A **quadratic curve** it is a curve representing a given quadratic equation. The curve y = ax^{2 }+ bx + c, with, is called the quadratic function of a variable. In general, the curve f (x, y) = Ax^{2 }+ Bxy + Cxy^{2} + Dx + Ey + F is called **quadratic function**; the sign of the discriminant «B^{2} - 4AC ”of said function determines the shape of the curve: parabola, ellipse or circle, or hyperbola.

**Curve type identification**

- Expand and simplify the given quadratic equation.
- Rearrange the given equation and rewrite it as the general formula “Ax
^{2 }+ Bxy + Cxy^{2}+ Dx + Ey + F = 0 ”. - Compare the equation obtained in step 2 with the general formula to find y.
- Finding the discriminant «B
^{2}- 4AC ”:

- If B
^{2}- 4AC = 0, then the given equation represents a**Parabola**. - If B
^{2}- 4AC <0, then the given equation represents a**ellipse**or a**circle.** - If B
^{2}- 4AC> 0, then the given equation represents a**hyperbola**.

**Example 1: **Identify the type of curve for the equation y^{2 }- 4y - 2x + 3 = 0.

Rearranging the equation in the general formula “Ax^{2 }+ Bxy + Cxy^{2} + Dx + Ey + F = 0 ”:

0x^{2 }- 0xy + (1) (1) and^{2} - 2x - 4y + 3 = 0

Comparing both equations we determine that:

A = 0; B = 0; C = 1; D = - 2; E = - 4 and F = 3.

Now, when evaluating the discriminant we obtain that:

B^{2} - 4AC = (0)^{2} - 4 (0) (1) = 0

Therefore, the equation y^{2 }- 4y - 2x + 3 = 0 represents a parabola.

**Example 2: **Identify the type of curve for the 7x equation^{2 }+5 + 2xy + 4y^{2 }+ 3y - 22 = 0.

Rearranging the equation in the general formula “Ax^{2 }+ Bxy + Cxy^{2} + Dx + Ey + F = 0 ”:

7x^{2 }+ 2xy + (4) (1) and^{2 }- 5x + 3y - 22 = 0.

Comparing both equations we determine that:

A = 7; B = 2; C = 4; D = - 5; E = 3 and F = - 22.

Now, when evaluating the discriminant we obtain that:

B^{2} - 4AC = (7)^{2} - 4 (3) (4)> 0

Therefore, the 7x equation^{2 }+5 + 2xy + 4y^{2 }+ 3y - 22 = 0. represents either an ellipse or a circle.

**Example 3: **Identify the type of curve for the 8x equation^{2 }+ 3x + 4y^{2 }+ 12y = 0.

Rearranging the equation in the general formula “Ax^{2 }+ Bxy + Cxy^{2} + Dx + Ey + F = 0 ”:

8x^{2 }- 0xy + (4) (1) and^{2} + 3x + 12y + 0 = 0.

Comparing both equations we determine that:

A = 8; B = 0; C = 4; D = 3; E = 12 and F = 0.

Now, when evaluating the discriminant we obtain that:

B^{2} - 4AC = (0)^{2} - 4 (8) (4) <0

Therefore, the 8x equation^{2 }+ 3x + 4y^{2 }+ 12y = 0 represents a hyperbola.