GCSE Maths: Real-life curved graphs (Higher)
Part of the GCSE Maths quadratic and other graphs chapter (Higher). Chapter overview · Foundation version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Real-life curved graphs
Many real situations are described by curved graphs: the path of a ball, the area of a field, the profit of a business, the depth of water draining from a tank. You meet them in exams as a graph that is already drawn (or a rule from which you draw it), and you answer questions by reading values and by interpreting what the graph tells you.
Always check which axis is which, and the units. To find the value on one axis from a value on the other, draw a straight line from the given value to the curve, then straight across or down to the other axis. Use a ruler and use the grid carefully.
Some special points on a curve have a meaning in the real situation:
- The highest point (maximum) of the ball graph tells you the greatest height reached: 20 m after 2 seconds.
- Where the curve meets the horizontal axis the quantity is 0: the ball is on the ground at 0 s (start) and 4 s (landing).
- Where a horizontal line crosses the curve, the quantity equals that value. A height of 15 m happens twice, once going up (1 s) and once coming down (3 s).
- The curve is symmetrical: going up takes as long as coming down.
If you are given the rule you can also find the exact value by substitution. For example when t = 1: h = 20 × 1 − 5 × 1² = 15. A reciprocal graph also appears in real life. If a coach costing £240 is shared equally between n people, each person pays C = 240 ÷ n pounds. The more people, the less each pays, but the cost never reaches 0.
Key ideas
- Read the axis labels and units first.
- To read a value: go from the known value to the curve, then to the other axis, using straight lines.
- The highest or lowest point of the curve gives the maximum or minimum.
- A horizontal line can meet a curve twice: look for both answers.
- Where the curve meets the horizontal axis the quantity is 0.
- Readings are approximate; check with the rule if you are given one.
Key words. To read from a graph means going from one axis to the curve and across to the other axis. Readings are approximate, so give your answer with its unit. The maximum is the highest value and the minimum is the lowest value.
Method. Go up from the time to the curve, then across to the height.
- Find 1 s on the horizontal axis. The horizontal axis shows the time.Locate t = 1.
- Go straight up to the curve, then straight across to the vertical axis. The vertical axis shows the height. Use a ruler to keep the lines straight.The curve is at height 15.
- Read the height. Read the number on the vertical axis and write the unit.15 m
Check. Use the rule: t = 1: h = −5 × (1)2 + 20 × 1 = −5 + 20 = 15. ✓
Watch out. Read from the correct axis. The question tells you which quantity is given and which you need.
Key words. To read from a graph means going from one axis to the curve and across to the other axis. Readings are approximate, so give your answer with its unit. The maximum is the highest value and the minimum is the lowest value.
Method. Go across from the height to the curve, then down to the time. Check for more than one crossing.
- Find 15 m on the vertical axis. The vertical axis shows the height.Locate h = 15.
- Draw a horizontal line across to the curve. Use a ruler. Every point on this line has a height of 15 m. Mark every place where it meets the curve (there may be more than one).The line meets the curve at (1, 15) and (3, 15).
- Go straight down from each point to the horizontal axis and read the time. The horizontal axis shows the time.1 and 3 s
Check. Use the rule: t = 1: h = −5 × (1)2 + 20 × 1 = −5 + 20 = 15; t = 3: h = −5 × (3)2 + 20 × 3 = −45 + 60 = 15. ✓
Watch out. A horizontal line can cross a curve twice. Look for both crossing points.
Key words. To read from a graph means going from one axis to the curve and across to the other axis. Readings are approximate, so give your answer with its unit. The maximum is the highest value and the minimum is the lowest value.
Method. The maximum is at the turning point of the curve.
- Find the highest point of the curve. This is the turning point. A U-shaped curve has a lowest point; an n-shaped curve has a highest point.The turning point is (2, 20).
- Read both coordinates. The horizontal coordinate is the time at which it happens. The vertical coordinate is the maximum height.Time = 2 s, height = 20 m
- Write the answer in words. Always say what the numbers mean and give units.The maximum height is 20 m, at 2 s.
Check. Use the rule: t = 2: h = −5 × (2)2 + 20 × 2 = −20 + 40 = 20. ✓
Watch out. The question may ask for the maximum value (a height) or for when it happens (a time). Read the right coordinate.
