### Parts of a Parabola

A parabola is a conic section defined in terms of distance to a fixed point and a fixed line.

The graphs of quadratic functions are parabolas. However, a parabola is defined differently when it is considered as a conic section. A **parabola as a conic section** is the set of points such that the distance from a fixed point, the focus, is equal to the distance to a fixed line, the **directrix**.

The properties of a parabola include:

- For any point on a parabola, the distance from the point to the focus is equal to the distance from that same point to the directrix.
- The
**vertex of a parabola**is the midpoint of the segment from the focus to the directrix. - The axis of symmetry of the parabola divides the parabola into two halves that are mirror images. It passes through both the focus and the vertex.

$y=\left(\frac{1}{4a}\right)x^2$

$x^2=4ay$

- If $a>0$, the parabola opens upward.
- If $a<0$, the parabola opens downward.
- The focus is at $(0,a)$, and the directrix is at $y=-a$.

### Parabolas with a Vertical Axis of Symmetry

Opens Upward | Opens Downward |
---|---|

$x^2=8y$ | $x^2=-8y$ |

Focus: $(0,2)$ | Focus: $(0,-2)$ |

Directrix: $y = -2$ | Directrix: $y=2$ |

A parabola with a horizontal axis of symmetry and its vertex at the origin has an equation that can be written in the form:
It can also be rewritten as:

$x=\left(\frac{1}{4a}\right)y^2$

$y^2=4ax$

- If $a>0$, the parabola opens to the right.
- If $a<0$, the parabola opens to the left.
- The focus is at $(a,0)$, and the directrix is at $x=-a$.

### Parabolas with a Horizontal Axis of Symmetry

Opens Right | Opens Left |
---|---|

$y^2=8x$ | $y^2=-8x$ |

Focus: $(2,0)$ | Focus: $(-2,0)$ |

Directrix: $x=-2$ | Directrix: $x=2$ |

### Graphing Parabolas

A parabola with a given equation can be graphed in the coordinate plane by locating the vertex, focus, and directrix.

A parabola may or may not have its vertex at the origin. A parabola with a vertex at the point (

*h*,*k*) represents a horizontal translation by*h*units and a vertical translation by*k*units of a parabola with its vertex at the origin.### Equations of Parabolas with Vertex at $(h,k)$

Equation | Axis of Symmetry | Focus | Directrix |
---|---|---|---|

$(x-h)^2=4a(y-k)$ |
Vertical: $x=h$ |
$(h,k+a)$ |
$y = k-a$ |

$(y-k)^2=4a(x-h)$ |
Horizontal: $y=k$ |
$(h+a,k)$ |
$x = h-a$ |

Step-By-Step Example

Graphing a Parabola

A parabola has the equation:
Calculate the vertex, focus, and directrix of the parabola. Sketch a graph of the parabola.

$8(y-2)=(x+5)^2$

Step 1

Write the equation in standard form.

The squared variable in the equation is $x$, so the parabola has a vertical axis of symmetry and the standard form of its equation is:$(x-h)^2=4a(y-k)$

$\begin{aligned}8(y-2)&=(x+5)^2\\(x+5)^2&=8(y-2)\\(x-(-5))^2&=8(y-2)\\(x-(-5))^2&=4(2)(y-2)\end{aligned}$

Step 2

Use the standard form of the equation to identify the values of $h$, $k$, and $a$.
So, $h=-5$, $k=2$, and $a=2$.

$\begin{aligned}(x-h)^2&=4a(y-k)\\(x-(-5))^2&=4(2)(y-2)\end{aligned}$

Step 3

Determine the vertex, focus, and directrix.

The vertex $(h, k)$ is $(-5, 2)$.

The coordinates of the focus are $(h, k+a)$.

The focus is $(-5, 2+2)$, or $(-5, 4)$.

The equation of the directrix is:$y = k-a$

$y=2-2=0$

Step 4

Plot the focus and vertex, and graph the directrix.

To locate another point on the parabola, draw a square with one vertex at the focus and one side along the directrix. The vertex $(-1, 4)$ of this square is a point on the parabola. Use symmetry to locate the point $(-9,4)$.Solution

Draw the parabola through the three points.