How to Find the X-Intercepts and Y-Intercepts
x-intercept of a Line
x-intercepts of a Parabola
y-intercept of a Line
y-intercepts of a Parabola
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 = x2 − 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 = 3x2 + 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.