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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