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 |