How to find the x and y intercepts of a graph

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.