## Parabolas Not Centered at the Origin

This section will focus on graphing parabolas given equations not centered at the origin.

### 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 units horizontally and units vertically, the vertex will be . This translation results in the standard form of the equation we saw previously with replaced by and replaced by .

To graph parabolas with a vertex other than the origin, we use the standard form for parabolas that have an axis of symmetry parallel to the -axis, and for parabolas that have an axis of symmetry parallel to the -axis. These standard forms are given below, along with their general graphs and key features.

### STANDARD FORMS OF PARABOLAS WITH VERTEX

Table 2 and Figure 9 summarize the standard features of parabolas with a vertex at a point .

Axis of Symmetry |
Equation |
Focus | Directrix | Endpoints of Latus Rectum |
---|---|---|---|---|

**Table 2**

**Figure 9** (a) When , the parabola opens right. (b) When , the parabola opens left. (c) When , the parabola opens up. (d) When , the parabola opens down.

### HOW TO

**Given a standard form equation for a parabola centered at , sketch the graph.**

- Determine which of the standard forms applies to the given equation: or .
- 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 , then:
- use the given equation to identify and for the vertex,
- use the value of to determine the axis of symmetry,
- set equal to the coefficient of in the given equation to solve for . If , the parabola opens right. If , the parabola opens left.
- use , and to find the coordinates of the focus,
- use and to find the equation of the directrix,
- use , and to find the endpoints of the latus rectum,

- If the equation is in the form , then:
- use the given equation to identify and for the vertex,
- use the value of to determine the axis of symmetry,
- set equal to the coefficient of in the given equation to solve for . If , the parabola opens up. If , the parabola opens down.
- use , and to find the coordinates of the focus,
- use and to find the equation of the directrix,
- use , and to find the endpoints of the latus rectum,

- If the equation is in the form , then:
- Plot the vertex, axis of symmetry, focus, directrix, and latus rectum, and draw a smooth curve to form the parabola.

### EXAMPLE 4

#### Graphing a Parabola with Vertex and Axis of Symmetry Parallel to the -axis

Graph . Identify and label the vertex, axis of symmetry, focus, directrix, and endpoints of the latus rectum.

#### Solution

The standard form that applies to the given equation is . Thus, the axis of symmetry is parallel to the -axis. It follows that:

- the vertex is
- the axis of symmetry is
- , so . Since , the parabola opens left.
- the coordinates of the focus are
- the equation of the directrix is
- the endpoints of the latus rectum are , or and

Next we plot the vertex, axis of symmetry, focus, directrix, and latus rectum, and draw a smooth curve to form the parabola. See Figure 10.

**Figure 10**

### TRY IT #4

Graph . Identify and label the vertex, axis of symmetry, focus, directrix, and endpoints of the latus rectum.

### EXAMPLE 5

#### Graphing a Parabola from an Equation Given in General Form

Graph . Identify and label the vertex, axis of symmetry, focus, directrix, and endpoints of the latus rectum.

#### Solution

Start by writing the equation of the parabola in standard form. The standard form that applies to the given equation is . Thus, the axis of symmetry is parallel to the -axis. To express the equation of the parabola in this form, we begin by isolating the terms that contain the variable in order to complete the square.

It follows that:

since and so the parabola opens up

the coordinates of the focus are

the equation of the directrix is

the endpoints of the latus rectum are , or and

Next we plot the vertex, axis of symmetry, focus, directrix, and latus rectum, and draw a smooth curve to form the parabola. See Figure 11.

**Figure 11**

### TRY IT #5

Source: Rice University, https://openstax.org/books/college-algebra/pages/8-3-the-parabola

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