GCSE Maths: Real-life graphs and rates of change (Foundation)
Part of the GCSE Maths straight-line graphs and coordinates chapter (Foundation). Chapter overview · Higher version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Real-life graphs and rates of change
Straight-line graphs are used to describe real situations where one quantity changes steadily with another: converting between units, the cost of a taxi, a phone tariff, water filling a tank. The graph is a quick way to read off values, and the equation y = mx + c lets you calculate them exactly.
What the gradient and intercept mean
In a real-life graph the gradient is the rate of change: how much the quantity on the vertical axis changes for each 1 unit on the horizontal axis. Its units are “vertical units per horizontal unit”. In a conversion graph from miles to km the gradient is 1.6 km per mile. In a taxi fare graph it is the cost per mile. A negative gradient means the quantity is decreasing, like water draining from a tank.
The y-intercept is the starting value, when the horizontal value is 0. For a taxi, it is the fixed charge you pay before travelling any distance. For a graph that goes through the origin, the starting value is 0: this is a graph of direct proportion, y = kx, where doubling x doubles y.
Reading values from a graph
To find a value, go from the known value on one axis to the line, then to the other axis. Use a ruler and dotted lines. First check the scale: work out what each small square is worth on each axis. To find an exact value for an equation such as C = 2n + 3, substitute into the formula instead.
Comparing two graphs
When two companies charge differently, draw both lines on the same axes. The point where the lines cross is where the two costs are equal. Before that point one company is cheaper, after it the other is. A line with a smaller starting charge but a bigger gradient is cheaper for small amounts and more expensive for large amounts.
Journey graphs
In a distance–time graph the gradient is the speed (distance ÷ time). A steeper line means faster travel; a horizontal line means the object is not moving (the distance is not changing); a line going back down means returning towards the start. You will meet these again, with curves, in Chapter 13, and speed in Chapter 16.
Direct and inverse proportion graphs
If y is directly proportional to x, the graph is a straight line through the origin (graph A below). A straight line that does not go through the origin (B) is a linear relationship but not direct proportion. If y is inversely proportional to x, the graph is a curve that falls as x rises (C): see Chapters 13 and 16.
Key ideas
- The gradient of a real-life graph is the rate of change: units are (vertical units) per (horizontal unit).
- The y-intercept is the starting value (when the horizontal value is 0), for example a fixed charge.
- Direct proportion: a straight line through the origin, y = kx. Inverse proportion: a curve.
- Where two lines cross, the two quantities are equal.
- On a distance–time graph, the gradient is the speed and a horizontal line means stopped.
- Read the scale of each axis carefully before you read values.
Key words. To read off a value from a graph, draw guide lines from one axis to the line and then across to the other axis.
Method. Read across from the graph: go up from the horizontal value to the line, then across to the vertical axis.
- Find 30 miles on the horizontal axis. The horizontal axis shows the distance in miles.distance in miles: 30 miles
- Go straight up to the line. This finds the point on the graph that has this horizontal value. Use a ruler and a dotted line.
- Go across to the vertical axis and read the value. The vertical axis shows the distance in km. The dotted line meets it at the answer.distance in km: 48 km
Check. Use the formula k = 1.6m: k = 1.6 × 30 = 48. ✓
Watch out. Read the scale carefully: count how much each small square is worth on each axis.
Key words. The equation y = mx + c describes a straight line. m is the gradient (steepness) and c is the y-intercept (the value of y where the line crosses the y-axis, which is where x = 0).
Method. Find the y-intercept and the gradient, then write the equation in the form y = mx + c with the letters given in the question.
- Find the y-intercept c. Read where the line crosses the vertical axis.c = 3
- Find the gradient m. Use rise ÷ run on two points of the line.(23 − 3) ÷ 10 = 20 ÷ 10 = 2
- Write y = mx + c using the letters from the question. Replace x by n and y by C, since these stand for the real quantities.C = 2n + 3
Check. Put n = 10 into the formula: 2 × 10 + 3 = 23, which is the height of the line there. ✓
Watch out. Use the letters in the question, not x and y, if the question defines other letters.
Key words. The gradient of a line is how steep it is: rise (the change up or down) divided by run (the change across, to the right). A line sloping up from left to right has a positive gradient; sloping down has a negative gradient. In a real-life graph the gradient is the rate of change.
Method. The gradient of a real-life graph is rise ÷ run, and it is the rate of change: how much the vertical quantity changes for each 1 on the horizontal axis.
- Choose two points on the line. Choose points that are easy to read and far apart.(0, 3) and (10, 23)
- Draw a triangle and work out the rise and the run. Rise = change in the vertical value; run = change in the horizontal value.rise = 23 − 3 = 20
run = 10 − 0 = 10 - Divide the rise by the run. Gradient = rise ÷ run.gradient = 20 ÷ 10 = 2
- Say what the gradient means. The gradient is the change in the cost for each 1 unit increase in the distance. It is a rate of change.The gradient is £2 per mile.