Key words. To read from a graph means going from one axis to the curve and across to the other axis. Readings are approximate, so give your answer with its unit. The maximum is the highest value and the minimum is the lowest value.
Method. Go across from the cost per person to the curve, then down to the number of people. Check for more than one crossing.
- Find £15 on the vertical axis. The vertical axis shows the cost per person.Locate C = 15.
- Draw a horizontal line across to the curve. Use a ruler. Every point on this line has a cost per person of £15. Mark every place where it meets the curve (there may be more than one).The line meets the curve at (16, 15).
- Go straight down from each point to the horizontal axis and read the number of people. The horizontal axis shows the number of people.16 people
Check. Use the rule: n = 16: C = 240 ÷ 16 = 15. ✓
Watch out. A horizontal line can cross a curve twice. Look for both crossing points.
Watch out
Check what each axis shows. On the ball graph the horizontal axis is time and the vertical axis is height. The highest point is (2, 20): the ball is highest after 2 seconds, and the greatest height is 20 m. Do not give only one answer when the question asks when and a horizontal line crosses the curve twice. Always include the unit.Exam tip
Draw your reading lines on the diagram with a ruler (the examiner can then give credit even if the final number is slightly out), and write the units in your answer.Exercise 13F Real-life graphs
Non-calculatorKey words. To read from a graph means going from one axis to the curve and across to the other axis. Readings are approximate, so give your answer with its unit. The maximum is the highest value and the minimum is the lowest value.
Method. Go up from the time to the curve, then across to the height.
- Find 1 s on the horizontal axis. The horizontal axis shows the time.Locate t = 1.
- Go straight up to the curve, then straight across to the vertical axis. The vertical axis shows the height. Use a ruler to keep the lines straight.The curve is at height 15.
- Read the height. Read the number on the vertical axis and write the unit.15 m
Check. Use the rule: t = 1: h = −5 × (1)2 + 20 × 1 = −5 + 20 = 15. ✓
Watch out. Read from the correct axis. The question tells you which quantity is given and which you need.
Key words. To read from a graph means going from one axis to the curve and across to the other axis. Readings are approximate, so give your answer with its unit. The maximum is the highest value and the minimum is the lowest value.
Method. Go up from the time to the curve, then across to the height.
- Find 3 s on the horizontal axis. The horizontal axis shows the time.Locate t = 3.
- Go straight up to the curve, then straight across to the vertical axis. The vertical axis shows the height. Use a ruler to keep the lines straight.The curve is at height 15.
- Read the height. Read the number on the vertical axis and write the unit.15 m
Check. Use the rule: t = 3: h = −5 × (3)2 + 20 × 3 = −45 + 60 = 15. ✓
Watch out. Read from the correct axis. The question tells you which quantity is given and which you need.
Key words. To read from a graph means going from one axis to the curve and across to the other axis. Readings are approximate, so give your answer with its unit. The maximum is the highest value and the minimum is the lowest value.
Method. The maximum is at the turning point of the curve.
- Find the highest point of the curve. This is the turning point. A U-shaped curve has a lowest point; an n-shaped curve has a highest point.The turning point is (2, 20).
- Read both coordinates. The horizontal coordinate is the time at which it happens. The vertical coordinate is the maximum height.Time = 2 s, height = 20 m
- Write the answer in words. Always say what the numbers mean and give units.The maximum height is 20 m, at 2 s.
Check. Use the rule: t = 2: h = −5 × (2)2 + 20 × 2 = −20 + 40 = 20. ✓
Watch out. The question may ask for the maximum value (a height) or for when it happens (a time). Read the right coordinate.
Key words. To read from a graph means going from one axis to the curve and across to the other axis. Readings are approximate, so give your answer with its unit. The maximum is the highest value and the minimum is the lowest value.
Method. The time in the air is the time between the two points where the height is 0.
- Find where the height is 0. The curve meets the horizontal axis at the start and at the end.At 0 s and 4 s.
- Subtract to find how long it lasts. The time between the two is the end minus the start.4 − 0 = 4 s
Check. The curve is symmetrical about time 2 s, the middle of 0 and 4. ✓
Watch out. Give the length of time, not the two readings.
Key words. To read from a graph means going from one axis to the curve and across to the other axis. Readings are approximate, so give your answer with its unit. The maximum is the highest value and the minimum is the lowest value.
Method. Go across from the height to the curve, then down to the time. Check for more than one crossing.
