Graphing Parabolas with Vertices Not at the Origin
Like other graphs we’ve worked with, the graph of a parabola can be translated. If a parabola is translated [latex]h[/latex] units horizontally and [latex]k[/latex] units vertically, the vertex will be [latex]\left(h,k\right)[/latex]. This translation results in the standard form of the equation we saw previously with [latex]x[/latex] replaced by [latex]\left(x-h\right)[/latex] and [latex]y[/latex] replaced by [latex]\left(y-k\right)[/latex].
To graph parabolas with a vertex [latex]\left(h,k\right)[/latex] other than the origin, we use the standard form [latex]{\left(y-k\right)}^{2}=4p\left(x-h\right)[/latex] for parabolas that have an axis of symmetry parallel to the x-axis, and [latex]{\left(x-h\right)}^{2}=4p\left(y-k\right)[/latex] for parabolas that have an axis of symmetry parallel to the y-axis. These standard forms are given below, along with their general graphs and key features.
parabolas with vertex (h,k)
| Axis of Symmetry | Equation | Focus | Directrix | Endpoints of Latus Rectum |
| [latex]y=k[/latex] | [latex]{\left(y-k\right)}^{2}=4p\left(x-h\right)[/latex] | [latex]\left(h+p,\text{ }k\right)[/latex] | [latex]x=h-p[/latex] | [latex]\left(h+p,\text{ }k\pm 2p\right)[/latex] |
| [latex]x=h[/latex] | [latex]{\left(x-h\right)}^{2}=4p\left(y-k\right)[/latex] | [latex]\left(h,\text{ }k+p\right)[/latex] | [latex]y=k-p[/latex] | [latex]\left(h\pm 2p,\text{ }k+p\right)[/latex] |
- Determine which of the standard forms applies to the given equation: [latex]{\left(y-k\right)}^{2}=4p\left(x-h\right)[/latex] or [latex]{\left(x-h\right)}^{2}=4p\left(y-k\right)[/latex].
- Use the standard form identified in Step 1 to determine the vertex, axis of symmetry, focus, equation of the directrix, and endpoints of the latus rectum.
- If the equation is in the form [latex]{\left(y-k\right)}^{2}=4p\left(x-h\right)[/latex], then:
- use the given equation to identify [latex]h[/latex] and [latex]k[/latex] for the vertex, [latex]\left(h,k\right)[/latex]
- use the value of [latex]k[/latex] to determine the axis of symmetry, [latex]y=k[/latex]
- set [latex]4p[/latex] equal to the coefficient of [latex]\left(x-h\right)[/latex] in the given equation to solve for [latex]p[/latex]. If [latex]p>0[/latex], the parabola opens right. If [latex]p<0[/latex], the parabola opens left.
- use [latex]h,k[/latex], and [latex]p[/latex] to find the coordinates of the focus, [latex]\left(h+p,\text{ }k\right)[/latex]
- use [latex]h[/latex] and [latex]p[/latex] to find the equation of the directrix, [latex]x=h-p[/latex]
- use [latex]h,k[/latex], and [latex]p[/latex] to find the endpoints of the latus rectum, [latex]\left(h+p,k\pm 2p\right)[/latex]
- If the equation is in the form [latex]{\left(x-h\right)}^{2}=4p\left(y-k\right)[/latex], then:
- use the given equation to identify [latex]h[/latex] and [latex]k[/latex] for the vertex, [latex]\left(h,k\right)[/latex]
- use the value of [latex]h[/latex] to determine the axis of symmetry, [latex]x=h[/latex]
- set [latex]4p[/latex] equal to the coefficient of [latex]\left(y-k\right)[/latex] in the given equation to solve for [latex]p[/latex]. If [latex]p>0[/latex], the parabola opens up. If [latex]p<0[/latex], the parabola opens down.
- use [latex]h,k[/latex], and [latex]p[/latex] to find the coordinates of the focus, [latex]\left(h,\text{ }k+p\right)[/latex]
- use [latex]k[/latex] and [latex]p[/latex] to find the equation of the directrix, [latex]y=k-p[/latex]
- use [latex]h,k[/latex], and [latex]p[/latex] to find the endpoints of the latus rectum, [latex]\left(h\pm 2p,\text{ }k+p\right)[/latex]
- If the equation is in the form [latex]{\left(y-k\right)}^{2}=4p\left(x-h\right)[/latex], then:
- Plot the vertex, axis of symmetry, focus, directrix, and latus rectum, and draw a smooth curve to form the parabola.
Graph [latex]{\left(y+1\right)}^{2}=4\left(x - 8\right)[/latex]. Identify and label the vertex, axis of symmetry, focus, directrix, and endpoints of the latus rectum.