Check. For each 1 across, the line goes up 2, which matches the gradient. ✓
Watch out. Give the units of the gradient: it is (vertical units) per (horizontal unit), for example £2 per mile.
Method. The y-intercept is where the line meets the vertical axis. In a real-life graph it is the value at the start (when the horizontal value is 0).
- Find where the line crosses the vertical axis. This is where the horizontal value is 0: the very start.
- Read the value. The line meets the vertical axis at 3.cost = £3 when distance = 0 miles
- Say what it means. This is the fixed charge: £3 is paid before any miles are travelled.The y-intercept is 3.
Check. The graph really does start at that height when the horizontal value is 0. ✓
Watch out. Describe the meaning in words: “the fixed charge” or “the starting amount”, not just the number.
Method. Put the known value into the formula and solve the equation for the letter you do not know.
- Write the formula and put in the known value. We know the vertical value, so replace the letter for it.15 = 2n + 3
- Subtract 3 from both sides. This removes the fixed amount so only the term with the letter is left.12 = 2n
- Divide both sides by 2. This leaves n on its own.n = 12 ÷ 2 = 6
Check. Substitute back: 2 × 6 + 3 = 15. ✓
Watch out. Undo the addition first, then the multiplication (reverse order).
Watch out
Read the scale. The small squares on a real-life graph are often worth 2, 5 or 10, not 1. Also give units with the gradient: write “£2 per mile”, not just “2”. And a line that does not start at the origin is not direct proportion even though it is straight.Exercise 12K Conversion graphs, rates and proportion
Calculator allowedKey words. To read off a value from a graph, draw guide lines from one axis to the line and then across to the other axis.
Method. Read across from the graph: go up from the horizontal value to the line, then across to the vertical axis.
- Find 30 miles on the horizontal axis. The horizontal axis shows the distance in miles.distance in miles: 30 miles
- Go straight up to the line. This finds the point on the graph that has this horizontal value. Use a ruler and a dotted line.
- Go across to the vertical axis and read the value. The vertical axis shows the distance in km. The dotted line meets it at the answer.distance in km: 48 km
Check. Use the formula k = 1.6m: k = 1.6 × 30 = 48. ✓
Watch out. Read the scale carefully: count how much each small square is worth on each axis.
Method. Read from the graph in the other direction: go across from the vertical value to the line, then down to the horizontal axis.
- Find 72 km on the vertical axis. The vertical axis shows the distance in km.distance in km: 72 km
- Go across to the line. This finds the point on the graph with this vertical value. Use a ruler and a dotted line.
- Go straight down to the horizontal axis and read the value. The horizontal axis shows the distance in miles.distance in miles: 45 miles
Check. Use the formula k = 1.6m: 72 = 1.6m, so m = 45. ✓
Watch out. Start from the correct axis: for a vertical value, go across first.
Key words. To read off a value from a graph, draw guide lines from one axis to the line and then across to the other axis.
Method. Read across from the graph: go up from the horizontal value to the line, then across to the vertical axis.
- Find 25 miles on the horizontal axis. The horizontal axis shows the distance in miles.distance in miles: 25 miles
- Go straight up to the line. This finds the point on the graph that has this horizontal value. Use a ruler and a dotted line.
- Go across to the vertical axis and read the value. The vertical axis shows the distance in km. The dotted line meets it at the answer.distance in km: 40 km
Check. Use the formula k = 1.6m: k = 1.6 × 25 = 40. ✓
Watch out. Read the scale carefully: count how much each small square is worth on each axis.
Key words. The gradient of a line is how steep it is: rise (the change up or down) divided by run (the change across, to the right). A line sloping up from left to right has a positive gradient; sloping down has a negative gradient. In a real-life graph the gradient is the rate of change.
Method. The gradient of a real-life graph is rise ÷ run, and it is the rate of change: how much the vertical quantity changes for each 1 on the horizontal axis.
- Choose two points on the line. Choose points that are easy to read and far apart.(0, 0) and (50, 80)
- Draw a triangle and work out the rise and the run. Rise = change in the vertical value; run = change in the horizontal value.rise = 80 − 0 = 80
run = 50 − 0 = 50 - Divide the rise by the run. Gradient = rise ÷ run.gradient = 80 ÷ 50 = 1.6
- Say what the gradient means. The gradient is the change in the distance in km for each 1 unit increase in the distance in miles. It is a rate of change.The gradient is 1.6 km per mile.
Check. For each 1 across, the line goes up 1.6, which matches the gradient. ✓
Watch out. Give the units of the gradient: it is (vertical units) per (horizontal unit), for example 1.6 km per mile.
Key words. The equation y = mx + c describes a straight line. m is the gradient (steepness) and c is the y-intercept (the value of y where the line crosses the y-axis, which is where x = 0).
Method. Find the y-intercept and the gradient, then write the equation in the form y = mx + c with the letters given in the question.