- Find 15 m on the vertical axis. The vertical axis shows the height.Locate h = 15.
- Draw a horizontal line across to the curve. Use a ruler. Every point on this line has a height of 15 m. Mark every place where it meets the curve (there may be more than one).The line meets the curve at (1, 15) and (3, 15).
- Go straight down from each point to the horizontal axis and read the time. The horizontal axis shows the time.1 and 3 s
Check. Use the rule: t = 1: h = −5 × (1)2 + 20 × 1 = −5 + 20 = 15; t = 3: h = −5 × (3)2 + 20 × 3 = −45 + 60 = 15. ✓
Watch out. A horizontal line can cross a curve twice. Look for both crossing points.
Key words. To read from a graph means going from one axis to the curve and across to the other axis. Readings are approximate, so give your answer with its unit. The maximum is the highest value and the minimum is the lowest value.
Method. Go across from the height to the curve, then down to the time. Check for more than one crossing.
- Find 20 m on the vertical axis. The vertical axis shows the height.Locate h = 20.
- Draw a horizontal line across to the curve. Use a ruler. Every point on this line has a height of 20 m. Mark every place where it meets the curve (there may be more than one).The line meets the curve at (2, 20).
- Go straight down from each point to the horizontal axis and read the time. The horizontal axis shows the time.2 s
Check. Use the rule: t = 2: h = −5 × (2)2 + 20 × 2 = −20 + 40 = 20. ✓
Watch out. A horizontal line can cross a curve twice. Look for both crossing points.
Key words. To read from a graph means going from one axis to the curve and across to the other axis. Readings are approximate, so give your answer with its unit. The maximum is the highest value and the minimum is the lowest value.
Method. Go across from the height to the curve, then down to the time. Check for more than one crossing.
- Find 10 m on the vertical axis. The vertical axis shows the height.Locate h = 10.
- Draw a horizontal line across to the curve. Use a ruler. Every point on this line has a height of 10 m. Mark every place where it meets the curve (there may be more than one).The line meets the curve at (0.6, 10) and (3.4, 10).
- Go straight down from each point to the horizontal axis and read the time. The horizontal axis shows the time.0.6 and 3.4 s (approximately)
Check. Put the readings into the rule: t = 0.6 gives h = 10.0; t = 3.4 gives h = 10.0, close to the target. (Readings from a graph are approximate.) ✓
Watch out. A horizontal line can cross a curve twice. Look for both crossing points.
Key words. To read from a graph means going from one axis to the curve and across to the other axis. Readings are approximate, so give your answer with its unit. The maximum is the highest value and the minimum is the lowest value.
Method. Go up from the width to the curve, then across to the area.
- Find 3 m on the horizontal axis. The horizontal axis shows the width.Locate x = 3.
- Go straight up to the curve, then straight across to the vertical axis. The vertical axis shows the area. Use a ruler to keep the lines straight.The curve is at height 27.
- Read the area. Read the number on the vertical axis and write the unit.27 m²
Check. Use the rule: x = 3: A = −(3)2 + 12 × 3 = −9 + 36 = 27. ✓
Watch out. Read from the correct axis. The question tells you which quantity is given and which you need.
Key words. To read from a graph means going from one axis to the curve and across to the other axis. Readings are approximate, so give your answer with its unit. The maximum is the highest value and the minimum is the lowest value.
Method. The maximum is at the turning point of the curve.
- Find the highest point of the curve. This is the turning point. A U-shaped curve has a lowest point; an n-shaped curve has a highest point.The turning point is (6, 36).
- Read both coordinates. The horizontal coordinate is the width at which it happens. The vertical coordinate is the maximum area.Width = 6 m, area = 36 m²
- Write the answer in words. Always say what the numbers mean and give units.The maximum area is 36 m², at 6 m.
Check. Use the rule: x = 6: A = −(6)2 + 12 × 6 = −36 + 72 = 36. ✓
Watch out. The question may ask for the maximum value (a area) or for when it happens (a width). Read the right coordinate.
Key words. To read from a graph means going from one axis to the curve and across to the other axis. Readings are approximate, so give your answer with its unit. The maximum is the highest value and the minimum is the lowest value.
Method. Go across from the area to the curve, then down to the width. Check for more than one crossing.
- Find 20 m² on the vertical axis. The vertical axis shows the area.Locate A = 20.
