Year 12 Functions and relations: Intersection of two graphs

An intersection point is where two graphs share the same x-value and the same y-value. That means solving an intersection problem usually comes down to setting the two function rules equal and solving the resulting equation. The graph gives the geometric meaning, but the algebra gives the exact answer. Year 12 questions often use this idea to connect graphing, solving equations, and interpretation in one step.

Work through the sections in order, then try the linked Year 12 skill quizzes after each section.

If the graphs of

y = f (x)

and

y = g (x)

intersect, then

f (x) = g (x)

at those intersection x-values.

Main method

Set the two expressions equal.
Solve the resulting equation for x.
Substitute back to find the matching y-value.
Interpret how many intersections the graphs have.

Skill quiz: Intersection of two graphs

Practise the skills used in this section before continuing.

Intersection of two graphs

Worked example 1

Problem: Find the points of intersection of

y = x^{2}

and

y = x + 2

Set the functions equal: $x^{2} = x + 2$ .
Rearrange: $x^{2} - x - 2 = 0$ .
Factor: $(x - 2) (x + 1) = 0$ .
So $x = 2$ or $x = - 1$ .
Substitute into $y = x + 2$ .
For $x = 2$ , $y = 4$ .
For $x = - 1$ , $y = 1$ .

Answer: the intersection points are

(2, 4)

and

(- 1, 1)

Skill quiz: Intersection of two graphs

Try these questions using the method from the worked example.

Intersection of two graphs

Worked example 2

Problem: Determine whether

y = x^{2} + 1

and

y = - x + 1

intersect.

Set the functions equal: $x^{2} + 1 = - x + 1$ .
Simplify: $x^{2} + x = 0$ .
Factor: $x (x + 1) = 0$ .
So $x = 0$ or $x = - 1$ .
Substitute into $y = - x + 1$ .
This gives points $(0, 1)$ and $(- 1, 2)$ .

Answer: yes, the graphs intersect at

(0, 1)

and

(- 1, 2)

Skill quiz: Intersection of two graphs

Try these questions using the method from the worked example.

Intersection of two graphs

Tangent intersections and repeated solutions

Not every intersection question produces two different points. Sometimes one graph just touches another before turning away. Algebraically, this often appears as a repeated factor or repeated root. Graphically, it means the two functions share a point without properly crossing there. Recognising this possibility is important because it explains why an equation can have one repeated solution even when both graphs are still meeting.

Skill quiz: Intersection of two graphs

Practise the skills used in this section before continuing.

Intersection of two graphs

Worked example 3

Problem: Find the intersection of

y = x^{2}

and

y = 2 x - 1

Set the rules equal: $x^{2} = 2 x - 1$ .
Rearrange: $x^{2} - 2 x + 1 = 0$ .
Factor: ${(x - 1)}^{2} = 0$ .
So the only x-value is $x = 1$ .
Substitute back: $y = 2 (1) - 1 = 1$ .
The repeated factor shows the line is tangent to the parabola at that point.

Answer: the graphs touch at exactly one point,

(1, 1)

Skill quiz: Intersection of two graphs

Try these questions using the method from the worked example.

Intersection of two graphs

Common traps

Solving for x but forgetting to find the corresponding y-values.
Substituting into the wrong formula and creating an arithmetic inconsistency.
Reading a graph as if the intersection count must match what "looks" right without algebra.

Skill quiz: Intersection of two graphs

Use these questions to check the errors described above.

Intersection of two graphs

Revision focus

Intersection questions are one of the clearest examples of functions moving between graph and equation. The graph tells you how many solutions may exist, while the equation gives the exact coordinates. A good solution respects both views. If the algebra gives two intersections but your sketch only allows one, one of them needs to be checked again.

It is also useful to remember that intersection problems are often disguised equation problems. If two curves meet, they are sharing a common output. Setting those outputs equal is the whole idea.

Skill quiz: Intersection of two graphs

Apply the checks from this section while working through the quiz.

Intersection of two graphs