Intercepts of a Graph
The intercepts of a graph are the points where the curve crosses the coordinate axes. The $x$-intercepts are where the graph crosses the $x$-axis, and the $y$-intercept is where it crosses the $y$-axis. Intercepts are among the most informative features of any graph.
Quick Reference
| Intercept | How to Find | What It Means |
|---|---|---|
| $x$-intercept | Set $y = 0$, solve for $x$ | Where the graph meets the $x$-axis |
| $y$-intercept | Set $x = 0$, solve for $y$ | Where the graph meets the $y$-axis |
Definition and Geometric Meaning
Let $f(x)$ be an expression in $x$, for example $f(x) = 3x + 2$ or $f(x) = 8x^2 - \sqrt{x} + 5$.
Finding Intercepts
- $x$-intercept: Set $y = 0$ and solve for $x$. The $x$-intercepts of $y = f(x)$ are the solutions to $f(x) = 0$. These are also called the roots or zeros of $f$.
- $y$-intercept: Substitute $x = 0$ into $y = f(x)$ and evaluate. The $y$-intercept is the value $f(0)$.
Worked Examples
Example 1. Find the $x$- and $y$-intercepts of $y = 2x - 4$.
Solution.
$y$-intercept: Set $x = 0$:
$y = 2(0) - 4 = -4.$The $y$-intercept is $(0, -4)$.
$x$-intercept: Set $y = 0$:
$0 = 2x - 4 \implies x = 2.$The $x$-intercept is $(2, 0)$.
Example 2. Find the $x$- and $y$-intercepts of $y = x^2 - 4$.
Solution.
$y$-intercept: Set $x = 0$:
$y = 0^2 - 4 = -4.$The $y$-intercept is $(0, -4)$.
$x$-intercepts: Set $y = 0$:
$0 = x^2 - 4 \implies x^2 = 4 \implies x = \pm 2.$The $x$-intercepts are $(-2, 0)$ and $(2, 0)$.
Example 3. Find the intercepts of the circle $(x+1)^2 + (y-1)^2 = 25$.
Solution.
$y$-intercepts: Set $x = 0$:
$\begin{aligned} (0+1)^2 + (y-1)^2 &= 25 \\ 1 + (y-1)^2 &= 25 \\ (y-1)^2 &= 24 \\ y - 1 &= \pm\sqrt{24} = \pm 2\sqrt{6} \\ y &= 1 \pm 2\sqrt{6}. \end{aligned}$The $y$-intercepts are $\left(0,\; 1 + 2\sqrt{6}\right)$ and $\left(0,\; 1 - 2\sqrt{6}\right)$.
$x$-intercepts: Set $y = 0$:
$\begin{aligned} (x+1)^2 + (0-1)^2 &= 25 \\ (x+1)^2 + 1 &= 25 \\ (x+1)^2 &= 24 \\ x + 1 &= \pm 2\sqrt{6} \\ x &= -1 \pm 2\sqrt{6}. \end{aligned}$The $x$-intercepts are $\left(-1 + 2\sqrt{6},\; 0\right)$ and $\left(-1 - 2\sqrt{6},\; 0\right)$.
Frequently Asked Questions
What is an x-intercept?
An $x$-intercept is a point where the graph of an equation crosses the $x$-axis. At such a point, the $y$-coordinate is zero, so we find $x$-intercepts by setting $y = 0$ and solving for $x$. A curve can have zero, one, or many $x$-intercepts.
What is a y-intercept?
The $y$-intercept is the point where the graph crosses the $y$-axis. At that point, the $x$-coordinate is zero, so we find it by setting $x = 0$ and computing $y = f(0)$. A function can have at most one $y$-intercept (since a function gives a unique output for each input), but a general equation can have multiple $y$-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 $f(x)$ is a value of $x$ satisfying $f(x) = 0$.
- A zero of $f$ is the same thing: an $x$ value where the function equals zero.
- An $x$-intercept is the corresponding point $(x, 0)$ on the graph.
