1.4.2 Intersection points with axes

What you'll learn

What an intersection point with an axis means.
How to find where a curve meets the $x$ -axis by setting $y = 0$ .
How to find where a curve meets the $y$ -axis by setting $x = 0$ .
How to handle repeated roots, restrictions and curves given implicitly.

Coordinates and axes

A point in the coordinate plane is written in the form $(x, y)$ , where:

the $x$ -coordinate gives its horizontal position;
the $y$ -coordinate gives its vertical position.

Every point on the $x$ -axis has a vertical position of zero, so its $y$ -coordinate is zero. Similarly, every point on the $y$ -axis has a horizontal position of zero, so its $x$ -coordinate is zero.

Definition

Axis intersection

An intersection point with an axis is a point where a graph and that coordinate axis have a point in common.

The graph below meets the $x$ -axis at points of the form $(a, 0)$ , $(b, 0)$ and $(c, 0)$ . It meets the $y$ -axis at a point of the form $(0, d)$ .

A curve crossing the x-axis at three marked points and the y-axis at one marked point

Key Idea

The two substitutions

To find intersections with the $x$ -axis, set $y = 0$ and solve for $x$ .
To find intersections with the $y$ -axis, set $x = 0$ and solve for $y$ .

Intersections with the y-axis

Suppose a curve is given in the form

y = f (x) .

On the $y$ -axis, $x = 0$ . Substituting this into the equation gives

y = f (0) .

Therefore, provided $f (0)$ is defined, the curve meets the $y$ -axis at $(0, f (0))$ .

For a polynomial, this is particularly quick: substituting $x = 0$ removes every term containing $x$ , leaving only the constant term.

Example

Finding a y-axis intersection

Find where the curve

y=2x^3-5x^2+7x-4

meets the $y$ -axis.

A point on the $y$ -axis has $x = 0$ , so substitute this into the equation:
$y=2(0)^3-5(0)^2+7(0)-4.$
Simplifying gives the equation $y = - 4$ .
The intersection point therefore satisfies both $x = 0$ and $y = - 4$ , so its coordinates are $(0, - 4)$ .

Tip

Polynomial shortcut

For a polynomial written as $y = f (x)$ , the constant term is the $y$ -coordinate of its intersection with the $y$ -axis.

Intersections with the x-axis

On the $x$ -axis, $y = 0$ . For a curve $y = f (x)$ , its $x$ -axis intersections therefore satisfy

f (x) = 0.

The values of $x$ that satisfy this equation are called the roots or zeros of the function.

Definition

Root of a function

A root of $f (x)$ is a value of $x$ for which $f (x) = 0$ . Each real root gives an intersection point with the $x$ -axis.

Finding these intersections may require factorising, using the quadratic formula, completing the square, or applying another equation-solving technique.

Example

Finding two x-axis intersections

Find where the curve

y=x^2-5x+6

meets the $x$ -axis.

Points on the $x$ -axis have $y = 0$ , so solve
$x^2-5x+6=0.$
Factorise the quadratic:
$(x - 2) (x - 3) = 0.$
A product is zero when at least one factor is zero, giving $x = 2$ or $x = 3$ .
In each case the $y$ -coordinate is zero, so the intersection points are $(2, 0)$ and $(3, 0)$ .

Common Mistake

Giving roots instead of points

The values $x = 2$ and $x = 3$ are the roots, but the intersection points are $(2, 0)$ and $(3, 0)$ . Check whether the question asks for values of $x$ or coordinates.

How many x-axis intersections are there?

A curve may have:

no real intersections with the $x$ -axis;
one intersection;
two or more intersections, depending on the type of function.

For a quadratic curve, the discriminant $b^2-4ac$ determines the number of real roots of

ax^2+bx+c=0.

If $b^2-4ac>0$ , there are two distinct $x$ -axis intersections.
If $b^2-4ac=0$ , there is one repeated root and the curve touches the axis.
If $b^2-4ac<0$ , there are no real $x$ -axis intersections.

