x- and y-Intercepts: Finding Where a Graph Crosses the Axes

Most of a graph is hard to pin down to exact numbers. The places where it crosses the axes are the exception — those points almost always have clean coordinates, and they are the ones a graph gets labelled with. They are called the intercepts.

There are two to find. The -intercept is where the graph crosses the horizontal axis. The -intercept is where it crosses the vertical axis. Both come from the same trick: on an axis, one of the coordinates is zero, so you set that coordinate to zero and solve for the other.

The Intercept Rules

The -intercept is a point of the form ; the -intercept is a point of the form .

Why Setting a Variable to Zero Works

Every point on the -axis has height zero — that is what makes it the -axis. So the point where a graph touches the -axis is the point on the graph with . Substituting into the equation and solving for finds it.

The -axis works the same way in the other direction: every point on it has . Substitute , solve for , and you have the -intercept.

Finding Intercepts Algebraically

Take .

-intercept — set :

-intercept — set :

Two points, and for a line that is all you need. Plot and , draw the straight line through them, and the graph is done.

Intercepts of an equation in standard form

When the equation is written as , intercepts are especially quick. For :

• Set : , so . -intercept .
• Set : , so . -intercept .

Each intercept only involves one term, which is why standard form is convenient for sketching.

Reading Intercepts off a Graph

Going the other way, if the graph is already drawn:

• The -intercept(s) are wherever the curve touches or crosses the horizontal axis. Read the -value there; the -value is .
• The -intercept is where the curve touches the vertical axis. Read the -value there; the -value is .

A function has at most one -intercept — you can only put in once and get one output. It can have many -intercepts.

When There Are None, or Many

No -intercept. A horizontal line like never comes down to the axis. Setting gives , which is false — the algebra is telling you there is no crossing.

No -intercept. A vertical line like never reaches the -axis. It is also not a function, and setting contradicts the equation.

Two -intercepts. The parabola crosses the axis twice. Setting : , so or . Intercepts and . These are also the roots of the equation , which is why “find the -intercepts” and “solve for the roots” are the same task.

One -intercept. The parabola touches the axis at and nowhere else.

Infinitely many. The line is the -axis, so every one of its points is an -intercept. This case is a curiosity more than a hazard.

Worked Example A: Line From Slope-Intercept Form

Find both intercepts of .

-intercept — set : , so . (In the -intercept is always , so this is a read-off.)

-intercept — set :

So . Plot the two points and connect them.

Worked Example B: Line in Standard Form

Find the intercepts of .

Set : , so → .

Set : , so → .

Worked Example C: A Parabola

Find the intercepts of .

-intercept — set : → .

-intercepts — set :

So and . Three labelled points is enough to sketch this parabola.

Worked Example D: No Real x-Intercept

Does have an -intercept?

Set : , so . No real number squares to , so there is no -intercept. The parabola sits entirely above the axis, lowest point . Its only intercept is the -intercept .

Worked Example E: Intercepts of a Horizontal and a Vertical Line

• : every point has . Setting gives the -intercept . Setting gives , false — no -intercept.
• : every point has . Setting gives the -intercept . Setting contradicts — no -intercept.

Common Mistakes to Avoid

• Swapping the two rules. -intercept comes from ; -intercept comes from . Set the other variable to zero.
• Writing an intercept as a single number. An intercept is a point. The -intercept of is , not just , though “the -intercept is at ” is acceptable shorthand.
• Stopping after one intercept on a parabola. Quadratics can cross the -axis twice. Solve the whole equation.
• Declaring “no intercept” too early. is solvable — an -intercept exists. Only a contradiction like means there is none.
• Forgetting the -intercept is free in . It is — no work required.
• Mixing up roots and the -intercept. The -intercepts are the roots of the equation; the -intercept is the constant term when .

Where Intercepts Show Up Next

• Graphing lines. Two intercepts define a line, so this is the fastest sketch method for most linear equations.
• Slope-intercept form. The in is the -intercept, read directly from the equation.
• Solving equations. Finding -intercepts of is the same as solving .
• Quadratics. The -intercepts of a parabola are its roots; the discriminant predicts how many there are.
• Rational and higher-degree functions. Intercepts are among the first features you locate before sketching.

Practice Problems

Work each one before opening the answer.

Problem 1. Find the - and -intercepts of .

Problem 2. Find the intercepts of .

Problem 3. Find the intercepts of .

Problem 4. Find the intercepts of .

Problem 5. State the -intercept of without solving.

Problem 6. Find the intercepts of .

Problem 7. Find the intercepts of .

Problem 8. Does have any -intercepts?

Problem 9. What are the intercepts of the line ?

Problem 10. What are the intercepts of the line ?

Problem 11. A line has -intercept and -intercept . Sketch it and estimate its slope.

Problem 12. The -intercepts of are and . What are the solutions of ?

Quick Reference

Intercepts are the fastest points to plot when graphing equations, and the in slope-intercept form links straight to Lines and Slope. More Algebra lessons are available too.