Parabola Formulas

 

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Parabolas: Standard Form (Vertex Form)

Vertical:

y = a (x - h)2 + k

If "a" is positive the parabola opens up and has a minimum value.

If "a" is negative the parabola opens down and has a maximum value.

|a| = 1; normal width

|a| > 1; narrow width (vertical stretch)

|a| < 1; wider width (vertical shrink)

vertex (h, k)

axis of symmetry: x = h

The minimum or maximum is the same as the "y" value of the vertex.

The vertex is the midpoint between the focus and the directrix.

The eccentricity of a parabola is the distance from the focus to any point on the graph divided by the distance from that same point on the graph to the directrix.  Since this the same for a parabola, the eccentricity, e = 1.

 

  

 

 

Horizontal:

x = a (y - k)2 + h

If a is positive the parabola opens right

If a is negative the parabola opens left

|a| = 1 normal width

|a| > 1 narrow width (horizontal stretch)

|a| < 1 wider width (horizontal shrink)

vertex (h, k)

axis of symmetry: y = k

The vertex is the midpoint between the focus and the directrix.

The eccentricity of a parabola is the distance from the focus to any point on the graph divided by the distance from that same point on the graph to the directrix.  Since this the same for a parabola, the eccentricity, e = 1.

  

 

 

Parabolas in General Form:Ax² + By² + Cx + Dy + E = 0

If either A = 0 or B = 0 then the equation defines a parabola (x² or y² is missing).  Isolate the variable that is not squared and use the completing the square method to convert the equation to that of a Parabola in Standard Form. 

Hint: Solve for y if there is no y² in the equation or solve for x if there is no x² in the equation.

 

 

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