Example

Recognising a point of contact

Find the intersections of

y=x^2-6x+9

with the coordinate axes.

For the $x$ -axis, set $y = 0$ and factorise:
$x^2-6x+9=(x-3)^2=0.$
The only root is $x = 3$ . It is a repeated root, so the curve touches the $x$ -axis at $(3, 0)$ rather than crossing it there.
For the $y$ -axis, set $x = 0$ :
$y=(0)^2-6(0)+9=9.$
The intersections are therefore $(3, 0)$ and $(0, 9)$ .

Key Idea

Touching still counts

A curve does not need to cross an axis to intersect it. A point where the curve just touches the axis is still an intersection point.

Curves not written as y equals f of x

Some curves are given by an implicit equation, in which $x$ and $y$ appear together rather than with $y$ isolated. For example,

x^2+xy+y^2=12.

The same axis rules still apply: substitute $y = 0$ for the $x$ -axis and $x = 0$ for the $y$ -axis.

Example

Intersections of an implicit curve

Find all intersections of

x^2+xy+3y^2=12

with the coordinate axes.

On the $x$ -axis, $y = 0$ , so the equation becomes
$x^2=12.$
Hence $x=±23x=\pm 2\sqrt{3}$ , giving the points $(−23,0)(-2\sqrt{3},0)$ and $(23,0)(2\sqrt{3},0)$ .
On the $y$ -axis, $x = 0$ , so the equation becomes
$3y^2=12.$
Therefore $y^2=4$ , so $y=±2y=\pm 2$ .
The complete set of axis intersections is
$(−23,0),(23,0),(0,−2),(0,2).(-2\sqrt{3},0),\quad (2\sqrt{3},0),\quad (0,-2),\quad (0,2).$

Common Mistake

Losing a negative solution

When an equation simplifies to $x^2=k$ with $k > 0$ , both $x=kx=\sqrt{k}$ and $x=−kx=-\sqrt{k}$ must be considered. Missing the negative root loses an intersection.

Restrictions and undefined values

You must check that each proposed coordinate is allowed by the original equation. This matters especially for rational functions, square roots and logarithms.

For example, the function

y=x+1xy=\frac{x+1}{x}

has no $y$ -axis intersection because substituting $x = 0$ would require division by zero. The graph is not defined on the $y$ -axis.

Common Mistake

Substitution may be invalid

Setting $x = 0$ does not guarantee a $y$ -axis intersection. If the original function is undefined at $x = 0$ , the graph cannot meet the $y$ -axis.

Intersections at the origin

The origin is the point $(0, 0)$ , where the two coordinate axes meet.

If a curve passes through the origin, that single point is an intersection with both axes. It should not be listed twice when you give the distinct intersection points.

For a function $y = f (x)$ , the graph passes through the origin exactly when

f (0) = 0.

Exam technique

In the exam

Decide which axis is involved: use $y = 0$ for the $x$ -axis and $x = 0$ for the $y$ -axis.
Solve the resulting equation completely, including negative, repeated or exact roots where appropriate.
Return to the original equation to reject undefined values, then give coordinates rather than only roots when points are requested.
Check your answers quickly: every $x$ -axis point must have $y = 0$ , and every $y$ -axis point must have $x = 0$ .

Self review

Check yourself

Where does $y=x^2+x-6$ meet each coordinate axis?
How many $x$ -axis intersections does $y=2x^2+4x+5$ have?
Why does $y=3x−2y=\frac{3}{x-2}$ have no $x$ -axis intersection?

What you'll learn

What an intersection point with an axis means.
How to find where a curve meets the $x$ -axis by setting $y = 0$ .
How to find where a curve meets the $y$ -axis by setting $x = 0$ .
How to handle repeated roots, restrictions and curves given implicitly.

Coordinates and axes

A point in the coordinate plane is written in the form $(x, y)$ , where:

the $x$ -coordinate gives its horizontal position;
the $y$ -coordinate gives its vertical position.

Definition

Axis intersection

An intersection point with an axis is a point where a graph and that coordinate axis have a point in common.