- Draw a horizontal line across to the curve. Use a ruler. Every point on this line has a area of 20 m². Mark every place where it meets the curve (there may be more than one).The line meets the curve at (2, 20) and (10, 20).
- Go straight down from each point to the horizontal axis and read the width. The horizontal axis shows the width.2 and 10 m
Check. Use the rule: x = 2: A = −(2)2 + 12 × 2 = −4 + 24 = 20; x = 10: A = −(10)2 + 12 × 10 = −100 + 120 = 20. ✓
Watch out. A horizontal line can cross a curve twice. Look for both crossing points.
Key words. To read from a graph means going from one axis to the curve and across to the other axis. Readings are approximate, so give your answer with its unit. The maximum is the highest value and the minimum is the lowest value.
Method. Go up from the time to the curve, then across to the depth of water.
- Find 4 minutes on the horizontal axis. The horizontal axis shows the time.Locate t = 4.
- Go straight up to the curve, then straight across to the vertical axis. The vertical axis shows the depth of water. Use a ruler to keep the lines straight.The curve is at height 36.
- Read the depth of water. Read the number on the vertical axis and write the unit.36 cm
Check. Use the rule: t = 4: d = 42 − 20 × 4 + 100 = 16 − 80 + 100 = 36. ✓
Watch out. Read from the correct axis. The question tells you which quantity is given and which you need.
Key words. To read from a graph means going from one axis to the curve and across to the other axis. Readings are approximate, so give your answer with its unit. The maximum is the highest value and the minimum is the lowest value.
Method. Go up from the time to the curve, then across to the depth of water.
- Find 2 minutes on the horizontal axis. The horizontal axis shows the time.Locate t = 2.
- Go straight up to the curve, then straight across to the vertical axis. The vertical axis shows the depth of water. Use a ruler to keep the lines straight.The curve is at height 64.
- Read the depth of water. Read the number on the vertical axis and write the unit.64 cm
Check. Use the rule: t = 2: d = 22 − 20 × 2 + 100 = 4 − 40 + 100 = 64. ✓
Watch out. Read from the correct axis. The question tells you which quantity is given and which you need.
Key words. To read from a graph means going from one axis to the curve and across to the other axis. Readings are approximate, so give your answer with its unit. The maximum is the highest value and the minimum is the lowest value.
Method. Go across from the depth of water to the curve, then down to the time. Check for more than one crossing.
- Find 25 cm on the vertical axis. The vertical axis shows the depth of water.Locate d = 25.
- Draw a horizontal line across to the curve. Use a ruler. Every point on this line has a depth of water of 25 cm. Mark every place where it meets the curve (there may be more than one).The line meets the curve at (5, 25).
- Go straight down from each point to the horizontal axis and read the time. The horizontal axis shows the time.5 minutes
Check. Use the rule: t = 5: d = 52 − 20 × 5 + 100 = 25 − 100 + 100 = 25. ✓
Watch out. A horizontal line can cross a curve twice. Look for both crossing points.
Key words. To read from a graph means going from one axis to the curve and across to the other axis. Readings are approximate, so give your answer with its unit. The maximum is the highest value and the minimum is the lowest value.
Method. Where the curve meets the horizontal axis, the depth of water is 0.
- Find where the depth of water is 0. A depth of water of 0 means the curve is on the horizontal axis.Look along the horizontal axis for the points where the curve touches it.
- Read the crossing points. Read the numbers along the horizontal axis at those points.The curve meets the axis at 10 minutes.
Check. Use the rule: t = 10: d = 102 − 20 × 10 + 100 = 100 − 200 + 100 = 0. ✓
Watch out. Roots are values on the horizontal axis (the x-axis).
Key words. To read from a graph means going from one axis to the curve and across to the other axis. Readings are approximate, so give your answer with its unit. The maximum is the highest value and the minimum is the lowest value.
Method. Go up from the number of people to the curve, then across to the cost per person.
- Find 8 people on the horizontal axis. The horizontal axis shows the number of people.Locate n = 8.
- Go straight up to the curve, then straight across to the vertical axis. The vertical axis shows the cost per person. Use a ruler to keep the lines straight.The curve is at height 30.
- Read the cost per person. Read the number on the vertical axis and write the unit.£30
Check. Use the rule: n = 8: C = 240 ÷ 8 = 30. ✓
Watch out. Read from the correct axis. The question tells you which quantity is given and which you need.
