# How to Find the X-Intercepts and Y-Intercepts

## The X-Intercepts

The x-intercepts are points where the graph of a function or an equation crosses or “touches” the $x$-axis of the Cartesian Plane. You may think of this as a point with $y$-value of zero.

• To find the $x$-intercepts of an equation, let $y = 0$ then solve for $x$.
• In a point notation, it is written as $\left( {x,0} \right)$.

x-intercept of a Linear Function or a Straight Line

x-intercepts of a Quadratic Function or Parabola

## The Y-Intercepts

The y-intercepts are points where the graph of a function or an equation crosses or “touches” the $y$-axis of the Cartesian Plane. You may think of this as a point with $x$-value of zero.

• To find the $y$-intercepts of an equation, let $x = 0$ then solve for $y$.
• In a point notation, it is written as $\left( {0,y} \right)$.

y-intercept of a Linear Function or a Straight Line

y-intercept of a Quadratic Function or Parabola

### Examples of How to Find the x and y-intercepts of a Line, Parabola, and Circle

Example 1: From the graph, describe the $x$ and $y$-intercepts using point notation.

The graph crosses the $x$-axis at $x= 1$ and $x= 3$, therefore, we can write the $x$-intercepts as points $(1,0)$ and $(–3, 0)$.

Similarly, the graph crosses the $y$-axis at $y=3$. Its y-intercept can be written as the point $(0,3)$.

Example 2: Find the x and y-intercepts of the line $y = – 2x + 4$.

To find the x-intercepts algebraically, we let $y=0$ in the equation and then solve for values of $x$. In the same manner, to find for $y$-intercepts algebraically, we let $x=0$ in the equation and then solve for $y$.

Here’s the graph to verify that our answers are correct.

Example 3: Find the $x$ and $y$-intercepts of the quadratic equation $y = {x^2} – 2x – 3$.

The graph of this quadratic equation is a parabola. We expect it to have a “U” shape where it would either open up or down.

To solve for the $x$-intercept of this problem, you will factor a simple trinomial. Then you set each binomial factor equal to zero and solve for $x$.

Our solved values for both $x$ and $y$-intercepts match with the graphical solution.

Example 4: Find the $x$ and $y$-intercepts of the quadratic equation $y = 3{x^2} + 1$.

This is an example where the graph of the equation has a y-intercept but without an $x$-intercept.

• Let’s find the $y$-intercept first because it’s extremely easy! Plug in $x = 0$ then solve for $y$.
• Now for the $x$-intercept. Plug in $y = 0$, and solve for $x$.

The square root of a negative number is imaginary. This suggests that this equation does not have an $x$-intercept!

The graph can verify what’s going on. Notice that the graph crossed the $y$-axis at $(0,1)$, but never did with the $x$-axis.

Example 5: Find the x and y intercepts of the circle ${\left( {x + 4} \right)^2} + {\left( {y + 2} \right)^2} = 8$.

This is a good example to illustrate that it is possible for the graph of an equation to have $x$-intercepts but without $y$-intercepts.

When solving for $y$, we arrived at the situation of trying to get the square root of a negative number. The answer is imaginary, thus, no solution. That means the equation doesn’t have any $y$-intercepts.

The graph verifies that we are right for the values of our $x$-intercepts, and it has no $y$-intercepts.