- Find the y-intercept c. Read where the line crosses the vertical axis.c = 0
- Find the gradient m. Use rise ÷ run on two points of the line.(80 − 0) ÷ 50 = 80 ÷ 50 = 1.6
- Write y = mx + c using the letters from the question. Replace x by m and y by k, since these stand for the real quantities.k = 1.6m
Check. Put m = 50 into the formula: 85 × 50 = 80, which is the height of the line there. ✓
Watch out. Use the letters in the question, not x and y, if the question defines other letters.
Key words. To read off a value from a graph, draw guide lines from one axis to the line and then across to the other axis.
Method. Read across from the graph: go up from the horizontal value to the line, then across to the vertical axis.
- Find £60 on the horizontal axis. The horizontal axis shows the amount in pounds.amount in pounds: £60
- Go straight up to the line. This finds the point on the graph that has this horizontal value. Use a ruler and a dotted line.
- Go across to the vertical axis and read the value. The vertical axis shows the amount in dollars. The dotted line meets it at the answer.amount in dollars: $75
Check. Use the formula D = 1.25p: D = 1.25 × 60 = 75. ✓
Watch out. Read the scale carefully: count how much each small square is worth on each axis.
Method. Read from the graph in the other direction: go across from the vertical value to the line, then down to the horizontal axis.
- Find $50 on the vertical axis. The vertical axis shows the amount in dollars.amount in dollars: $50
- Go across to the line. This finds the point on the graph with this vertical value. Use a ruler and a dotted line.
- Go straight down to the horizontal axis and read the value. The horizontal axis shows the amount in pounds.amount in pounds: £40
Check. Use the formula D = 1.25p: 50 = 1.25p, so p = 40. ✓
Watch out. Start from the correct axis: for a vertical value, go across first.
Key words. To read off a value from a graph, draw guide lines from one axis to the line and then across to the other axis.
Method. Read across from the graph: go up from the horizontal value to the line, then across to the vertical axis.
- Find £30 on the horizontal axis. The horizontal axis shows the amount in pounds.amount in pounds: £30
- Go straight up to the line. This finds the point on the graph that has this horizontal value. Use a ruler and a dotted line.
- Go across to the vertical axis and read the value. The vertical axis shows the amount in dollars. The dotted line meets it at the answer.amount in dollars: $37.5
Check. Use the formula D = 1.25p: D = 1.25 × 30 = 37.5. ✓
Watch out. Read the scale carefully: count how much each small square is worth on each axis.
Method. Find the gradient (the exchange rate), write the formula, then substitute 45.
- Find the formula. The graph passes through the origin, so c = 0. The gradient is the number of dollars for each £1.rise ÷ run = 100 ÷ 80 = 1.25, so D = 1.25p
- Substitute p = 45. Replace the letter by the number.D = 1.25 × 45 = 56.25
- Write the answer with its unit. Money is written to 2 decimal places.$56.25
Check. Using the graph: £40 is $50 and £5 is $6.25, so £45 is $56.25. ✓
Watch out. The gradient of a conversion graph is the exchange rate: dollars per pound.
Key words. To read off a value from a graph, draw guide lines from one axis to the line and then across to the other axis.
Method. Read across from the graph: go up from the horizontal value to the line, then across to the vertical axis.
- Find £10 on the horizontal axis. The horizontal axis shows the amount in pounds.amount in pounds: £10
- Go straight up to the line. This finds the point on the graph that has this horizontal value. Use a ruler and a dotted line.
- Go across to the vertical axis and read the value. The vertical axis shows the amount in dollars. The dotted line meets it at the answer.amount in dollars: $12.5
Check. Use the formula D = 1.25p: D = 1.25 × 10 = 12.5. ✓
Watch out. Read the scale carefully: count how much each small square is worth on each axis.
Key words. To read off a value from a graph, draw guide lines from one axis to the line and then across to the other axis.
Method. Read across from the graph: go up from the horizontal value to the line, then across to the vertical axis.
- Find £30 on the horizontal axis. The horizontal axis shows the amount in pounds.amount in pounds: £30
- Go straight up to the line. This finds the point on the graph that has this horizontal value. Use a ruler and a dotted line.
- Go across to the vertical axis and read the value. The vertical axis shows the amount in dollars. The dotted line meets it at the answer.amount in dollars: $37.5
Check. Use the formula D = 1.25p: D = 1.25 × 30 = 37.5. ✓
Watch out. Read the scale carefully: count how much each small square is worth on each axis.
Key words. To read off a value from a graph, draw guide lines from one axis to the line and then across to the other axis.
Method. Read across from the graph: go up from the horizontal value to the line, then across to the vertical axis.
- Find £50 on the horizontal axis. The horizontal axis shows the amount in pounds.amount in pounds: £50
- Go straight up to the line. This finds the point on the graph that has this horizontal value. Use a ruler and a dotted line.