Key words. To read from a graph means going from one axis to the curve and across to the other axis. Readings are approximate, so give your answer with its unit. The maximum is the highest value and the minimum is the lowest value.
Method. Go up from the number of people to the curve, then across to the cost per person.
- Find 15 people on the horizontal axis. The horizontal axis shows the number of people.Locate n = 15.
- Go straight up to the curve, then straight across to the vertical axis. The vertical axis shows the cost per person. Use a ruler to keep the lines straight.The curve is at height 16.
- Read the cost per person. Read the number on the vertical axis and write the unit.£16
Check. Use the rule: n = 15: C = 240 ÷ 15 = 16. ✓
Watch out. Read from the correct axis. The question tells you which quantity is given and which you need.
Key words. To read from a graph means going from one axis to the curve and across to the other axis. Readings are approximate, so give your answer with its unit. The maximum is the highest value and the minimum is the lowest value.
Method. Go across from the cost per person to the curve, then down to the number of people. Check for more than one crossing.
- Find £12 on the vertical axis. The vertical axis shows the cost per person.Locate C = 12.
- Draw a horizontal line across to the curve. Use a ruler. Every point on this line has a cost per person of £12. Mark every place where it meets the curve (there may be more than one).The line meets the curve at (20, 12).
- Go straight down from each point to the horizontal axis and read the number of people. The horizontal axis shows the number of people.20 people
Check. Use the rule: n = 20: C = 240 ÷ 20 = 12. ✓
Watch out. A horizontal line can cross a curve twice. Look for both crossing points.
Key words. To read from a graph means going from one axis to the curve and across to the other axis. Readings are approximate, so give your answer with its unit. The maximum is the highest value and the minimum is the lowest value.
Method. Go across from the cost per person to the curve, then down to the number of people. Check for more than one crossing.
- Find £10 on the vertical axis. The vertical axis shows the cost per person.Locate C = 10.
- Draw a horizontal line across to the curve. Use a ruler. Every point on this line has a cost per person of £10. Mark every place where it meets the curve (there may be more than one).The line meets the curve at (24, 10).
- Go straight down from each point to the horizontal axis and read the number of people. The horizontal axis shows the number of people.24 people
Check. Use the rule: n = 24: C = 240 ÷ 24 = 10. ✓
Watch out. A horizontal line can cross a curve twice. Look for both crossing points.
Key words. To read from a graph means going from one axis to the curve and across to the other axis. Readings are approximate, so give your answer with its unit. The maximum is the highest value and the minimum is the lowest value.
Method. Go up from the ticket price to the curve, then across to the profit.
- Find £3 on the horizontal axis. The horizontal axis shows the ticket price.Locate x = 3.
- Go straight up to the curve, then straight across to the vertical axis. The vertical axis shows the profit. Use a ruler to keep the lines straight.The curve is at height 25.
- Read the profit. Read the number on the vertical axis and write the unit.£25
Check. Use the rule: x = 3: P = −5 × (3)2 + 50 × 3 − 80 = −45 + 150 − 80 = 25. ✓
Watch out. Read from the correct axis. The question tells you which quantity is given and which you need.
Key words. To read from a graph means going from one axis to the curve and across to the other axis. Readings are approximate, so give your answer with its unit. The maximum is the highest value and the minimum is the lowest value.
Method. The maximum is at the turning point of the curve.
- Find the highest point of the curve. This is the turning point. A U-shaped curve has a lowest point; an n-shaped curve has a highest point.The turning point is (5, 45).
- Read both coordinates. The horizontal coordinate is the ticket price at which it happens. The vertical coordinate is the maximum profit.Ticket price = £5, profit = £45
- Write the answer in words. Always say what the numbers mean and give units.The maximum profit is £45, at £5.
Check. Use the rule: x = 5: P = −5 × (5)2 + 50 × 5 − 80 = −125 + 250 − 80 = 45. ✓
Watch out. The question may ask for the maximum value (a profit) or for when it happens (a ticket price). Read the right coordinate.
Key words. To read from a graph means going from one axis to the curve and across to the other axis. Readings are approximate, so give your answer with its unit. The maximum is the highest value and the minimum is the lowest value.
Method. Where the curve meets the horizontal axis, the profit is 0.
- Find where the profit is 0. A profit of 0 means the curve is on the horizontal axis.Look along the horizontal axis for the points where the curve touches it.
