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.
standard forms of parabolas with vertex [latex](h, k)[/latex]
The table summarizes the standard features of parabolas with a vertex at a point [latex]\left(h,k\right)[/latex].
| Axis of Symmetry | Equation | Focus | Directrix | Endpoints of focal diameter |
| [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 focal diameter.
- 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 focal diameter, [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 focal diameter, [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 focal diameter, and draw a smooth curve to form the parabola.
Sometimes, our equations for parabolas might not initially appear in the standard form, which can make it challenging to identify their key features like the vertex, focus, and directrix. To address this, we often need to convert the given equation into the standard form by completing the square. This transformation allows us to rewrite the equation in a more familiar format, making it easier to analyze and graph the parabola. Completing the square helps us identify important components of the parabola and better understand its geometric properties.
