How to Find x and y Intercept of a Parabola

To find the x-intercepts of a parabola, set y = 0 in the equation and solve for x. To find the y-intercept, set x = 0 and solve for y. For example, if the parabola is given by y = ax² + bx + c, solving for x gives the x-intercepts, and plugging x = 0 gives the y-intercept at y = c.

Let's discuss this in detail with examples.

What Is a Parabola?

A parabola is a U-shaped curve that can open upward or downward. It is one of the most common shapes in algebra and geometry and can be seen in many real-life situations, like the path of a thrown ball.

A parabola is a symmetrical curve where any point is at an equal distance from a fixed point called the "focus" and a fixed line called the "directrix."

Standard Form of a Parabola Equation

The standard form of a parabola’s equation is: y = ax² + bx + c Here, a, b, and c are constants, and x and y are the variables. The shape and position of the parabola are determined by the values of these constants.

In general, if the directrix is parallel to the y-axis in the standard equation of a parabola is given as

y² = 4ax

If the parabola is sideways, i.e., the directrix is parallel to the x-axis, the standard equation of a parabola becomes

x² = 4ay

How to Find the x-Intercept of a Parabola

To find the x-intercept, you need to set in the quadratic equation and solve for x. Here’s how you can do it:

1. Start with the equation of the parabola in the form .
2. Set in the equation because the x-intercept occurs where .
3. Solve the resulting quadratic equation for x using factoring, the quadratic formula, or completing the square.
4. The solutions for x are the x-intercepts of the parabola.

Example: Find x-intercept of parabola with equation .

Solution:

• Set y = 0: .
• Solve the quadratic equation:
  1. Divide by 2:
  2. Factor the equation:
• The solutions are and .

So the x-intercepts are at (3, 0) and (-1, 0).

How to Find the y-intercept of a Parabola?

To find y-intercept of a parabola we can use the following steps:

1. Start with the equation of the parabola .
2. Set x = 0 in the equation because the y-intercept occurs where x = 0.
3. Simplify the equation to find the value of y.

Example: Find the y-intercept for equation .

Solution:

• Set x = 0: .
• Simplify to get .

The y-intercept is at (0, -6).

Read More,

• Parabola
• Standard Equation of a Parabola
• Vertex of a Parabola
• Axis of Symmetry of a Parabola
• Focus and Directrix of a Parabola
• Applications of Parabola in Real-Life
• X and Y Intercept