M.4-Graphs, gradients, intercepts and areas

Identify x and y quantities with units, use most of both axes with simple scales, plot accurately, then choose a straight best-fit line or smooth curve that represents the trend rather than joining points dot-to-dot.

$gradient=Δy/Δx units=(y-axis unit)/(x-axis unit) Use two well-separated points on the best-fit line, not necessarily measured data points.$

For an instantaneous rate on a curve, draw a tangent at the required point and calculate its gradient using a large triangle. An intersection is where two plotted quantities satisfy both relationships; an intercept is the model value when the other axis variable is zero.

$area under y against x ≈ Σ y Δx units=(y-axis unit)(x-axis unit) It represents a physical quantity only when the governing relation makes that product meaningful.$

A straight line indicates y=mx+c; proportionality requires a straight line through the origin. Curvature may signal powers, reciprocals, exponentials, changing rate or a limited model—use physics and any requested transformation to distinguish them.

A visually straight trend is not automatically direct proportionality. Always test the intercept within uncertainty and attach compound units to gradient and area.

ConceptA-Level CAIE Physics A2