# Identifying the Graph of an Equation Without Plotting

Being familiar of the graph of common equation is very helpful specially when dealing with complex problem. One important application is for us to easily identify if an equation is a function or not. The graph is the soul of an equation.

General Equation:

$Ax^2+By^2+Cx+Dy+E=0$   this is also the general equations of conic sections. Where {A,B,C,D,E} are real numbers.

Case 1: A=B

$Ax^2+Ay^2+Cx+Dy+E=0$  ,  this is an equation of a CIRCLE.

Example:    $x^2+y^2+2x+y-4=0$

But there are cases that this is an equation of an EMPTY SET.

If      $\displaystyle\frac{C^2+D^2}{4A}-E<0$.

Case 2: A & B are of the same sign but not equal.

$Ax^2+By^2+Cx+Dy+E=0$   or  $-x^2-By^2+Cx+Dy+E=0$ ,     this is an equation of an ELLIPSE

Example:     $2 x^2+3y^2+-4x+5y-2=0$

Case 3: A and B are of different signs. Whether A and B equal or unequal.

$Ax^2-By^2+Cx+Dy+E=0$ ,   this is an equation of a HYPERBOLA

Example:    $x^2-y^2-2x+2y-2=0$

Case 4: A=0 or B=0

$By^2+Cx+Dy+E=0$  or $Ax^2+Cx+Dy+E=0$ ,   this is an equation of PARABOLA.

Example (A=0)  :    $y^2-3x+4y+5=0$

Example (B=0): $3x^2-2x+2y+4=0$

Case 5: A=0 and B=0

$Cx+Dy+E=0$ ,  this is an equation of a SLANT STRAIGHT LINE

Example:    $2x-3y+5=0$

Case 6. A=0, B=0,C=0

$Dy+E=0$ ,  this is an equation of a HORIZONTAL LINE

Example: $3y+4=0$

Case 7: A=0,B=0,D=0

$Cx+E=0$ ,  this is an equation of a VERTICAL LINE

Example:     $2x-1=0$

Practice Problems:

Direction: Identify the equation if it is a graph of parabola, hyperbola, circle, ellipse,Empty set, slant straight line, vertical line or horizontal line.

1.   $3x^2+4y^2+2x-3y+9=0$

2.   $x^2-2y^2+4x+5y-6=0$

3.   $3x-2y+1=0$

4.   $x^2+y^2-4x+3y-6=0$

5.   $5y+2=0$

6.   $4x^2+8x-3y=0$

7.   $2y^2-3x+4y+7=0$

8.   $4x^2+4y^2+1=0$

9.   $3x^2+4y^2+5=0$

10.   $x^2+y^2+4x-4y+9=0$