Conic Sections

How to identify and graph them.

Identifying Conic Sections

A quadratic relationship is a relation specified by an equation or inequality of the form:

Ax2 +Bxy + Cy2 + Dx + Ey + F = 0Where A, B, C, D, E, & F are constants. The following information assumes that B=0.

Therefore there is no xy-term.

To IDENTIFY what conic section you have:

Look at the coefficients of x2 and y2.

There are five types of conic sections you need to worry

about.

Circles

The coefficients of x2 and y2 have the same sign and same value.

An example of a circle is x2 + y2 = 16.

In this example x2

and y2 have coefficients equal to positive 1.

x

y

-5 0 5

-5

0

5

Ellipses

The coefficients of x2 and y2 have the same sign but different values.

An example of an ellipses is 9x2 + 25y2 = 225

In this example the coefficient of x2 is positive 9. The coefficient of y2 is positive 25.

x

y

-5 0 5

-5

0

5

Hyperbolas

The coefficients of x2 and y2 have different signs and different values.

An example of a hyperbola is 16x2 - 9y2 = 144

In this example the coefficient of x2 is positive 16. The coefficient of y2 is negative 9.

x

y

-5 0 5

-5

0

5

Parabolas

Parabolas are “special” conic sections. There are two types

parabolas that you will need to graph.

Y-Direction Parabolas

Y-Direction Parabolas open in the y-direction.

Y-Direction Parabolas are defined by the general formula

y = ax2 + bx + c An example of a Y-

Direction Parabola is:

y = 2x2+4x-3

x

y

-5 0 5

-5

0

5

X-Direction Parabolas

X-Direction Parabolas open in the x-direction.

X-Direction Parabolas are defined by the general formula

x = ay2 + by + c An example of a X-

Direction Parabola is:

x = 4y2+yx-2

x

y

-5 0 5

-5

0

5

That’s all folks!