Parabola Calculator

How to use the Parabola Calculator

1. 1Enter a, b and c.
2. 2Pick the units or options from the menus if the parabola tool offers them.
3. 3Read the parabola in the large result box.
4. 4Look at the extra rows for related figures.
5. 5Press Copy result to keep the answer.

When you need the parabola

Students use it for algebra homework and graphing. Engineers use the focus property in satellite dishes and headlights, where a parabola sends parallel rays to a single point.

The sign of a says which way the parabola opens. A larger value of a makes the curve narrower, and a smaller value makes it wider. When the discriminant b² − 4ac is negative, the curve never touches the x-axis.

How the parabola is worked out

The vertex is at x = −b ÷ 2a, with y found by putting x back into the equation. The focus is 1 ÷ 4a above the vertex and the directrix is 1 ÷ 4a below it. The roots come from (−b ± √(b² − 4ac)) ÷ 2a. For y = x² − 6x + 5, the vertex is at (3, −4), the axis is x = 3, the focus is (3, −3.75), the directrix is y = −4.25, and the roots are 1 and 5.

Every answer on this page comes from that one rule, so you can check any result by hand with a calculator. The figures shown underneath, such as other units, are conversions of the same answer.

• Vertex, focus and directrix
• Roots and intercepts
• Opening direction

Mistakes to avoid with parabola

The value a must not be 0, since then the equation is a straight line. Also take care with signs: for y = x² − 6x + 5, b is −6 and not 6. Entering the wrong sign moves the vertex to the wrong side.

On phone and computer

On a phone, type into the boxes one after another and read the parabola underneath; Copy result puts it on the clipboard for a message or a note. On a computer the inputs sit on the left and the answer on the right, so you can change one figure and watch the parabola move. The parabola calculator works the same on both.

Checking your parabola by hand

A good habit is to test the parabola calculator with a case you already know. For y = x² − 6x + 5, the vertex is at (3, −4), the axis is x = 3, the focus is (3, −3.75), the directrix is y = −4.25, and the roots are 1 and 5.

If your own figures give a surprising answer, repeat the working yourself: The vertex is at x = −b ÷ 2a, with y found by putting x back into the equation. The focus is 1 ÷ 4a above the vertex and the directrix is 1 ÷ 4a below it. The roots come from (−b ± √(b² − 4ac)) ÷ 2a. Small differences usually come from rounding, from mixing units, or from a typing slip in one of the inputs. A parabola comes from a quadratic equation, so check the roots as well. Use the Quadratic Equation Solver for the roots, the Slope Calculator for the steepness at a point, and the Distance Calculator to measure between two points.

Features

• a cannot be zero
• Watch the sign of b
• Negative discriminant means no roots
• Clear messages when the parabola inputs make no sense
• Copy the result with one tap
• Private: nothing is uploaded

Frequently asked questions