Applied graphs, gradients and areas

Algebra • Graphs

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Why is choosing clear points important when estimating gradients from graphs?

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Clear points reduce reading error and produce a more accurate rise/run calculation; avoid fractional or poorly defined grid intersections.

Key concepts

What you'll likely be quizzed about

Graphs in real contexts

Graphs map an independent variable on the horizontal axis to a dependent variable on the vertical axis. Distance–time graphs display position versus time; velocity–time graphs display speed (with sign) versus time; financial rate graphs display flow rates such as income per month versus time. Domain restrictions and units define where the graph applies and how results translate to the real world. Graph shape determines interpretation. Straight-line segments represent constant rate. Curved sections represent changing rates. Points where the curve crosses an axis or has asymptotes indicate sign changes, undefined values or long-term behaviour.

Reciprocal and exponential graphs

Reciprocal graphs follow y = k/x and form hyperbolas with vertical and horizontal asymptotes. As x approaches zero, |y| grows large and the model becomes invalid if the real context forbids division by zero. Reciprocal graphs model quantities that decrease rapidly as the independent variable increases and that cannot take zero for x. Exponential graphs follow y = A·B^x (B>0). Growth occurs if B>1 and decay if 0<B<1. Exponential models represent compound growth, radioactive decay or depreciation. Multiplicative changes over equal intervals imply exponential behaviour, and extrapolation far beyond data points risks large errors.

Gradients as rates of change

Gradient equals rise over run: change in vertical value divided by change in horizontal value. For a straight line through two points (x1, y1) and (x2, y2), gradient = (y2 − y1)/(x2 − x1). Units of gradient combine the units of the axes (e.g., metres per second for a distance–time gradient). For curves, the average gradient on an interval equals the gradient of the secant line between two points. Instantaneous rate estimation uses a small interval around the point and treats the chord as an approximation of the tangent. No calculus methods are required; accuracy improves as the interval shrinks.

Areas under graphs as accumulation

Area under a rate-versus-time graph equals accumulated quantity. For a velocity–time graph, area under the curve between two times equals displacement (taking sign into account). For a flow-rate graph such as income per month, area equals total income over the interval. Exact areas for simple shapes (rectangles, triangles, trapezia) compute accumulated quantities precisely. For curved regions, numerical methods such as the trapezium rule approximate the area by dividing the interval into small sections and summing the areas of trapezia.

Estimation methods and limitations

Gradient estimation uses two clearly read points to compute rise/run. Smaller intervals give closer approximations of instantaneous rate. Reading errors, coarse scales and rounded values reduce accuracy. Avoid estimating gradients at steep points using far-apart points. Area estimation uses equal or unequal subdivisions; the trapezium rule provides a simple and effective approximation. Errors arise when sections are large or the function varies rapidly. Interpretation requires checking units and sign conventions (negative area indicates opposite direction or loss).

Key notes

Important points to keep in mind

Gradient units equal vertical units divided by horizontal units; always state units with answers.

Average gradient on [a,b] equals (y(b) − y(a)) ÷ (b − a); use small intervals to approximate instantaneous rate.

Area under a rate graph equals total accumulated quantity; use signed area for direction-sensitive quantities.

Use the trapezium rule for curved areas: area ≈ Σ (h/2)·(y_left + y_right).

Identify asymptotes and domain restrictions for reciprocal and shifted reciprocal graphs.

Recognise exponential growth (B>1) and decay (0<B<1) from curve shape and doubling/halving behaviour.

Split intervals at axis crossings before summing areas to handle sign changes correctly.

Avoid excessive extrapolation beyond measured data; check model assumptions before extending a graph.

Choose points on clear grid intersections when calculating gradients to minimise reading error.

Express displacement from velocity–time graphs as net area; use separate positive and negative areas if needed.