height 2 and it rises to the right.graph A
Check. Growth graphs rise to the right, the decay graph falls. ✓
Watch out. The steepest rising graph has the biggest base.
Method. Work out the value of y when x = 1 and compare it with the diagram.
- Substitute x = 1. The graphs cross the line x = 1 at different heights.y = 31 = 3
- Decide which graph. At x = 1 the orange graph is at height 3, the highest, and it rises to the right.graph B
Check. Growth graphs rise to the right, the decay graph falls. ✓
Watch out. The steepest rising graph has the biggest base.
Method. Work out the value of y when x = 1 and compare it with the diagram.
- Substitute x = 1. The graphs cross the line x = 1 at different heights.y = 1/21 = 0.5
- Decide which graph. At x = 1 the green graph is at height 0.5, and it falls to the right.graph C
Check. Growth graphs rise to the right, the decay graph falls. ✓
Watch out. The steepest rising graph has the biggest base.
Method. Compare the base with 1: bigger than 1 means growth, between 0 and 1 means decay.
- Look at the base. The base is 1.05.1.05 > 1
- Decide. Each time x goes up by 1, y is multiplied by the base: bigger each time.exponential growth
Check. Try x = 0 and x = 1: y = 1 then 1.05, which is bigger. ✓
Watch out. A base between 0 and 1, such as 0.9, means decay even though it is a positive number.
Method. Compare the base with 1: bigger than 1 means growth, between 0 and 1 means decay.
- Look at the base. The base is 0.9.0.9 < 1
- Decide. Each time x goes up by 1, y is multiplied by the base: smaller each time.exponential decay
Check. Try x = 0 and x = 1: y = 1 then 0.9, which is smaller. ✓
Watch out. A base between 0 and 1, such as 0.9, means decay even though it is a positive number.
Method. Compare the base with 1: bigger than 1 means growth, between 0 and 1 means decay.
- Look at the base. The base is 7.7 > 1
- Decide. Each time x goes up by 1, y is multiplied by the base: bigger each time.exponential growth
Check. Try x = 0 and x = 1: y = 1 then 7, which is bigger. ✓
Watch out. A base between 0 and 1, such as 0.9, means decay even though it is a positive number.
Method. Compare the base with 1: bigger than 1 means growth, between 0 and 1 means decay.
- Look at the base. The base is 0.75.0.75 < 1
- Decide. Each time x goes up by 1, y is multiplied by the base: smaller each time.exponential decay
Check. Try x = 0 and x = 1: y = 1 then 0.75, which is smaller. ✓
Watch out. A base between 0 and 1, such as 0.9, means decay even though it is a positive number.
Method. Compare the base with 1: bigger than 1 means growth, between 0 and 1 means decay.
- Look at the base. The base is 0.99.0.99 < 1
- Decide. Each time x goes up by 1, y is multiplied by the base: smaller each time.exponential decay
Check. Try x = 0 and x = 1: y = 1 then 0.99, which is smaller. ✓
Watch out. A base between 0 and 1, such as 0.9, means decay even though it is a positive number.
Method. Try whole-number powers of 2 until the value matches 32. A power 0 gives 1, a negative power is 1 divided by the positive power.
- Write the equation. We need the power (the index) that turns the base into the target.2x = 32
- Work out powers of the base. Start near the answer. Remember that going up one power multiplies by the base.20 = 1
21 = 2
22 = 4
23 = 8
24 = 16
25 = 32 - Pick the matching power. x = 5
Check. 25 = 32. ✓
Watch out. A negative power means a reciprocal: 5−2 = 1/25, not −25.
Method. Try whole-number powers of 3 until the value matches 81. A power 0 gives 1, a negative power is 1 divided by the positive power.
- Write the equation. We need the power (the index) that turns the base into the target.3x = 81
- Work out powers of the base. Start near the answer. Remember that going up one power multiplies by the base.3−1 = 13
30 = 1
31 = 3
32 = 9
33 = 27
34 = 81 - Pick the matching power. x = 4
Check. 34 = 81. ✓
Watch out. A negative power means a reciprocal: 5−2 = 1/25, not −25.
Method. Try whole-number powers of 5 until the value matches 0.04. A power 0 gives 1, a negative power is 1 divided by the positive power.
- Write the equation. We need the power (the index) that turns the base into the target.5x = 0.04
- Work out powers of the base. Start near the answer. Remember that going up one power multiplies by the base.5−3 = 0.008
5−2 = 0.04 - Pick the matching power. x = −2
Check. 5−2 = 0.04. ✓
Watch out. A negative power means a reciprocal: 5−2 = 1/25, not −25.
Method. Try whole-number powers of 0.5 until the value matches 64. A power 0 gives 1, a negative power is 1 divided by the positive power.
