How to Find x- and y-Intercepts of a Function
Intercepts are where a graph meets the axes, and they are usually the first thing a question asks for. The idea is simple; the marks are lost in the details, particularly in telling a genuine crossing from a curve that only approaches an axis.
What each intercept means
An x-intercept is a point where the curve crosses or
touches the x-axis. At every such point the height is zero, so the
coordinates are (something, 0).
A y-intercept is where the curve crosses the y-axis. That
happens at x = 0, so the coordinates are
(0, something).
That gives the two rules everything else follows from:
To find the y-intercept, set
x = 0.
To find the x-intercepts, sety = 0.
Finding the y-intercept
This is the easy one. Substitute x = 0 and evaluate. Every term
containing x disappears, so usually only the constant survives.
| Function | Substitute x = 0 | y-intercept |
|---|---|---|
y = 3x + 7 | 3(0) + 7 | (0, 7) |
y = x² - 2x - 3 | 0 - 0 - 3 | (0, -3) |
y = 2ˣ | 2⁰ = 1 | (0, 1) |
y = 1/x | 1/0, undefined | None |
That last row matters. A function has at most one y-intercept, and
it has none when the function is undefined at zero. ln(x) and
1/x both fall into that category.
Finding x-intercepts
Set the function equal to zero and solve. How you solve depends on what kind of function it is.
Linear
One step of algebra. For y = 3x + 7, set
3x + 7 = 0, so x = -7/3.
Quadratic
Factor if you can, otherwise use the quadratic formula.
x² - 2x - 3 = 0 factors to (x - 3)(x + 1) = 0, so
x = 3 or x = -1.
The discriminant b² - 4ac tells you how many to expect before
you start: positive gives two, zero gives one (the curve touches the axis
and turns), negative gives none.
Anything else
Most functions cannot be solved by hand. y = x³ - 4x + 1 has
three real roots and none of them is a neat number. For these, a numerical
method is not a shortcut, it is the only option, and it is exactly what a
graphing calculator is for.
Getting exact values, not estimates
Reading an intercept off a printed graph gives you roughly one decimal place, which is rarely enough. There are two better routes.
Use the analysis tool
- Enter your function in the calculator.
- Make sure the intercepts you want are inside the visible window.
- Open the Analyse tab and press the button.
- Every root, turning point and y-intercept in view is listed to six figures, with a copy button.
Zoom in and read the trace
Hovering over a curve shows its exact coordinates at that x-value. Zoom in near a crossing and move along until the y-value is as close to zero as you need. Slower than the analysis tool, but it builds a feel for what the number means.
When there is no intercept at all
A curve that approaches but never touches
y = 1/x gets closer and closer to the x-axis as x
grows, but never reaches it. There is no x for which
1/x = 0. The x-axis is a horizontal asymptote, not an
intercept.
This is the single most common false positive. A calculator that simply
looks for a change of sign will report a root at x = 0 for
1/x, because the function really does go from negative to
positive there. It is wrong: the function is undefined at that point, not
zero. Our calculator checks the magnitude of the function at the candidate
point and rejects it, which is why 1/x correctly reports no
roots.
A curve that sits entirely on one side
y = x² + 1 has a minimum value of 1, so it never reaches the
x-axis. y = 2ˣ is positive for every x. Both have
a y-intercept and no x-intercept.
Quick check before you answer. If a question asks for "the" x-intercept but you find two, or none, re-read it. Exam questions often specify a domain, and restricting the domain changes the answer.
Questions
How many x-intercepts can a function have?
Any number, including none. A line has at most one, a quadratic has zero, one or two, and a sine curve has infinitely many. The only limit is on y-intercepts: a function can have at most one, because a function returns a single value at x = 0.
What is the difference between a root, a zero and an x-intercept?
For a single-variable function they are three names for the same thing: a value of x where the output is zero. Root and zero are used when talking about the equation, x-intercept when talking about the graph.
Why does my graph cross the axis but the calculator finds no root?
Check whether it is really crossing or approaching an asymptote. 1/x gets arbitrarily close to the x-axis but never touches it, so it has no root. A good calculator distinguishes the two; a naive sign-change search reports a false root at the asymptote.
Can a function have a y-intercept but no x-intercept?
Yes, and it is common. y = x² + 1 has a y-intercept at (0, 1) and never reaches the x-axis, because the smallest value it takes is 1.