- Go across to the vertical axis and read the value. The vertical axis shows the amount in dollars. The dotted line meets it at the answer.amount in dollars: $62.5
Check. Use the formula D = 1.25p: D = 1.25 × 50 = 62.5. ✓
Watch out. Read the scale carefully: count how much each small square is worth on each axis.
Key words. To read off a value from a graph, draw guide lines from one axis to the line and then across to the other axis.
Method. Read across from the graph: go up from the horizontal value to the line, then across to the vertical axis.
- Find 25°C on the horizontal axis. The horizontal axis shows the temperature in °C.temperature in °C: 25°C
- Go straight up to the line. This finds the point on the graph that has this horizontal value. Use a ruler and a dotted line.
- Go across to the vertical axis and read the value. The vertical axis shows the temperature in °F. The dotted line meets it at the answer.temperature in °F: 77°F
Check. Use the formula F = 1.8C + 32: F = 1.8 × 25 + 32 = 77. ✓
Watch out. Read the scale carefully: count how much each small square is worth on each axis.
Method. Read from the graph in the other direction: go across from the vertical value to the line, then down to the horizontal axis.
- Find 86°F on the vertical axis. The vertical axis shows the temperature in °F.temperature in °F: 86°F
- Go across to the line. This finds the point on the graph with this vertical value. Use a ruler and a dotted line.
- Go straight down to the horizontal axis and read the value. The horizontal axis shows the temperature in °C.temperature in °C: 30°C
Check. Use the formula F = 1.8C + 32: 86 = 1.8C + 32, so C = 30. ✓
Watch out. Start from the correct axis: for a vertical value, go across first.
Method. The y-intercept is where the line meets the vertical axis. In a real-life graph it is the value at the start (when the horizontal value is 0).
- Find where the line crosses the vertical axis. This is where the horizontal value is 0: the very start.
- Read the value. The line meets the vertical axis at 32.temperature in °F = 32°F when temperature in °C = 0°C
- Say what it means. Water freezes at 0°C, which is 32°F.The y-intercept is 32.
Check. The graph really does start at that height when the horizontal value is 0. ✓
Watch out. Describe the meaning in words: “the fixed charge” or “the starting amount”, not just the number.
Key words. The gradient of a line is how steep it is: rise (the change up or down) divided by run (the change across, to the right). A line sloping up from left to right has a positive gradient; sloping down has a negative gradient. In a real-life graph the gradient is the rate of change.
Method. The gradient of a real-life graph is rise ÷ run, and it is the rate of change: how much the vertical quantity changes for each 1 on the horizontal axis.
- Choose two points on the line. Choose points that are easy to read and far apart.(0, 32) and (40, 104)
- Draw a triangle and work out the rise and the run. Rise = change in the vertical value; run = change in the horizontal value.rise = 104 − 32 = 72
run = 40 − 0 = 40 - Divide the rise by the run. Gradient = rise ÷ run.gradient = 72 ÷ 40 = 1.8
- Say what the gradient means. The gradient is the change in the temperature in °F for each 1 unit increase in the temperature in °C. It is a rate of change.The gradient is 1.8°F per °C.
Check. For each 1 across, the line goes up 1.8, which matches the gradient. ✓
Watch out. Give the units of the gradient: it is (vertical units) per (horizontal unit), for example 1.8°F per °C.
Method. Find the formula from the intercept (32) and the gradient (1.8), then substitute.
- Find the intercept and gradient. The line crosses the vertical axis at 32. Rise ÷ run gives the gradient.c = 32, m = (104 − 32) ÷ 40 = 72 ÷ 40 = 1.8
- Write the formula. Use the letters C and F.F = 1.8C + 32
- Substitute C = 100. Multiply first, then add.F = 1.8 × 100 + 32 = 180 + 32 = 212
Check. Water boils at 100°C, which is 212°F. ✓
Watch out. This graph does not go through the origin, so it is not direct proportion.
Method. The y-intercept is where the line meets the vertical axis. In a real-life graph it is the value at the start (when the horizontal value is 0).
- Find where the line crosses the vertical axis. This is where the horizontal value is 0: the very start.
- Read the value. The line meets the vertical axis at 20.volume of water = 20 litres when time = 0 minutes
- Say what it means. The tank already holds 20 litres when the tap is first turned on.The y-intercept is 20.
Check. The graph really does start at that height when the horizontal value is 0. ✓
Watch out. Describe the meaning in words: “the fixed charge” or “the starting amount”, not just the number.
Key words. The gradient of a line is how steep it is: rise (the change up or down) divided by run (the change across, to the right). A line sloping up from left to right has a positive gradient; sloping down has a negative gradient. In a real-life graph the gradient is the rate of change.
Method. The gradient of a real-life graph is rise ÷ run, and it is the rate of change: how much the vertical quantity changes for each 1 on the horizontal axis.
- Choose two points on the line. Choose points that are easy to read and far apart.(0, 20) and (10, 60)
- Draw a triangle and work out the rise and the run. Rise = change in the vertical value; run = change in the horizontal value.rise = 60 − 20 = 40
run = 10 − 0 = 10 - Divide the rise by the run. Gradient = rise ÷ run.gradient = 40 ÷ 10 = 4
- Say what the gradient means. The gradient is the change in the volume of water for each 1 unit increase in the time. It is a rate of change.The gradient is 4 litres per minute.