The graph below meets the $x$ -axis at points of the form $(a, 0)$ , $(b, 0)$ and $(c, 0)$ . It meets the $y$ -axis at a point of the form $(0, d)$ .

A curve crossing the x-axis at three marked points and the y-axis at one marked point

Key Idea

The two substitutions

To find intersections with the $x$ -axis, set $y = 0$ and solve for $x$ .
To find intersections with the $y$ -axis, set $x = 0$ and solve for $y$ .

Intersections with the y-axis

Suppose a curve is given in the form

y = f (x) .

On the $y$ -axis, $x = 0$ . Substituting this into the equation gives

y = f (0) .

Therefore, provided $f (0)$ is defined, the curve meets the $y$ -axis at $(0, f (0))$ .

For a polynomial, this is particularly quick: substituting $x = 0$ removes every term containing $x$ , leaving only the constant term.

Example

Finding a y-axis intersection

Find where the curve

y=2x^3-5x^2+7x-4

meets the $y$ -axis.

A point on the $y$ -axis has $x = 0$ , so substitute this into the equation:
$y=2(0)^3-5(0)^2+7(0)-4.$
Simplifying gives the equation $y = - 4$ .
The intersection point therefore satisfies both $x = 0$ and $y = - 4$ , so its coordinates are $(0, - 4)$ .

Tip

Polynomial shortcut

For a polynomial written as $y = f (x)$ , the constant term is the $y$ -coordinate of its intersection with the $y$ -axis.

Intersections with the x-axis

On the $x$ -axis, $y = 0$ . For a curve $y = f (x)$ , its $x$ -axis intersections therefore satisfy

f (x) = 0.

The values of $x$ that satisfy this equation are called the roots or zeros of the function.

Definition

Root of a function

A root of $f (x)$ is a value of $x$ for which $f (x) = 0$ . Each real root gives an intersection point with the $x$ -axis.

Finding these intersections may require factorising, using the quadratic formula, completing the square, or applying another equation-solving technique.

Example

Finding two x-axis intersections

Find where the curve

y=x^2-5x+6

meets the $x$ -axis.

Points on the $x$ -axis have $y = 0$ , so solve
$x^2-5x+6=0.$
Factorise the quadratic:
$(x - 2) (x - 3) = 0.$
A product is zero when at least one factor is zero, giving $x = 2$ or $x = 3$ .
In each case the $y$ -coordinate is zero, so the intersection points are $(2, 0)$ and $(3, 0)$ .

Common Mistake

Giving roots instead of points

The values $x = 2$ and $x = 3$ are the roots, but the intersection points are $(2, 0)$ and $(3, 0)$ . Check whether the question asks for values of $x$ or coordinates.

How many x-axis intersections are there?

A curve may have:

no real intersections with the $x$ -axis;
one intersection;
two or more intersections, depending on the type of function.

For a quadratic curve, the discriminant $b^2-4ac$ determines the number of real roots of

ax^2+bx+c=0.

If $b^2-4ac>0$ , there are two distinct $x$ -axis intersections.
If $b^2-4ac=0$ , there is one repeated root and the curve touches the axis.
If $b^2-4ac<0$ , there are no real $x$ -axis intersections.

Example

Recognising a point of contact

Find the intersections of

y=x^2-6x+9

with the coordinate axes.

For the $x$ -axis, set $y = 0$ and factorise:
$x^2-6x+9=(x-3)^2=0.$
The only root is $x = 3$ . It is a repeated root, so the curve touches the $x$ -axis at $(3, 0)$ rather than crossing it there.
For the $y$ -axis, set $x = 0$ :
$y=(0)^2-6(0)+9=9.$
The intersections are therefore $(3, 0)$ and $(0, 9)$ .

Key Idea

Touching still counts

A curve does not need to cross an axis to intersect it. A point where the curve just touches the axis is still an intersection point.

Curves not written as y equals f of x

Some curves are given by an implicit equation, in which $x$ and $y$ appear together rather than with $y$ isolated. For example,

x^2+xy+y^2=12.

