Perfect Parabola: Definition & Explanation

Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

Perfect parabolas are created from formulas and can also be hand-drawn. Learn how you can draw your own and what the mathematical formulas look like. Also learn what makes a parabola perfect.

Definition

A perfect parabola is a curve where the distance between a fixed point and another fixed line is the same at all points on the curve. The fixed point is called the focus, and the fixed line is called the directrix. Drawing it out, it looks like this.

A perfect parabola with focus and directrix.
perfect parabola

The dashed blue lines above show the distance to a point from both the directrix and the focus. From the directrix, it's a straight up and down path, and from the focus, it's a direct path to the point. Both of these distances need to be equal for all points in a perfect parabola.

Knowing this can help us to draw a perfect parabola.

How to Draw

1. Start by drawing a straight line on your piece of paper. Then you place a big dot somewhere else on the paper that is not on the straight line. Your paper should look something like this.

A straight line with a big dot.
perfect parabola

Now you have your focus which is your big dot and you have your directrix which is your straight line.

2. Next, you take two rulers out and start measuring away. Place one ruler up and down for the directrix measurement and place the other ruler touching both the first ruler and the focus point. Play with the second ruler until you find a point that is the same distance to both the focus and the directrix. Mark this point with a small dot. Remember to keep the second ruler connected to the focus at all times.

Finding a point equidistant to both focus and directrix.
perfect parabola

Look carefully at the illustration above. Notice how I have found the point where both my rulers measure the same distance. That is where I place my small dot.

Move the first ruler along the directrix to a different position, keeping it straight up and down, and move the second ruler until you find another point that is equidistant from the focus and the directrix. Mark this new point with a small dot. Keep repeating this procedure until you have formed a parabola.

A perfect parabola taking shape.
perfect parabola

That is how you draw a perfect parabola. It does take time, but it is fun seeing the parabola take shape. Try it out for yourself and see. Try placing the focus dot at different distances from the directrix to see what happens to the parabola. You might be surprised!

Quadratics

Mathematically, a perfect parabola is expressed in the form of a quadratic equation. A quadratic equation is an equation where the degree is 2. The highest exponent you will see is a square. Examples of quadratics include the following:

Examples of quadratic equations that produce perfect parabolas.
perfect parabola

Either the x can be squared or the y. If the y is squared, you will get a sideways parabola pointing either to the left or right. When the x is squared you will get a parabola pointing either up or down. You can plot these parabolas yourself on the coordinate plane too by making a t-chart of different values. Try it out and see how the parabola changes with different numbers.

Different Equation Forms

There are two ways you can write your quadratic equation for a parabola. Each way provides useful information for you.

1. The first way is the standard form.

Standard form.
perfect parabola

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