Check. For each 1 across, the line goes up 4, which matches the gradient. ✓
Watch out. Give the units of the gradient: it is (vertical units) per (horizontal unit), for example 4 litres per minute.
Key words. The equation y = mx + c describes a straight line. m is the gradient (steepness) and c is the y-intercept (the value of y where the line crosses the y-axis, which is where x = 0).
Method. Find the y-intercept and the gradient, then write the equation in the form y = mx + c with the letters given in the question.
- Find the y-intercept c. Read where the line crosses the vertical axis.c = 20
- Find the gradient m. Use rise ÷ run on two points of the line.(60 − 20) ÷ 10 = 40 ÷ 10 = 4
- Write y = mx + c using the letters from the question. Replace x by t and y by V, since these stand for the real quantities.V = 4t + 20
Check. Put t = 10 into the formula: 4 × 10 + 20 = 60, which is the height of the line there. ✓
Watch out. Use the letters in the question, not x and y, if the question defines other letters.
Method. Put the known value into the formula and solve the equation for the letter you do not know.
- Write the formula and put in the known value. We know the vertical value, so replace the letter for it.50 = 4t + 20
- Subtract 20 from both sides. This removes the fixed amount so only the term with the letter is left.30 = 4t
- Divide both sides by 4. This leaves t on its own.t = 30 ÷ 4 = 7.5
Check. Substitute back: 4 × 7.5 + 20 = 50. ✓
Watch out. Undo the addition first, then the multiplication (reverse order).
Key words. The gradient of a line is how steep it is: rise (the change up or down) divided by run (the change across, to the right). A line sloping up from left to right has a positive gradient; sloping down has a negative gradient. In a real-life graph the gradient is the rate of change.
Method. The gradient of a real-life graph is rise ÷ run, and it is the rate of change: how much the vertical quantity changes for each 1 on the horizontal axis.
- Choose two points on the line. Choose points that are easy to read and far apart.(0, 80) and (16, 0)
- Draw a triangle and work out the rise and the run. Rise = change in the vertical value; run = change in the horizontal value.rise = 0 − 80 = −80
run = 16 − 0 = 16 - Divide the rise by the run. Gradient = rise ÷ run.gradient = −80 ÷ 16 = −5
- Say what the gradient means. The gradient is the change in the volume of water for each 1 unit increase in the time. It is a rate of change.The gradient is −5 litres per minute.
Check. For each 1 across, the line goes down 5, which matches the gradient. ✓
Watch out. Give the units of the gradient: it is (vertical units) per (horizontal unit), for example −5 litres per minute.
Method. Put the known value into the formula and solve the equation for the letter you do not know.
- Write the formula and put in the known value. We know the vertical value, so replace the letter for it.0 = −5t + 80
- Subtract 80 from both sides. This removes the fixed amount so only the term with the letter is left.−80 = −5t
- Divide both sides by −5. This leaves t on its own.t = −80 ÷ −5 = 16
Check. Substitute back: (−5) × 16 + 80 = 0. ✓
Watch out. Undo the addition first, then the multiplication (reverse order).
Method. Substitute t = 6 into the formula.
- Write the formula. It gives the volume left after t minutes.V = 80 − 5t
- Substitute. Replace t by 6. Multiply before subtracting.V = 80 − 5 × 6 = 80 − 30 = 50
- Write the answer with units. 50 litres
Check. Check with the graph: at 6 minutes the line is at 50. ✓
Watch out. Multiply first: 5 × 6 = 30, then subtract from 80.
Method. The speed is the gradient of the distance–time graph: distance ÷ time.
- Find the distance and time for the first section. From the start to A: read the values on the axes.distance = 12 km, time = 1 hour
- Divide. Speed = distance ÷ time.12 ÷ 1 = 12 km/h
Check. In 1 hour at 12 km/h you travel 12 km. ✓
Watch out. Speed is in km per hour: say “km/h”.
Method. The rider is stopped when the distance does not change, which is the horizontal section.
- Find the horizontal section. The line from A to B is flat: the distance stays at 12 km.A at 1 hour, B at 1.5 hours
- Subtract the times. 1.5 − 1 = 0.5 hours = 30 minutes
Check. The distance from home does not change between 1 and 1.5 hours, so she is resting. ✓
Watch out. Flat means stopped, not slow.
Method. On the way back, the distance falls from 12 km to 0 km. Speed = distance travelled ÷ time taken (ignore the sign).
- Find the time for the last section. From B (1.5 hours) to C (3 hours).3 − 1.5 = 1.5 hours
- Find the distance. She travels 12 km back home.12 km
- Divide. 12 ÷ 1.5 = 8 km/h
Check. At 8 km/h for 1.5 hours: 8 × 1.5 = 12 km. ✓
Watch out. The gradient is negative here (the distance is falling), but the speed is positive: 8 km/h.
