Understanding and finding y-intercepts

The y-intercept of a graph is the point where it crosses the y-axis, which is the vertical axis from the xy-coordinate plane. Below, we will see how to find the y-intercept of any function and why a function can have at most one y-intercept in general. You can also always scroll down to a video example.

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Seeing it on a graph

Before we go into detail, consider the graph below. As you can see, it is a linear function (the graph is a line) and it crosses the y-axis at the point (0, 3). This tells you that the y-intercept is 3.

graph showing line crossing y-axis at the point (0,3)

Since any point along the y-axis has an x-coordinate of 0, the form of any y-intercept is \((0, c)\) for some number \(c\).

Using algebra to find the y-intercept of a function

To find the y-intercept of a function, let \(x = 0\) and solve for \(y\). Consider the following example.

Example

Find the y-intercept of the function: \(y = x^2 + 4x – 1\)

Solution

Let \(x = 0\) and solve for \(y\).

\(\begin{align} y &= 0^2 + 4(0) – 1\\ &= \boxed{-1}\end{align}\)

Thus the y-intercept is –1 and is located at the point \((0, –1)\).

A closer look

Now that we have seen how to find them, there are two interesting questions that can come up:

  1. Can a function have more than one y intercept?
  2. Can a function have no y intercept?

In answering these, remember that by definition, a function can only have one output (y-value) for each input (x-value). A function having more than one y-intercept would violate this, since it would mean that there are two outputs for \(x = 0\). Therefore, it is not possible for a function to have more than one y-intercept.

What about no y intercept? Well, consider the graph below. This is a graph of the function: \(y = \dfrac{1}{x}\)

a graph of y = 1/x, which has no y-intercept

This function never crosses the y-axis because, since you can’t divide by zero, it is undefined at \(x = 0\). In fact, any time a function is undefined at 0, it will have no y-intercept.

Video example

In the video below, I show you three examples of how to find the y-intercept. As you will see, the idea is pretty straight-forward!

Summary

When working with any graph, two useful things to know are the location of any x-intercepts, and the location of the y-intercept, if it exists. With a linear function (a line) these two points are enough to quickly sketch a graph. For more complex functions however, finding intercepts is often part of a deeper analysis.

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Continue your study of graphing

You may find the following articles useful as you continue to study graphs:

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