4.4PureCore

Points of Intersection

To find where two graphs meet, set their equations equal and solve. The number of solutions tells you how many times the curves cross — connecting algebra to the picture once again.

25 min Graphs and Transformations

Edexcel AS Level Maths: 4.4 Points of IntersectionCurated by Kepler, not produced by us. Watch it before the notes below.

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What you'll be able to do

Find intersection points by setting equations equal
Solve the resulting equation (often a polynomial)
Find both coordinates of each intersection
Determine the number of intersections

Set the equations equal

If two curves both equal $y$ , then at an intersection their right-hand sides must be equal. Setting $f(x) = g(x)$ gives an equation whose solutions are the $x$ -coordinates of the crossings.

Example 1Find where

y = x^2

meets

y = x + 6

1Set equal:

x^2 = x + 6 \Rightarrow x^2 - x - 6 = 0

2Factorise:

(x - 3)(x + 2) = 0

, so

x = 3

x = -2

3Find

y

from

y = x + 6

y = 9

y = 4

Answer

(3, 9)

and

(-2, 4)

Tip — Use the simpler equation to find each y-coordinate once you have the x-values.

How many intersections?

The number of real solutions equals the number of intersection points. For a line and a curve this links back to the discriminant; for two curves it depends on the resulting equation.

Intersections with the axes

A special case is finding where a single curve meets the axes: set $y = 0$ for the $x$ -intercepts and $x = 0$ for the $y$ -intercept — exactly intersection with the lines $y = 0$ and $x = 0$ .

Formula recap

f(x) = g(x)

Set equal to find intersection x-values.

\#\text{solutions} = \#\text{intersections}

Count the crossings.

Common mistakes to avoid

Giving only the x-coordinates of the intersection.

Each intersection is a point — substitute back to find y too.

Setting the equations equal incorrectly when one is not in y = … form.

Rearrange both to make y the subject (or substitute) before equating.

Key takeaways

Set f(x) = g(x) and solve for the x-coordinates.
Substitute back to get each y-coordinate.
The number of solutions = number of intersection points.

Test yourself

Answer a few questions, then review anything you missed.

Next step

Check your understanding

Work through questions on Points of Intersection. Use the feedback to decide what needs another look.

5 past-paper questions · 13 marks

Practise this topic

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