Method. Find the distance at 2 hours by using the return speed from B (1.5 hours).
- Time since B. At 2 hours she has been returning for half an hour.2 − 1.5 = 0.5 hours
- Distance covered on the way back. Speed × time.8 × 0.5 = 4 km
- Distance from home. She started the return 12 km from home.12 − 4 = 8 km
Check. Read the graph: at 2 hours the line is at 8 km. ✓
Watch out. She is going towards home, so the distance falls.
Method. A direct proportion graph is a straight line through the origin. Check each graph for both features.
- Check graph A. Is it straight? Does it go through (0, 0)?A is a straight line through the origin.
- Check graph B. It is straight, but where does it cross the vertical axis?B crosses the vertical axis at 3, not 0, so it does not go through the origin.
- Check graph C. It is a curve.C is a curve, so it is not a straight line.
- Answer. Only A shows direct proportion. In A, doubling x doubles y.
Check. In A, y = 2x: when x = 3, y = 6; when x = 6, y = 12. Doubling x doubles y. ✓
Watch out. A straight line is not enough: it must also pass through the origin.
Key words. The gradient of a line is how steep it is: rise (the change up or down) divided by run (the change across, to the right). A line sloping up from left to right has a positive gradient; sloping down has a negative gradient. In a real-life graph the gradient is the rate of change.
Method. The gradient of a real-life graph is rise ÷ run, and it is the rate of change: how much the vertical quantity changes for each 1 on the horizontal axis.
- Choose two points on the line. Choose points that are easy to read and far apart.(0, 0) and (40, 56)
- Draw a triangle and work out the rise and the run. Rise = change in the vertical value; run = change in the horizontal value.rise = 56 − 0 = 56
run = 40 − 0 = 40 - Divide the rise by the run. Gradient = rise ÷ run.gradient = 56 ÷ 40 = 1.4
- Say what the gradient means. The gradient is the change in the cost for each 1 unit increase in the amount of petrol. It is a rate of change.The gradient is £1.40 per litre.
Check. For each 1 across, the line goes up 1.4, which matches the gradient. ✓
Watch out. Give the units of the gradient: it is (vertical units) per (horizontal unit), for example £1.40 per litre.
Key words. To read off a value from a graph, draw guide lines from one axis to the line and then across to the other axis.
Method. Read across from the graph: go up from the horizontal value to the line, then across to the vertical axis.
- Find 35 litres on the horizontal axis. The horizontal axis shows the amount of petrol.amount of petrol: 35 litres
- Go straight up to the line. This finds the point on the graph that has this horizontal value. Use a ruler and a dotted line.
- Go across to the vertical axis and read the value. The vertical axis shows the cost. The dotted line meets it at the answer.cost: £49
Check. Use the formula C = 1.4L: C = 1.4 × 35 = 49. ✓
Watch out. Read the scale carefully: count how much each small square is worth on each axis.
Key words. The equation y = mx + c describes a straight line. m is the gradient (steepness) and c is the y-intercept (the value of y where the line crosses the y-axis, which is where x = 0).
Method. Find the y-intercept and the gradient, then write the equation in the form y = mx + c with the letters given in the question.
- Find the y-intercept c. Read where the line crosses the vertical axis.c = 0
- Find the gradient m. Use rise ÷ run on two points of the line.(56 − 0) ÷ 40 = 56 ÷ 40 = 1.4
- Write y = mx + c using the letters from the question. Replace x by L and y by C, since these stand for the real quantities.C = 1.4L
Check. Put L = 40 into the formula: 75 × 40 = 56, which is the height of the line there. ✓
Watch out. Use the letters in the question, not x and y, if the question defines other letters.
Method. A conversion graph starts at the origin because zero in one unit is zero in the other.
- Think about the starting values. If you have £0 you have $0.£0 = $0
- Link this to the graph. The point (0, 0) is the origin, so the graph passes through it.The graph goes through the origin.
- Say what this tells you. The amount of dollars is a fixed multiple of the pounds, so doubling the pounds doubles the dollars: direct proportion.The two amounts are directly proportional, so D = 1.25p with no extra number added.
Check. A temperature graph does not do this because 0°C is 32°F, not 0°F. ✓
Watch out. Explain with a real value, not just “it is proportional”.
Exercise 12L Fares, tariffs and comparing lines
Calculator allowedKey words. To read off a value from a graph, draw guide lines from one axis to the line and then across to the other axis.
Method. Read across from the graph: go up from the horizontal value to the line, then across to the vertical axis.
- Find 4 miles on the horizontal axis. The horizontal axis shows the distance.distance: 4 miles
- Go straight up to the line. This finds the point on the graph that has this horizontal value. Use a ruler and a dotted line.
