Everything You (N)ever Wanted to Know About Parabolas

Posted March 7, 2012

by Lensyl Urbano

So that my students could more easily check their answers graphically, I put together a page with a more complete analysis of parabolas (click this link for more details).

Analyzing Parabolas

Standard Form Vertex Form
y = a x2 + b x + c y = a (x - h)2 + k
y = x2 + x +

y = ( x - ) 2 +

Intercepts: Vertex:
Focus: Directrix: Axis:
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Solution by Factoring:

y = x2 x

Converting the forms

The key relationships are the ones to convert from the standard form of the parabolic equation:

         y = a x^2 + b x + c (1)

to the vertex form:

         y = a (x - h)^2 + k (2)

If you multiply out the vertex equation form you get:

         y = a x2 - 2ah x + ah2 + k (3)

When you compare this equation to the standard form of the equation (Equation 1), if you look at the coefficients and the constants, you can see that:

To convert from the vertex to the standard form use:

         a = a (4)
         b = -2ah (5)
         c = ah^2 + k (6)

Going the other way,

To convert from the standard to the vertex form of parabolic equations use:

         a = a (7)
         h = \frac{-b}{2a} (8)
         k = c - ah^2 (9)

Although it is sometimes convenient to let k not depend on coefficients from its own equation:

         k = c - \frac{b^2}{4a} (10)

The Vertex and the Axis

The nice thing about the vertex form of the equation of the parabola is that if you want the find the coordinates of the vertex of the parabola, they're right there in the equation.

Specifically, the vertex is located at the point:

         (x_v, y_v) = (h, k) (11)

The axis of the parabola is the vertical line going through the vertex, so:

The equation for the axis of a parabola is:

         x = h (12)

Focus and Directrix

Finally, it's important to note that the distance (d) from the vertex of the parabola to its focus is given by:

         d = \frac{1}{4a} (13)

Which you can just add d on to the coordinates of the vertex (Equation 11) to get the location of the focus.

         (x_f, y_f) = (x_v, y_v + d)  (14)

The directrix is just the opposite, vertical distance away, so the equation for the directrix is the equation of the horizontal line at:

         y = y_v + d  (15)

References

There are already some excellent parabola references out there including:

Citing this post: Urbano, L., 2012. Everything You (N)ever Wanted to Know About Parabolas, Retrieved September 25th, 2017, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: Montessori Muddle; Hat tip: Montessori Muddle.

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