Learn4024Paper 1 · Non-calculatorCoordinate geometryGradients of curves and graphical solutions

Coordinate geometry · Subtopic 4 of 4

Gradients of Curves and Graphical Solutions

A curve's steepness changes, so its gradient at a point is the gradient of the tangent there: draw it with a ruler, pick two far-apart points on it, and find rise over run. Where two graphs cross, both equations are true at once, so the crossing points solve the equation formed by setting them equal.

Cambridge 4024 Paper 1Cambridge 4024 Paper 2Edexcel 4MA1 Papers 1H, 2HAQA 8300 Papers 1–3≈ 30 minRevision 6 · updated 2026-10-27

Section 1 of 14

Learning objectives

Draw a tangent to a curve and estimate its gradient.
Interpret the gradient as a rate of change (speed, acceleration).
Solve equations by reading where graphs cross.
Find the straight line to draw to solve a related equation.

Pause and think

A car's distance–time graph is curved. At one moment the curve is steep, later it's flat. What is the car doing at each moment?

Section 2 of 14

Core idea

Start hereWhere the graphs cross

Some equations are hard to solve with algebra. But if you draw both sides as graphs, the solutions are simply where the graphs meet. It's an estimate, but a quick and visual one.

The big idea

Solve $f(x)=g(x)$ by reading where $y=f(x)$ and $y=g(x)$ cross. Estimate a gradient by drawing a tangent.

1. Intersections

Solving x² − 2x = k by drawing y = k

k = -0.9

Each crossing gives a solution. When the line is below

-1

there are none.

Worked example $x^2-2x=3$ from the graph of $y=x^2-2x$

Solve graphically.

Goal: $x$ values.
Strategy: Draw $y=3$ .

$y=3$
Right-hand side as a line.
$x=-1,\ 3$
Read the crossings.

Answer

-1

and

3

✓ Check

(x-3)(x+1)=0

✓.

Worked exampleChoosing the line

Which line, drawn on $y=x^2-2x$ , solves $x^2-3x-1=0$ ?

Goal: Line.
Strategy: Rearrange to "curve = line".

$x^2-3x-1=0\iff x^2-2x=x+1$
Add $x+1$ to both sides.
$y=x+1$
The other side.

Answer

y=x+1

✓ Check

x^2-2x-(x+1)=x^2-3x-1

✓.

2. Gradient by tangent

Estimating a gradient with a tangent

Gradient

=\dfrac{8}{2}=4

. On paper you draw the tangent by eye, so answers are estimates.

Worked exampleEstimating a gradient

A tangent drawn to a curve passes through $(1,2)$ and $(4,11)$ .

Goal: Gradient of the curve there.
Strategy: Rise over run on the tangent.

$\frac{11-2}{4-1}$
Two far-apart points on the tangent.
$=3$
Divide.

AnswerAbout

3

✓ CheckFar-apart points reduce reading error ✓.

Common misconceptionA chord, not a tangent The line must touch the curve at one point only, not cut through it.

Pause and think

Why are graphical answers estimates?

Summary

Rearrange to "curve = line".
Tangent for gradients.

Tangent: Line touching a curve.

HistoryFermat's tangents. In the

1630

s Pierre de Fermat found tangents to curves by imagining two points on the curve sliding together. Fifty years later Newton and Leibniz turned this idea into calculus, which you'll meet at A Level.

Fun factThe fly on the ceiling. A popular story says René Descartes invented coordinates while lying in bed watching a fly, realising he could describe its position by two distances from the walls. True or not, his

1637

book joined algebra and geometry for good: equations became pictures.

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