- Go across to the vertical axis and read the value. The vertical axis shows the cost. The dotted line meets it at the answer.cost: £11
Check. Use the formula C = 2n + 3: C = 2 × 4 + 3 = 11. ✓
Watch out. Read the scale carefully: count how much each small square is worth on each axis.
Method. Read from the graph in the other direction: go across from the vertical value to the line, then down to the horizontal axis.
- Find £15 on the vertical axis. The vertical axis shows the cost.cost: £15
- Go across to the line. This finds the point on the graph with this vertical value. Use a ruler and a dotted line.
- Go straight down to the horizontal axis and read the value. The horizontal axis shows the distance.distance: 6 miles
Check. Use the formula C = 2n + 3: 15 = 2n + 3, so n = 6. ✓
Watch out. Start from the correct axis: for a vertical value, go across first.
Method. The y-intercept is where the line meets the vertical axis. In a real-life graph it is the value at the start (when the horizontal value is 0).
- Find where the line crosses the vertical axis. This is where the horizontal value is 0: the very start.
- Read the value. The line meets the vertical axis at 3.cost = £3 when distance = 0 miles
- Say what it means. This is the fixed charge: £3 is paid before any miles are travelled.The y-intercept is 3.
Check. The graph really does start at that height when the horizontal value is 0. ✓
Watch out. Describe the meaning in words: “the fixed charge” or “the starting amount”, not just the number.
Key words. The gradient of a line is how steep it is: rise (the change up or down) divided by run (the change across, to the right). A line sloping up from left to right has a positive gradient; sloping down has a negative gradient. In a real-life graph the gradient is the rate of change.
Method. The gradient of a real-life graph is rise ÷ run, and it is the rate of change: how much the vertical quantity changes for each 1 on the horizontal axis.
- Choose two points on the line. Choose points that are easy to read and far apart.(0, 3) and (10, 23)
- Draw a triangle and work out the rise and the run. Rise = change in the vertical value; run = change in the horizontal value.rise = 23 − 3 = 20
run = 10 − 0 = 10 - Divide the rise by the run. Gradient = rise ÷ run.gradient = 20 ÷ 10 = 2
- Say what the gradient means. The gradient is the change in the cost for each 1 unit increase in the distance. It is a rate of change.The gradient is £2 per mile.
Check. For each 1 across, the line goes up 2, which matches the gradient. ✓
Watch out. Give the units of the gradient: it is (vertical units) per (horizontal unit), for example £2 per mile.
Key words. The equation y = mx + c describes a straight line. m is the gradient (steepness) and c is the y-intercept (the value of y where the line crosses the y-axis, which is where x = 0).
Method. Find the y-intercept and the gradient, then write the equation in the form y = mx + c with the letters given in the question.
- Find the y-intercept c. Read where the line crosses the vertical axis.c = 3
- Find the gradient m. Use rise ÷ run on two points of the line.(23 − 3) ÷ 10 = 20 ÷ 10 = 2
- Write y = mx + c using the letters from the question. Replace x by n and y by C, since these stand for the real quantities.C = 2n + 3
Check. Put n = 10 into the formula: 2 × 10 + 3 = 23, which is the height of the line there. ✓
Watch out. Use the letters in the question, not x and y, if the question defines other letters.
Method. Read both costs at 4 GB, then compare. The lower cost is the cheaper plan.
- Go up from 4 GB to each line. Use a ruler and read across to the cost axis for each.Plan A: £18, Plan B: £16
- Compare. Cheaper means the smaller number.£16 < £18, so Plan B is cheaper.
Check. Formulas: A = 2 × 4 + 10 = 18; B = 3 × 4 + 4 = 16. ✓
Watch out. Read both costs for the same amount of data.
Method. Read both costs at 8 GB, then compare.
- Go up from 8 GB to each line. Read across to the cost axis for each.Plan A: £26, Plan B: £28
- Compare. £26 < £28, so Plan A is cheaper.
Check. Formulas: A = 2 × 8 + 10 = 26; B = 3 × 8 + 4 = 28. ✓
Watch out. The cheaper plan changes: B is cheaper for small amounts of data, A for large amounts.
Method. The plans cost the same where the lines cross.
- Find the crossing point of the two lines. Both costs are equal at this point.
- Read its coordinates. The lines cross at 6 GB and £22.6 GB, £22
Check. Plan A: 2 × 6 + 10 = 22. Plan B: 3 × 6 + 4 = 22. ✓
Watch out. Give both the amount of data and the cost.
Method. Use the gradients and intercepts. The bigger the gradient, the faster the cost grows.
- Compare the monthly fees (y-intercepts). Plan B has the smaller fixed charge.£4 for B, £10 for A
- Compare the cost per GB (gradients). This is the rate of change.£3 per GB for B, £2 per GB for A
- Think about a heavy user. For a lot of data the cost per GB matters more than the fixed fee.B’s cost grows faster, so it ends up more expensive than A.
- Answer. A heavy user should choose Plan A (a lower cost per GB), even though its monthly fee is higher.