- Read the crossing points. Read the numbers along the horizontal axis at those points.The curve meets the axis at 2 and 8 £.
Check. Use the rule: x = 2: P = −5 × (2)2 + 50 × 2 − 80 = −20 + 100 − 80 = 0; x = 8: P = −5 × (8)2 + 50 × 8 − 80 = −320 + 400 − 80 = 0. ✓
Watch out. Roots are values on the horizontal axis (the x-axis).
Key words. To read from a graph means going from one axis to the curve and across to the other axis. Readings are approximate, so give your answer with its unit. The maximum is the highest value and the minimum is the lowest value.
Method. Find where the profit equals 40, then decide which prices give more than that.
- Find where P = 40. Draw the horizontal line P = 40 across the graph.The line meets the curve at £4 and £6.
- Decide which part of the curve is above the line. The profit is at least 40 where the curve is on or above the line, which is between the two crossing points.Between x = 4 and x = 6
- Write the answer. Include the pound sign.From £4 to £6
Check. x = 4: P = −5 × (4)2 + 50 × 4 − 80 = −80 + 200 − 80 = 40; x = 5: P = −5 × (5)2 + 50 × 5 − 80 = −125 + 250 − 80 = 45: profit is at least 40 in between. ✓
Watch out. "At least 40" means 40 or more: the part of the curve above the line.
Key words. To read from a graph means going from one axis to the curve and across to the other axis. Readings are approximate, so give your answer with its unit. The maximum is the highest value and the minimum is the lowest value.
Method. Find the times when the height is 18 m, then work out how long the ball is above that.
- Find where h = 18. Draw a horizontal line across at 18 m. It meets the curve twice.1.4 and 2.6 s (accept answers within 0.1)
- Find the time between the two crossings. The ball is above 18 m between them.2.6 − 1.4 = 1.2 seconds
- Compare with 1 second. 1.2 s is more than 1 s, so Dan is right.
Check. Using the rule: h(1.5) = 30 − 11.25 = 18.75 and h(2.5) = 50 − 31.25 = 18.75, both above 18. ✓
Watch out. Give a reason with numbers.
Key words. To read from a graph means going from one axis to the curve and across to the other axis. Readings are approximate, so give your answer with its unit. The maximum is the highest value and the minimum is the lowest value.
Method. Find the number of people when the cost is exactly £20, then decide whether more or fewer people are needed.
- Find n when C = 20. Draw a horizontal line at £20 to the curve, then read down.12 people
- Decide. More people means a lower cost per person. We need the cost to be 20 or less, so 12 or more people.n ≥ 12
- Write the smallest number. 12 people
Check. 240 ÷ 12 = 20 ✓, and 240 ÷ 11 = 21.8, which is too much.
Watch out. Fewer people means each pays more.
Key words. To read from a graph means going from one axis to the curve and across to the other axis. Readings are approximate, so give your answer with its unit. The maximum is the highest value and the minimum is the lowest value.
Method. Average rate = change in depth ÷ time taken.
- Read the depths. After 0 minutes and after 5 minutes.100 cm and 25 cm
- Find the change in depth. Subtract.100 − 25 = 75 cm
- Divide by the time. The time is 5 minutes.75 ÷ 5 = 15 cm per minute
Check. 15 cm per minute for 5 minutes drops 75 cm. ✓
Watch out. The graph is a curve, so this is only an average rate for the whole 5 minutes.
Key words. To read from a graph means going from one axis to the curve and across to the other axis. Readings are approximate, so give your answer with its unit. The maximum is the highest value and the minimum is the lowest value.
Method. Use the line of symmetry through the highest point.
- Find the line of symmetry. It passes through the highest point (2, 20).t = 2
- Use the mirror image. The graph starts at (0, 0) on the ground. Its mirror image in t = 2 is at t = 4 (2 + 2).0 and 4 are the same distance from 2
- Conclude. So the ball is back on the ground at 4 seconds.Time in the air = 4 seconds.
Check. h(4) = 80 − 80 = 0. ✓
Watch out. Going up and coming down take equal times (ignoring air resistance).
Key words. To read from a graph means going from one axis to the curve and across to the other axis. Readings are approximate, so give your answer with its unit. The maximum is the highest value and the minimum is the lowest value.
Method. If the rate were constant, equal times would give equal drops in depth. Compare two equal time intervals.
