# Equations of a parabolas graph Cheat Sheet

Equations of a Parabola: Vertex at `(0,0)`

; Focus on Axis; `a > 0`

Vertex | Focus | Directdix | Equation | Description |

$$(0,0)$$ | $$(a,0)$$ | $$x=-a$$ | $${ y }^{ 2 }=4ax$$ | Axis of symmetry is the x-axis, opens right |

$$(0,0)$$ | $$(-a,0)$$ | $$x=a$$ | $${ y }^{ 2 }=-4ax$$ | Axis of symmetry is the x-axis,opens left |

$$(0,0)$$ | $$(0,a)$$ | $$y=-a$$ | $${ x }^{ 2 }=4ay$$ | Axis of symmetry is the y-axis,opens up |

$$(0,0)$$ | $$(0,-a)$$ | $$y=a$$ | $${ x }^{ 2 }=-4ay$$ | Axis of symmetry is the y-axis,opens down |

Equations of a Parabola: Vertex at `(h,k)`

; Axis of Symmetry Parallel to a Coordinate Axis; `a > 0`

Vertex | Focus | Directdix | Equation | Description |

$$(h,k)$$ | $$(h+a,k)$$ | $$x=h-a$$ | $${ (y-k) }^{ 2 }=4a(x-h)$$ | Axis of symmetry is parallel to the x-axis, opens right |

$$(h,k)$$ | $$(h-a,k)$$ | $$x=h+a$$ | $${ (y-k) }^{ 2 }=-4a(x-h)$$ | Axis of symmetry is parallel to the x-axis, opens left |

$$(h,k)$$ | $$(h,k+a)$$ | $$y=k-a$$ | $${ (x-h) }^{ 2 }=4a(y-k)$$ | Axis of symmetry is parallel to the y-axis, opens up |

$$(h,k)$$ | $$(h,k-a)$$ | $$y=k+a$$ | $${ (x-h) }^{ 2 }=-4a(y-k)$$ | Axis of symmetry is the y-axis,opens down |