# 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 **( x + 4)^{2} + (y + 2)^{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.