- Write the equation. We need the power (the index) that turns the base into the target.0.5x = 64
- Work out powers of the base. Remember the base is less than 1, so the values get smaller as the power increases and bigger when the power is negative.0.5−6 = 64
- Pick the matching power. x = −6
Check. 0.5−6 = 64. ✓
Watch out. A negative power means a reciprocal: 5−2 = 1/25, not −25.
Key words. The multiplier is the number you multiply by to apply a percentage change in one step. Compound means each change is applied to the new amount, not the original.
Method. Growth and decay problems use a multiplier. Find the multiplier, raise it to the power of the number of years, and multiply by the starting amount.
- Find the growth multiplier. An increase of 4% means we add 4% of the amount. So we are left with 100% + 4% = 104% of the amount, and as a decimal that is the multiplier.104% = 104 ÷ 100 = 1.04
- Write the formula. The amount is multiplied by 1.04 once for each year. After n years we multiply n times, which is the multiplier to the power n.V = (starting amount) × (multiplier)n
- Put in the numbers: start = £3000, multiplier = 1.04, n = 6. Replace each letter by its value.V = £3000 × 1.046
- Work it out on a calculator. Do the power 1.046 first, then multiply by 3000. Keep all the digits until the final rounding.1.046 = 1.26532
V = £3000 × 1.26532 = 3795.96 - Round sensibly. Money is rounded to the nearest penny (2 decimal places).V = £3795.96
Check. Check the direction: the amount increases, so the answer should be bigger than £3000. It is £3795.96. ✓
Watch out. Do not work out the percentage once and multiply by the number of years. That gives simple growth. Compound growth multiplies by the multiplier every period, so use a power.
Key words. The multiplier is the number you multiply by to apply a percentage change in one step. Compound means each change is applied to the new amount, not the original.
Method. Growth and decay problems use a multiplier. Find the multiplier, raise it to the power of the number of years, and multiply by the starting amount.
- Find the growth multiplier. An increase of 2.5% means we add 2.5% of the amount. So we are left with 100% + 2.5% = 102.5% of the amount, and as a decimal that is the multiplier.102.5% = 102.5 ÷ 100 = 1.025
- Write the formula. The amount is multiplied by 1.025 once for each year. After n years we multiply n times, which is the multiplier to the power n.P = (starting amount) × (multiplier)n
- Put in the numbers: start = 24000, multiplier = 1.025, n = 8. Replace each letter by its value.P = 24000 × 1.0258
- Work it out on a calculator. Do the power 1.0258 first, then multiply by 24000. Keep all the digits until the final rounding.1.0258 = 1.2184
P = 24000 × 1.2184 = 29241.7 - Round sensibly. A population or count must be a whole number, so round to the nearest whole number.P = 29242
Check. Check the direction: the amount increases, so the answer should be bigger than 24000. It is 29242. ✓
Watch out. Do not work out the percentage once and multiply by the number of years. That gives simple growth. Compound growth multiplies by the multiplier every period, so use a power.
Key words. The multiplier is the number you multiply by to apply a percentage change in one step. Compound means each change is applied to the new amount, not the original.
Method. Growth and decay problems use a multiplier. Find the multiplier, raise it to the power of the number of years, and multiply by the starting amount.
- Find the decay multiplier. A decrease of 12% means we take away 12% of the amount. So we are left with 100% − 12% = 88% of the amount, and as a decimal that is the multiplier.88% = 88 ÷ 100 = 0.88
- Write the formula. The amount is multiplied by 0.88 once for each year. After n years we multiply n times, which is the multiplier to the power n.V = (starting amount) × (multiplier)n
- Put in the numbers: start = £9500, multiplier = 0.88, n = 4. Replace each letter by its value.V = £9500 × 0.884
- Work it out on a calculator. Do the power 0.884 first, then multiply by 9500. Keep all the digits until the final rounding.0.884 = 0.599695
V = £9500 × 0.599695 = 5697.11 - Round sensibly. Money is rounded to the nearest penny (2 decimal places).V = £5697.11
Check. Check the direction: the amount decreases, so the answer should be smaller than £9500. It is £5697.11. ✓
Watch out. Do not work out the percentage once and multiply by the number of years. That gives simple growth. Compound growth multiplies by the multiplier every period, so use a power.
Key words. The multiplier is the number you multiply by to apply a percentage change in one step. Compound means each change is applied to the new amount, not the original.
Method. Growth and decay problems use a multiplier. Find the multiplier, raise it to the power of the number of days, and multiply by the starting amount.
