Intercepts of a Graph

The intercepts of a graph are the points where the curve crosses the coordinate axes. The -intercepts are where the graph crosses the -axis, and the -intercept is where it crosses the -axis. Intercepts are among the most informative features of any graph.

Quick Reference

Intercept	How to Find	What It Means
-intercept	Set , solve for	Where the graph meets the -axis
-intercept	Set , solve for	Where the graph meets the -axis

Definition and Geometric Meaning

Let be an expression in , for example or .

A graph of a function showing its intersection with the y-axis labeled as the y-intercept, and two intersections with the x-axis labeled as x-intercepts. — The x-intercepts and y-intercept of a curve y = f(x)

Finding Intercepts

-intercept: Set and solve for . The -intercepts of are the solutions to . These are also called the roots or zeros of .
-intercept: Substitute into and evaluate. The -intercept is the value .

Worked Examples

Example 1. Find the - and -intercepts of .

Solution.

-intercept: Set :

The -intercept is .

-intercept: Set :

The -intercept is .

Example 2. Find the - and -intercepts of .

Solution.

-intercept: Set :

The -intercept is .

-intercepts: Set :

The -intercepts are and .

Example 3. Find the intercepts of the circle .

Solution.

-intercepts: Set :

The -intercepts are and .

-intercepts: Set :

The -intercepts are and .

Frequently Asked Questions

What is an x-intercept?

An -intercept is a point where the graph of an equation crosses the -axis. At such a point, the -coordinate is zero, so we find -intercepts by setting and solving for . A curve can have zero, one, or many -intercepts.

What is a y-intercept?

The -intercept is the point where the graph crosses the

-axis. At that point, the

-coordinate is zero, so we find it by setting

and computing

. A function can have at most one

-intercept (since a function gives a unique output for each input), but a general equation can have multiple

-intercepts.

What is the difference between a root, a zero, and an x-intercept?

All three terms describe the same concept from different perspectives:

A root of is a value of satisfying .
A zero of is the same thing: an value where the function equals zero.
An -intercept is the corresponding point on the graph.

Can a graph have no x-intercepts?

Yes. For example,

has no

-intercepts because

for all real

, meaning the graph never touches the

-axis. Similarly, the circle

is centered far from the axes and may not cross them at all.

Why are intercepts useful?

Intercepts tell us key information about the behavior of a function or curve. The

-intercept shows the starting value when

, and the

-intercepts show where the quantity modeled by the equation equals zero. In applications, zeros often represent break-even points, equilibrium states, or transition points.

Intercepts of a Graph

Intercepts of a Graph

Quick Reference

Definition and Geometric Meaning

Worked Examples

Frequently Asked Questions

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