- First 2 minutes. Depth at 0 min is 100 cm and at 2 min is 64 cm.100 − 64 = 36 cm
- Last 2 minutes. Depth at 8 min is 4 cm and at 10 min is 0 cm.4 − 0 = 4 cm
- Compare. 36 cm is not equal to 4 cm, so the rate is not constant. The curve gets less steep, so the water drains more slowly as the tank empties.Not a constant rate.
Check. t = 2: d = 22 − 20 × 2 + 100 = 4 − 40 + 100 = 64; t = 8: d = 82 − 20 × 8 + 100 = 64 − 160 + 100 = 4 ✓
Watch out. A straight-line graph means constant rate; this curve is not straight.
Key words. To read from a graph means going from one axis to the curve and across to the other axis. Readings are approximate, so give your answer with its unit. The maximum is the highest value and the minimum is the lowest value.
Method. Go up from the distance along the track to the curve, then across to the height.
- Find 2 m on the horizontal axis. The horizontal axis shows the distance along the track.Locate x = 2.
- Go straight up to the curve, then straight across to the vertical axis. The vertical axis shows the height. Use a ruler to keep the lines straight.The curve is at height 2.
- Read the height. Read the number on the vertical axis and write the unit.2 cm
Check. Use the rule: x = 2: h = 23 − 6 × 22 + 9 × 2 = 8 − 24 + 18 = 2. ✓
Watch out. Read from the correct axis. The question tells you which quantity is given and which you need.
Key words. To read from a graph means going from one axis to the curve and across to the other axis. Readings are approximate, so give your answer with its unit. The maximum is the highest value and the minimum is the lowest value.
Method. Go up from the distance along the track to the curve, then across to the height.
- Find 4 m on the horizontal axis. The horizontal axis shows the distance along the track.Locate x = 4.
- Go straight up to the curve, then straight across to the vertical axis. The vertical axis shows the height. Use a ruler to keep the lines straight.The curve is at height 4.
- Read the height. Read the number on the vertical axis and write the unit.4 cm
Check. Use the rule: x = 4: h = 43 − 6 × 42 + 9 × 4 = 64 − 96 + 36 = 4. ✓
Watch out. Read from the correct axis. The question tells you which quantity is given and which you need.
Key words. To read from a graph means going from one axis to the curve and across to the other axis. Readings are approximate, so give your answer with its unit. The maximum is the highest value and the minimum is the lowest value.
Method. Go across from the height to the curve, then down to the distance along the track. Check for more than one crossing.
- Find 2 cm on the vertical axis. The vertical axis shows the height.Locate h = 2.
- Draw a horizontal line across to the curve. Use a ruler. Every point on this line has a height of 2 cm. Mark every place where it meets the curve (there may be more than one).The line meets the curve at (0.3, 2), (2.0, 2) and (3.7, 2).
- Go straight down from each point to the horizontal axis and read the distance along the track. The horizontal axis shows the distance along the track.0.3, 2.0 and 3.7 m (approximately)
Check. Put the readings into the rule: x = 0.3 gives h = 2.0; x = 2.0 gives h = 2.0; x = 3.7 gives h = 2.0, close to the target. (Readings from a graph are approximate.) ✓
Watch out. A horizontal line can cross a curve twice. Look for both crossing points.
Key words. To read from a graph means going from one axis to the curve and across to the other axis. Readings are approximate, so give your answer with its unit. The maximum is the highest value and the minimum is the lowest value.
Method. Go up from the time to the curve, then across to the height.
- Find 0.5 s on the horizontal axis. The horizontal axis shows the time.Locate t = 0.5.
- Go straight up to the curve, then straight across to the vertical axis. The vertical axis shows the height. Use a ruler to keep the lines straight.The curve is at height 8.75.
- Read the height. Read the number on the vertical axis and write the unit.8.75 m
Check. Use the rule: t = 0.5: h = −5 × (0.5)2 + 20 × 0.5 = −1.25 + 10 = 8.75. ✓
Watch out. Read from the correct axis. The question tells you which quantity is given and which you need.
Key words. To read from a graph means going from one axis to the curve and across to the other axis. Readings are approximate, so give your answer with its unit. The maximum is the highest value and the minimum is the lowest value.
Method. Go up from the time to the curve, then across to the height.
- Find 3.5 s on the horizontal axis. The horizontal axis shows the time.Locate t = 3.5.