Check. At 10 GB: A = £30, B = £34. ✓
Watch out. Compare gradients for large amounts and intercepts for small amounts.
Method. Make a table of values from the formula, plot the points and join them.
- Write the formula. A fixed call-out charge plus an hourly rate.C = 25h + 40
- Make a table of values. Substitute each h.h = 0: 25 × 0 + 40 = 40
h = 1: 25 × 1 + 40 = 65
h = 2: 25 × 2 + 40 = 90
h = 3: 25 × 3 + 40 = 115
h = 4: 25 × 4 + 40 = 140
h = 5: 25 × 5 + 40 = 165h 0 1 2 3 4 5 C (£) 40 65 90 115 140 165 - Plot the points and join them. Choose a sensible scale: 1 square for 1 hour and 25 for £25. The line starts at (0, 40) on the vertical axis.
Check. The points are in a straight line, and the line starts at (0, 40). ✓
Watch out. Start the line on the vertical axis at the call-out charge, not at the origin.
Method. The fixed charge is the intercept and the hourly rate is the gradient.
- Find the fixed charge. This is paid whatever the time: the starting value.c = 5
- Find the rate of change. The cost goes up by £3 for every extra hour.m = 3
- Write the formula. y = mx + c with the context letters.C = 3h + 5
Check. When h = 0 the cost is £5 (the fixed fee). ✓
Watch out. Write the letters used in the question.
Method. Substitute the value into the formula and work it out.
- Write the formula. The formula tells us how to get the vertical quantity from the horizontal quantity.C = 3h + 5
- Substitute h = 4. Replace the letter by the number.C = 3 × 4 + 5
- Calculate. Multiply first, then add.C = 17
- Write the answer with its units. £17
Check. The point should lie on the graph: it matches the value you can read from the line. ✓
Watch out. Do the multiplication before the addition.
Method. Put the known value into the formula and solve the equation for the letter you do not know.
- Write the formula and put in the known value. We know the vertical value, so replace the letter for it.29 = 3h + 5
- Subtract 5 from both sides. This removes the fixed amount so only the term with the letter is left.24 = 3h
- Divide both sides by 3. This leaves h on its own.h = 24 ÷ 3 = 8
Check. Substitute back: 3 × 8 + 5 = 29. ✓
Watch out. Undo the addition first, then the multiplication (reverse order).
Method. The unlock fee is the starting value and the cost per minute is the gradient. Remember to write 25p as £0.25.
- Change 25p to pounds. The formula must use one unit throughout: pounds.25p = £0.25
- Write the formula. Starting value £1, rate £0.25 per minute.C = 0.25m + 1
- Substitute m = 18. Multiply first, then add.C = 0.25 × 18 + 1 = 4.5 + 1 = 5.5
- Write the answer. Money to 2 decimal places.£5.50
Check. 18 minutes is 18 quarters of a pound: £4.50, plus the unlock fee £1 = £5.50. ✓
Watch out. Do not mix pence and pounds in one calculation.
Method. Standing charge = starting value (intercept); price per m³ = gradient.
- Find the standing charge. A fixed amount paid however much water is used.c = 20
- Find the rate. Each m³ costs £1.80.m = 1.8
- Write the formula. C = 1.8w + 20
Check. With no water used the bill is £20. ✓
Watch out. The standing charge is the number on its own.
Method. Substitute the value into the formula and work it out.
- Write the formula. The formula tells us how to get the vertical quantity from the horizontal quantity.C = 1.8w + 20
- Substitute w = 150. Replace the letter by the number.C = 95 × 150 + 20
- Calculate. Multiply first, then add.C = 290
- Write the answer with its units. £290
Check. The point should lie on the graph: it matches the value you can read from the line. ✓
Watch out. Do the multiplication before the addition.
Method. Put the known value into the formula and solve the equation for the letter you do not know.
- Write the formula and put in the known value. We know the vertical value, so replace the letter for it.110 = 1.8w + 20
- Subtract 20 from both sides. This removes the fixed amount so only the term with the letter is left.90 = 1.8w
- Divide both sides by 1.8. This leaves w on its own.w = 90 ÷ 1.8 = 50
Check. Substitute back: 95 × 50 + 20 = 110. ✓
Watch out. Undo the addition first, then the multiplication (reverse order).
Method. A taxi charges a fixed amount as soon as you get in, before you have travelled any distance.
- Think about a journey of 0 miles. If the graph went through the origin, the cost of 0 miles would be £0.Through the origin: 0 miles costs £0
- Compare with the taxi. The taxi graph starts at £3 when the distance is 0.It starts at (0, 3), so there is a fixed charge of £3.
- Say what a graph through the origin would mean. There would be no fixed charge, and the cost would be directly proportional to the distance.A taxi graph through the origin would mean no fixed charge: the cost would simply be £2 per mile.
Check. A journey of 5 miles costs £13, not 5 × 2 = £10: the extra £3 is the fixed charge. ✓
Watch out. Straight line is not the same as direct proportion.