📈 Positive trend
As one variable rises, the other rises. Length of a wire against its resistance: longer wire, larger resistance.
Maths Skills » Maths Skills: Graphs and Shapes
What you'll learn this session
Study time: 30 minutes
Edexcel spec: Mathematical skills (Appendix 4), third part
- Spot a pattern in a scatter diagram and plot your own graph correctly
- Find the slope and intercept of a straight line, and the slope of a tangent to a curve
- Measure the area between a curve and the x-axis by counting squares
- Use angles, areas of triangles and rectangles, and the surface area and volume of a cube
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Unlock This CourseScatter diagrams: spotting a pattern
This is the last maths skills lesson. The physics behind each graph is taught in other lessons. Here you practise the graph and shape skills themselves.
Key terms:
- Scatter diagram: a graph of points showing two variables measured together, used to see if there is a pattern or trend between them.
- Slope (gradient): how steep a line is, found by change in y ÷ change in x.
- Intercept: the value of y where the line crosses the y-axis (where x = 0).
- Tangent: a straight line that touches a curve at one point without cutting across it.
📉 Negative trend
As one variable rises, the other falls. Temperature of a thermistor against its resistance: hotter, smaller resistance.
❓ No pattern
The points are spread all over the grid. Shoe size against reaction time shows no link.
A trend does not prove that one variable causes the other. It only tells you they change together.
Plotting a graph and moving between table and graph
You often swap between a table of numbers and a graph. Plotting a table turns numbers into a graph. Reading values off a graph turns it back into numbers. Follow these steps every time.
- Put the variable you change on the x-axis and the variable you measure on the y-axis.
- Choose scales that use more than half of the grid in each direction, going up in 1s, 2s, 5s or 10s. Never use awkward steps such as 3s or 7s.
- Label each axis with the quantity and its unit, for example Time (s).
- Plot each point with a small neat cross, placed accurately.
- Draw a line of best fit with a sharp pencil (remember from Investigating Current and Voltage, it follows the pattern with points spread evenly either side): a straight line if the points look straight, a smooth curve if they curve. Do not join the dots with a ruler unless told to.
Data can be discrete or continuous. Discrete data can only take certain values, such as the number of coils in a spring, so plot separate points or bars. Continuous data can take any value, such as time or temperature, so plot points and draw a line or curve of best fit.
Describe a graph in words by saying what happens to y as x increases, and use numbers from the graph to back it up.
y = mx + c: the straight line
A straight-line graph has the equation y = mx + c. Here m is the slope and c is the intercept. If c = 0 the line goes through the origin and y is directly proportional to x.
Worked example: slope and intercept
A toy car moves along a track. Its distance d from a marker is measured as it moves.
| Time t (s) | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| Distance d (m) | 0.5 | 2.5 | 4.5 | 6.5 | 8.5 |
Plotted, the points lie on a straight line. It crosses the d-axis at 0.5 m, so the intercept c = 0.5 m.
Choose two points far apart on the line: (0, 0.5) and (4, 8.5).
slope = change in d ÷ change in t = (8.5 − 0.5) ÷ (4 − 0) = 8.0 ÷ 4 = 2.0 m/s
The equation is d = 2.0 t + 0.5.
The slope has a unit: the unit of y divided by the unit of x. Use points far apart, because small triangles make the answer less accurate.
The slope of a tangent to a curve
The slope of a curve changes from place to place. To find its steepness at one point, draw a tangent there. The slope of the tangent is the rate of change at that point.
- Place a ruler so it just touches the curve at the point you want.
- Adjust it until the curve is on both sides of the ruler, by the same amount, near the point. Draw the line.
- Make it long, then pick two points on the tangent, far apart.
- Calculate change in y ÷ change in x.
Worked example: a cooling curve
A cup of hot water cools. On its temperature-time curve, a tangent drawn at 10 minutes passes through (4 min, 70 °C) and (16 min, 46 °C).
slope = (46 − 70) ÷ (16 − 4) = −24 ÷ 12 = −2.0 °C/min
At 10 minutes the water is cooling at 2.0 °C every minute. The negative sign means the temperature is falling.
Area between a curve and the x-axis
On some graphs the area under the line has a physical meaning. Find what it means by multiplying the units of the two axes. For a speed-time graph, speed (m/s) × time (s) gives metres, so the area is distance travelled.
For a curve, count squares:
- Work out what one square is worth: multiply the value of one square across by the value of one square up.
- Count the whole squares under the curve.
- Count the part squares: either count each part square as a half, or count those that are half or more as one and ignore the smaller ones.
- Multiply the total number of squares by the value of one square.
Worked example: counting squares
On a speed-time graph, one square is 2 m/s up and 1 s across, so one square = 2 m. A curve encloses 14 whole squares and 8 part squares. Counting each part square as a half:
total = 14 + (8 ÷ 2) = 18 squares
distance = 18 × 2 = 36 m
Angles, areas and volumes
Angles are measured in degrees (°). A full turn is 360° and a right angle is 90°. Use a protractor from the zero line, and read the scale that starts at 0. In optics, angles are measured from the normal, not from the surface.
Drawings on paper are 2D. A 3D object can be shown as a 2D drawing in different ways, such as the view from the front, the view from above, or a cube sketched with its hidden edges dashed. Practise picturing the object turning around.
📏 Area
Rectangle: area = length × width.
Triangle: area = ½ × base × height. The height is measured at right angles to the base.
🎲 Cube of side L
Volume = L × L × L = L3.
Surface area = 6 × L2, because a cube has six identical square faces.
Worked example: areas
A rectangular solar panel is 2.0 m long and 0.50 m wide.
area = 2.0 × 0.50 = 1.0 m2
A triangular sail has base 0.60 m and height 0.40 m.
area = ½ × 0.60 × 0.40 = 0.12 m2
Worked example: a cube
A cube has sides of 0.20 m.
volume = 0.20 × 0.20 × 0.20 = 0.0080 m3
surface area = 6 × 0.20 × 0.20 = 0.24 m2
Common mistakes
Common mistakes
- Joining dots with a ruler instead of drawing one smooth line of best fit.
- Forgetting units on axes or on the slope.
- Using points on the line that are too close together, or reading values from the table instead of from the line.
- Reading the intercept at the wrong place: it is where the line meets the y-axis, at x = 0.
- Drawing a tangent that cuts the curve instead of just touching it.
- Using the length of a slanted side as the height in the triangle formula. The height is at right angles to the base.
- Writing L2 for the volume of a cube, or forgetting the 6 in the surface area.
Exam-style question
A trolley moves away from a sensor. Its distance d from the sensor against time t gives a straight line on a graph. The line passes through (0 s, 1.5 m) and (4.0 s, 7.5 m).
(a) State the intercept of the line. [1 mark]
(b) Calculate the slope of the line. Include the unit. [2 marks]
(c) Write the equation of the line in the form y = mx + c, using d and t. [1 mark]
(d) A cube has sides of 3.0 cm. Calculate its volume. [2 marks]
Model answer
(a) 1.5 m (1)
(b) slope = (7.5 − 1.5) ÷ (4.0 − 0) (1) = 1.5 m/s (1)
(c) d = 1.5 t + 1.5 (1)
(d) volume = 3.0 × 3.0 × 3.0 (1) = 27 cm3 (1)
Exam tip
When a question gives two points on a line, you do not need the graph: change in y ÷ change in x gives the slope, and the y-value at x = 0 is the intercept. Always show the subtraction and write the unit.