﻿ Finding the x-intercepts of a function - Math Bootcamps

# Finding and understanding x-intercepts

Given the graph of any function, an x-intercept is simply the point, or points where the graph crosses the x-axis. There might be just one such point, no such point, or many, meaning a function can have several x-intercepts. As you will see below, we can use a graph or a simple algebra rule to find the x-intercept or x-intercepts of any function. You can also scroll down to a video example below.

## Finding the x-intercept or x-intercepts using a graph

As mentioned above, functions may have one, zero, or even many x-intercepts. These can be found by looking at where the graph of a function crosses the x-axis, which is the horizontal axis in the xy-coordinate plane. You can see this on the graph below. This function has a single x-intercept. In the graph below, the function has two x-intercepts. Notice that the form of the point is always $$(c, 0)$$ for some number $$c$$. Finally, the following graph shows a function with no x-intercepts. You can see this because it does not cross the x-axis at any point. You can see a more advanced discussion of these ideas here: The zeros of a polynomial.

## Finding the x-intercept or intercepts using algebra

The general rule for finding the x-intercept or intercepts of any function is to let $$y = 0$$ and solve for $$x$$. This may be somewhat easy or really difficult, depending on the function. Let’s look at some examples to see why this may be the case.

### Example

Find the x-intercept of the function: $$y = 3x – 9$$

### Solution

Let $$y = 0$$ and solve for $$x$$.

\begin{align}0 &= 3x – 9\\ -3x &= -9\\x &= 3\end{align}

Answer: Therefore the x-intercept is 3. You could also write it as a point: $$(3,0)$$

A more complicated example would be one where the equation representing the function itself is more complex. For these situations, you need to know a little more algebra in order to find any intercepts.

### Example

Find the x-intercepts for the function: $$y = x^2 + 2x – 8$$

### Solution

As before, let $$y = 0$$ and solve for $$x$$. This time, you have a quadratic equation to solve.

\begin{align} 0 &= x^2 + 2x – 8\\ 0 &= (x + 4)(x – 2)\\ x &= -4, 2\end{align}

Answer: This function has two x-intercepts: –4 and 2. These are located at $$(–4, 0)$$ and $$(2, 0)$$.

For equations more complex than this, a graphing calculator is often useful for at least estimating the location of any intercepts.

## Video examples

In the following video, you can see how to find the x-intercepts of three different functions. This also includes an example where there are no x-intercepts.

## Continue your study of graphs

You can continue your study of graphing with the following articles. ## Subscribe to our Newsletter!

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