- Go straight up to the curve, then straight across to the vertical axis. The vertical axis shows the height. Use a ruler to keep the lines straight.The curve is at height 8.75.
- Read the height. Read the number on the vertical axis and write the unit.8.75 m
Check. Use the rule: t = 3.5: h = −5 × (3.5)2 + 20 × 3.5 = −61.25 + 70 = 8.75. ✓
Watch out. Read from the correct axis. The question tells you which quantity is given and which you need.
Key words. To read from a graph means going from one axis to the curve and across to the other axis. Readings are approximate, so give your answer with its unit. The maximum is the highest value and the minimum is the lowest value.
Method. Compare the times with the line of symmetry.
- Find the distances from t = 2. 0.5 is 1.5 below 2 and 3.5 is 1.5 above 2.2 − 0.5 = 1.5 and 3.5 − 2 = 1.5
- Use symmetry. Points equally far either side of the line of symmetry have the same height.Same height by symmetry.
Check. Both heights are 8.75 m. ✓
Watch out. Equal distances from the line of symmetry give equal values.
Key words. To read from a graph means going from one axis to the curve and across to the other axis. Readings are approximate, so give your answer with its unit. The maximum is the highest value and the minimum is the lowest value.
Method. Go across from the area to the curve, then down to the width. Check for more than one crossing.
- Find 30 m² on the vertical axis. The vertical axis shows the area.Locate A = 30.
- Draw a horizontal line across to the curve. Use a ruler. Every point on this line has a area of 30 m². Mark every place where it meets the curve (there may be more than one).The line meets the curve at (3.6, 30) and (8.4, 30).
- Go straight down from each point to the horizontal axis and read the width. The horizontal axis shows the width.3.6 and 8.4 m (approximately)
Check. Put the readings into the rule: x = 3.6 gives A = 30.0; x = 8.4 gives A = 30.0, close to the target. (Readings from a graph are approximate.) ✓
Watch out. A horizontal line can cross a curve twice. Look for both crossing points.
Key words. To read from a graph means going from one axis to the curve and across to the other axis. Readings are approximate, so give your answer with its unit. The maximum is the highest value and the minimum is the lowest value.
Method. Think about the pen: a width of x means a length of 12 − x.
- Write the area. A = x(12 − x).width × length
- Swap the width and the length. A pen of width 12 − x has length x, and the same area.x(12 − x) = (12 − x)x
- Conclude. So widths 3.6 and 8.4 (which add to 12) give the same area.The two answers add up to 12.
Check. 3.6 + 8.4 = 12. ✓
Watch out. Symmetry about x = 6 comes from this.
Key words. To read from a graph means going from one axis to the curve and across to the other axis. Readings are approximate, so give your answer with its unit. The maximum is the highest value and the minimum is the lowest value.
Method. The ball is on the ground when h = 0. Put h = 0 and factorise.
- Put h = 0. The height is 0 on the ground.30t − 5t² = 0
- Factorise. Take out the common factor 5t.5t(6 − t) = 0
- Solve. Either 5t = 0 (the start) or 6 − t = 0.t = 0 or t = 6, so it lands after 6 s
Check. 30 × 6 − 5 × 36 = 180 − 180 = 0. ✓
Watch out. t = 0 is the start, not the landing.
Key words. To read from a graph means going from one axis to the curve and across to the other axis. Readings are approximate, so give your answer with its unit. The maximum is the highest value and the minimum is the lowest value.
Method. The highest point is halfway between the times when h = 0.
- Use the roots 0 and 6. By symmetry the highest point is midway.(0 + 6) ÷ 2 = 3
- Write the answer. Include the unit.The ball is highest at t = 3 s
Check. The flight lasts 6 s and 3 is half of 6. ✓
Watch out. Symmetry saves a lot of work.
Key words. To read from a graph means going from one axis to the curve and across to the other axis. Readings are approximate, so give your answer with its unit. The maximum is the highest value and the minimum is the lowest value.
Method. Substitute the time of the highest point into the rule.
- Substitute t = 3. Use h = 30t − 5t².h = 30 × 3 − 5 × 3² = 90 − 45
- Write the answer. Include the unit.45 m
Check. Using h = 5t(6 − t): 5 × 3 × 3 = 45. ✓
Watch out. Square first: 3² = 9, then multiply by 5.