The same axis rules still apply: substitute $y = 0$ for the $x$ -axis and $x = 0$ for the $y$ -axis.

Example

Intersections of an implicit curve

Find all intersections of

x^2+xy+3y^2=12

with the coordinate axes.

On the $x$ -axis, $y = 0$ , so the equation becomes
$x^2=12.$
Hence $x=±23x=\pm 2\sqrt{3}$ , giving the points $(−23,0)(-2\sqrt{3},0)$ and $(23,0)(2\sqrt{3},0)$ .
On the $y$ -axis, $x = 0$ , so the equation becomes
$3y^2=12.$
Therefore $y^2=4$ , so $y=±2y=\pm 2$ .
The complete set of axis intersections is
$(−23,0),(23,0),(0,−2),(0,2).(-2\sqrt{3},0),\quad (2\sqrt{3},0),\quad (0,-2),\quad (0,2).$

Common Mistake

Losing a negative solution

When an equation simplifies to $x^2=k$ with $k > 0$ , both $x=kx=\sqrt{k}$ and $x=−kx=-\sqrt{k}$ must be considered. Missing the negative root loses an intersection.

Restrictions and undefined values

You must check that each proposed coordinate is allowed by the original equation. This matters especially for rational functions, square roots and logarithms.

For example, the function

y=x+1xy=\frac{x+1}{x}

has no $y$ -axis intersection because substituting $x = 0$ would require division by zero. The graph is not defined on the $y$ -axis.

Common Mistake

Substitution may be invalid

Setting $x = 0$ does not guarantee a $y$ -axis intersection. If the original function is undefined at $x = 0$ , the graph cannot meet the $y$ -axis.

Intersections at the origin

The origin is the point $(0, 0)$ , where the two coordinate axes meet.

If a curve passes through the origin, that single point is an intersection with both axes. It should not be listed twice when you give the distinct intersection points.

For a function $y = f (x)$ , the graph passes through the origin exactly when

f (0) = 0.

Exam technique

In the exam

Decide which axis is involved: use $y = 0$ for the $x$ -axis and $x = 0$ for the $y$ -axis.
Solve the resulting equation completely, including negative, repeated or exact roots where appropriate.
Return to the original equation to reject undefined values, then give coordinates rather than only roots when points are requested.
Check your answers quickly: every $x$ -axis point must have $y = 0$ , and every $y$ -axis point must have $x = 0$ .

Self review

Check yourself

Where does $y=x^2+x-6$ meet each coordinate axis?
How many $x$ -axis intersections does $y=2x^2+4x+5$ have?
Why does $y=3x−2y=\frac{3}{x-2}$ have no $x$ -axis intersection?

What you'll learn

Coordinates and axes

Intersections with the y-axis

Intersections with the x-axis

How many x-axis intersections are there?

Curves not written as y equals f of x

Restrictions and undefined values

Intersections at the origin

Review 1.4.2 Intersection points with axes by teaching Genie

1.4.2 Intersection points with axes

What you'll learn

Coordinates and axes

Intersections with the y-axis

Intersections with the x-axis

How many x-axis intersections are there?

Curves not written as y equals f of x

Restrictions and undefined values

Intersections at the origin

Review 1.4.2 Intersection points with axes by teaching Genie

1.4.2 Intersection points with axes

What you'll learn

Coordinates and axes

Intersections with the y-axis

Intersections with the x-axis

How many x-axis intersections are there?

Curves not written as y equals f of x

Restrictions and undefined values

Intersections at the origin

Review 1.4.2 Intersection points with axes by teaching Genie

1.4.2 Intersection points with axes Revision Guide

1.4.2 Intersection points with axes

What you'll learn

Coordinates and axes

Intersections with the y-axis

Intersections with the x-axis

How many x-axis intersections are there?

Curves not written as y equals f of x

Restrictions and undefined values

Intersections at the origin

Review 1.4.2 Intersection points with axes by teaching Genie

1.4.2 Intersection points with axes Revision Guide