4 Graphs

Learning objectives

44A Graphical and numerical formsTranslate information between graphical and numerical form.44B Linear relationshipsUnderstand that y = mx + c represents a linear relationship.44C Plot two variablesPlot two variables, discrete or continuous, from experimental or other data.44D Slope and interceptDetermine the slope and intercept of a linear graph.44E Tangents and rate of changeUnderstand, draw and use a curve tangent's slope as a measure of rate of change.44F Area under a curveUnderstand the physical significance of the area between a curve and the x-axis and measure it by counting squares where appropriate.

A graph and a numerical table can represent the same paired measurements. Each plotted point has an $x$ -coordinate and a $y$ -coordinate, so translating between the forms means preserving each pair, its scale and its units.

Direction	Reliable method
Table $\to$ graph	Read one row as $(x,y)$ , locate $x$ horizontally, then $y$ vertically, and mark their intersection.
Graph $\to$ number	Start at the required axis value, move to the plotted point or curve, then project to the other axis and read its scale.

If a temperature–time graph passes through $(40\,\mathrm{s},32\,^{\circ}\mathrm{C})$ , the matching numerical statement is: at $40\,\mathrm{s}$ , the temperature is $32\,^{\circ}\mathrm{C}$ . A value read between labelled divisions must use the size of each small division, not merely the nearest printed label.

A value read from a curve is normally an estimate. Interpolation uses a value inside the measured range; extending the trend beyond the data is extrapolation and is less secure. Do not swap the coordinates or drop their units when translating.

The equation $y=mx+c$ represents a straight-line relationship. Its gradient $m$ is constant: every equal increase in $x$ produces the same change in $y$ . The intercept $c$ is the value of $y$ when $x=0$ .

$\Delta y=m\,\Delta x,\qquad y(0)=c$

For $y=2.5x+4$ , increasing $x$ by 3 increases $y$ by $2.5\times3=7.5$ , wherever the interval begins. The line crosses the $y$ -axis at 4. In a physical graph, $m$ carries the units of the $y$ -quantity divided by the units of the $x$ -quantity, while $c$ carries the units of $y$ .

A positive gradient rises from left to right, a negative gradient falls, and zero gradient gives a horizontal line. Changing $c$ shifts the line vertically without changing its gradient.

A straight line is not automatically direct proportion. Direct proportion has the special form $y=mx$ , so $c=0$ and the line passes through the origin. A non-zero intercept still gives a linear relationship, but not direct proportionality.

A two-variable graph places every data pair at one coordinate. Usually the independent or chosen variable goes on the $x$ -axis and the measured response on the $y$ -axis; the labels must name each quantity and its unit.

Choose simple, continuous linear scales that cover the whole data range and use more than half the available grid in both directions. Mark equal numerical steps at equal distances, plot each coordinate to the nearest half-small-square where the grid permits, and check every point against the table before adding any line or curve.

For wire length in mm and force in N, a row $(15,0.92)$ is plotted by moving to $15\,\mathrm{mm}$ on the horizontal axis and $0.92\,\mathrm{N}$ on the vertical axis. Discrete values occur as separate allowed values; continuous measurements can take values between recorded readings. Both still require correctly scaled axes.

A line of best fit represents the overall trend and need not pass through every point; a dot-to-dot join instead treats every fluctuation as meaningful. Do not use an irregular scale, omit units, or force a line through the origin unless the evidence or physical model requires it.

The gradient of a straight graph measures how much the vertical quantity changes per unit change in the horizontal quantity. The intercept is where the line crosses an axis; in $y=mx+c$ , $c$ is specifically the $y$ -intercept at $x=0$ .

$m=\frac{\Delta y}{\Delta x}=\frac{y_2-y_1}{x_2-x_1}$

Choose two well-separated points on the straight line, not automatically two experimental points. Draw a large right-angled gradient triangle, read both coordinate pairs from the axis scales, subtract in the same direction, and divide vertical change by horizontal change. State the sign and the compound unit.

If a velocity–time line passes through $(2\,\mathrm{s},5\,\mathrm{m\,s^{-1}})$ and $(8\,\mathrm{s},17\,\mathrm{m\,s^{-1}})$ , then $m=(17-5)/(8-2)=2\,\mathrm{m\,s^{-2}}$ . Extending or reading the line at $t=0$ gives the intercept $c=1\,\mathrm{m\,s^{-1}}$ .

The triangle dimensions are changes, not the coordinate values themselves. Reversing only one subtraction gives the wrong sign; reversing both leaves the gradient unchanged. A larger triangle usually reduces the percentage reading uncertainty.

A curve has a changing gradient, so its rate of change depends on position. A tangent drawn at one point has the same local direction as the curve there; the tangent's gradient estimates the instantaneous rate at that point.

Mark the point of interest and place a straight line so that it just follows the curve's direction there, with the curve lying evenly around the line locally. Extend the tangent, choose two well-separated points on the tangent, then calculate $\Delta y/\Delta x$ using the axis scales and units.

$\text{instantaneous rate at }x_0\approx\frac{y_2-y_1}{x_2-x_1}\quad\text{using two points on the tangent at }x_0$

On a distance–time graph, the tangent gradient has units $\mathrm{m/s}$ and estimates speed at that instant. A steeper positive tangent means a larger positive rate; a horizontal tangent gives zero rate; a downward tangent gives a negative rate under the chosen sign convention.

A chord joining two points on the curve gives an average rate over an interval, not the instantaneous rate at one point. Calculate with points on the tangent—even if they are not data points—and do not assume a tangent must touch the curve only once globally.

The area between a curve and the $x$ -axis combines the two plotted quantities. Its physical meaning and unit come from multiplying the vertical-axis quantity by the horizontal-axis quantity; the context decides what that product represents.

$\text{area unit}=(\text{$y$-axis unit})(\text{$x$-axis unit})$

For a velocity–time graph, $(\mathrm{m/s})(\mathrm{s})=\mathrm{m}$ , so area represents displacement. A constant velocity of $6\,\mathrm{m/s}$ for $4\,\mathrm{s}$ gives a rectangular area $6\times4=24\,\mathrm{m}$ . A triangular or trapezium section can be calculated geometrically.

For an irregular curve, find the value represented by one small grid square, count complete squares, combine partial squares into approximate wholes, and multiply the total by the value per square. Using smaller squares or sensible trapezia can improve the estimate while keeping the curve boundary visible.

The shaded shape has no automatic physical label: determine it from the axis product. Area below the $x$ -axis is negative in a signed total, so on a velocity–time graph it represents displacement in the opposite direction; total distance would require adding the magnitudes of the separate areas.

Next Topic

—5 Geometry and trigonometryRead

4 Graphs

Learning objectives

Move between a graph and its numbers

Read the straight-line model $y=mx+c$

Plot two measured variables accurately

Find gradient and intercept from a straight line

Use a tangent for an instantaneous rate

Give physical meaning to area under a graph

Next Topic