- Find the decay multiplier. A decrease of 6% means we take away 6% of the amount. So we are left with 100% − 6% = 94% of the amount, and as a decimal that is the multiplier.94% = 94 ÷ 100 = 0.94
- Write the formula. The amount is multiplied by 0.94 once for each day. After n days we multiply n times, which is the multiplier to the power n.M = (starting amount) × (multiplier)n
- Put in the numbers: start = 500 g, multiplier = 0.94, n = 10. Replace each letter by its value.M = 500 × 0.9410
- Work it out on a calculator. Do the power 0.9410 first, then multiply by 500. Keep all the digits until the final rounding.0.9410 = 0.538615
M = 500 × 0.538615 = 269.308 - Round sensibly. Round to 3 significant figures.M = 269 g
Check. Check the direction: the amount decreases, so the answer should be smaller than 500. It is 269. ✓
Watch out. Do not work out the percentage once and multiply by the number of years. That gives simple growth. Compound growth multiplies by the multiplier every period, so use a power.
Method. Use the multiplier repeatedly, building a table, until the amount reaches the target.
- Find the multiplier. Each year the amount is multiplied by the same number.103% = 1.03
- Work out the amount at the end of each year. Multiply by 1.03 each time. Stop as soon as the amount reaches or passes £1000.
After n years Amount 0 £800.00 1 £824.00 2 £848.72 3 £874.18 4 £900.41 5 £927.42 6 £955.24 7 £983.90 8 £1013.42 - Read off the answer. The first row that reaches or passes £1000 is after 8 years.n = 8
Check. Use the formula: £800 × 1.038 = £1013.42 (above £1000) but after 7 years it is £983.90 (not yet there). ✓
Watch out. Count the years carefully. The row marked 0 is the start, before any year has passed.
Method. Use the multiplier repeatedly, building a table, until the amount falls to the target.
- Find the multiplier. Each hour the amount is multiplied by the same number.80% = 0.8
- Work out the amount at the end of each hour. Multiply by 0.8 each time. Stop as soon as the amount falls to or below 50 mg.
After n hours Amount 0 200.0 mg 1 160.0 mg 2 128.0 mg 3 102.4 mg 4 81.9 mg 5 65.5 mg 6 52.4 mg 7 41.9 mg - Read off the answer. The first row that is at or below 50 mg is after 7 hours.n = 7
Check. Use the formula: 200 × 0.87 = 41.9 mg (below 50) but after 6 hours it is 52.4 mg (not yet there). ✓
Watch out. Count the hours carefully. The row marked 0 is the start, before any hour has passed.
Method. Put t in and work out the power. Each 5 days halves the mass.
- Find t ÷ 5. This is the number of halvings.5 ÷ 5 = 1
- Halve that many times. (1/2)1 = 0.5M = 80 × 0.5 = 40
Check. Halve 80 repeatedly: 80 → 40. ✓
Watch out. Halving three times means (1/2)3 = 1/8, not 3 × 1/2.
Method. Put t in and work out the power. Each 5 days halves the mass.
- Find t ÷ 5. This is the number of halvings.15 ÷ 5 = 3
- Halve that many times. (1/2)3 = 0.125M = 80 × 0.125 = 10
Check. Halve 80 repeatedly: 80 → 40 → 20 → 10. ✓
Watch out. Halving three times means (1/2)3 = 1/8, not 3 × 1/2.
Method. Put t in and work out the power. Each 5 days halves the mass.
- Find t ÷ 5. This is the number of halvings.25 ÷ 5 = 5
- Halve that many times. (1/2)5 = 0.03125M = 80 × 0.03125 = 2.5
Check. Halve 80 repeatedly: 80 → 40 → 20 → 10 → 5 → 2.5. ✓
Watch out. Halving three times means (1/2)3 = 1/8, not 3 × 1/2.
Method. Use the point on the y-axis to find a, then use the second point to find k.
- Write the general form. An exponential graph through the given points has the form y = a × kx, where a is the starting value and k is the multiplier.y = a × kx
- Use the point with x = 0 to find a. Any number to the power 0 is 1, so when x = 0 we get y = a × 1 = a.(0, 3) ⇒ a = 3
- Use the point (4, 48) to find k. Substitute x and y and then divide by a.48 = 3 × k4
k4 = 48 ÷ 3 = 16 - Find the number that gives this value when raised to the power. Ask: which number multiplied by itself 4 times gives 16?24 = 2 × 2 × 2 × 2 = 16, so k = 2
- Write the equation. y = 3 × 2x
Check. Test the second point: x = 4 gives 3 × 24 = 3 × 16 = 48. ✓
Watch out. Divide by a before looking for the power. Do not square-root or cube-root until you have kn alone.
Method. Use the point on the y-axis to find a, then use the second point to find k.
- Write the general form. An exponential graph through the given points has the form y = a × kx, where a is the starting value and k is the multiplier.y = a × kx
- Use the point with x = 0 to find a. Any number to the power 0 is 1, so when x = 0 we get y = a × 1 = a.(0, 5) ⇒ a = 5
- Use the point (3, 135) to find k. Substitute x and y and then divide by a.135 = 5 × k3
k3 = 135 ÷ 5 = 27 - Find the number that gives this value when raised to the power. Ask: which number multiplied by itself 3 times gives 27?33 = 3 × 3 × 3 = 27, so k = 3
- Write the equation. y = 5 × 3x
Check. Test the second point: x = 3 gives 5 × 33 = 5 × 27 = 135. ✓
Watch out. Divide by a before looking for the power. Do not square-root or cube-root until you have kn alone.
Method. Use the point on the y-axis to find a, then use the second point to find k.
- Write the general form. An exponential graph through the given points has the form y = a × kx, where a is the starting value and k is the multiplier.y = a × kx
- Use the point with x = 0 to find a. Any number to the power 0 is 1, so when x = 0 we get y = a × 1 = a.(0, 2) ⇒ a = 2
- Use the point (2, 50) to find k. Substitute x and y and then divide by a.50 = 2 × k2
k2 = 50 ÷ 2 = 25 - Find the number that gives this value when raised to the power. Ask: which number multiplied by itself 2 times gives 25?52 = 5 × 5 = 25, so k = 5
- Write the equation. y = 2 × 5x
Check. Test the second point: x = 2 gives 2 × 52 = 2 × 25 = 50. ✓
Watch out. Divide by a before looking for the power. Do not square-root or cube-root until you have kn alone.
Method. Write the multiplier as m, form the equation 1000 × m3 = 1331, and find m.
- Write the equation. After 3 years the amount is 1000 × m3.1000 × m3 = 1331
- Divide both sides by 1000. m3 = 1.331
- Find the cube root. Try 1.1 × 1.1 × 1.1: 1.1 × 1.1 = 1.21 and 1.21 × 1.1 = 1.331.m = 1.1
- Convert the multiplier to a rate. 1.1 = 110%, which is 100% plus 10%.rate = 10%
Check. 1000 × 1.1 = 1100, × 1.1 = 1210, × 1.1 = 1331. ✓
Watch out. The rate is the multiplier minus 1, written as a percentage: 1.1 means 10%.
Method. Work out both balances after 20 years using a formula, and compare.
- Plan A adds the same amount each year. This is a straight-line pattern: balance = 100 + 10 × years.After 20 years: 100 + 10 × 20 = £300
- Plan B multiplies by 1.1 each year. This is exponential growth: balance = 100 × 1.1years.After 20 years: 100 × 1.120 = £672.75
- Compare. £672.75 − £300 = £372.75
- Look at a table to see why the gap grows. In plan B the interest is paid on the interest as well, so each year’s increase is bigger than the last.
years Plan A Plan B 0 £100 £100.00 1 £110 £110.00 2 £120 £121.00 5 £150 £161.05 10 £200 £259.37 20 £300 £672.75
Check. Plan B’s increase in year 20 is about £61.16, much more than Plan A’s £10. ✓
Watch out. Linear growth adds the same amount each time; exponential growth multiplies, so its increases keep getting bigger.
Method. Average rate of change = change in y ÷ change in x.
- From x = 0 to x = 1. 20 = 1 and 21 = 2.(2 − 1) ÷ (1 − 0) = 1
- From x = 3 to x = 4. 23 = 8 and 24 = 16.(16 − 8) ÷ (4 − 3) = 8
- Compare. The second gradient is 8 times the first. The graph of an exponential growth function is not a straight line: it gets steeper and steeper.1 compared with 8
Check. Check: each step right doubles y, so the increase doubles each time: 1, 2, 4, 8. ✓
Watch out. A curve has a different gradient at different points, so a single number only describes an interval.
Method. Work out the value of 2x for more and more negative x.
- Work out some values. A negative power is a reciprocal.
x 2x −1 1/2 = 0.5 −2 1/4 = 0.25 −5 1/32 = 0.03125 −10 1/1024 = 0.00097656 −20 1/1048576 = 0.00000095 - Think about 1 divided by a huge number. It gets very close to 0 but is never exactly 0, and never negative.So y > 0 for every x.
- Conclude. The x-axis is an asymptote: the curve gets as close as you like but never touches it.Mia is wrong.
Check. If 2x = 0 then no power of 2 could ever give 0, because 2 × 2 × … is never 0. ✓
Watch out. “Asymptote” means very close but never touching.
Method. Write one equation for each point. Divide them to cancel a.
- Use (1, 6). Put x = 1, y = 6.ab = 6
- Use (3, 24). Put x = 3, y = 24.ab3 = 24
- Divide the second equation by the first. a cancels, and b3 ÷ b = b2.b2 = 24 ÷ 6 = 4
- Find b. b must be positive.b = 2
- Find a. Put b = 2 into ab = 6.a × 2 = 6, so a = 3
Check. y = 3 × 2x: at x = 1, 3 × 2 = 6; at x = 3, 3 × 8 = 24. ✓
Watch out. Dividing the two equations is the quick way to remove a.