# 8.3 The parabola

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In this section, you will:
• Graph parabolas with vertices at the origin.
• Write equations of parabolas in standard form.
• Graph parabolas with vertices not at the origin.
• Solve applied problems involving parabolas.

Did you know that the Olympic torch is lit several months before the start of the games? The ceremonial method for lighting the flame is the same as in ancient times. The ceremony takes place at the Temple of Hera in Olympia, Greece, and is rooted in Greek mythology, paying tribute to Prometheus, who stole fire from Zeus to give to all humans. One of eleven acting priestesses places the torch at the focus of a parabolic mirror (see [link] ), which focuses light rays from the sun to ignite the flame.

Parabolic mirrors (or reflectors) are able to capture energy and focus it to a single point. The advantages of this property are evidenced by the vast list of parabolic objects we use every day: satellite dishes, suspension bridges, telescopes, microphones, spotlights, and car headlights, to name a few. Parabolic reflectors are also used in alternative energy devices, such as solar cookers and water heaters, because they are inexpensive to manufacture and need little maintenance. In this section we will explore the parabola and its uses, including low-cost, energy-efficient solar designs.

## Graphing parabolas with vertices at the origin

In The Ellipse , we saw that an ellipse    is formed when a plane cuts through a right circular cone. If the plane is parallel to the edge of the cone, an unbounded curve is formed. This curve is a parabola    . See [link] .

Like the ellipse and hyperbola    , the parabola can also be defined by a set of points in the coordinate plane. A parabola is the set of all points $\text{\hspace{0.17em}}\left(x,y\right)$ in a plane that are the same distance from a fixed line, called the directrix    , and a fixed point (the focus ) not on the directrix.

In Quadratic Functions , we learned about a parabola’s vertex and axis of symmetry. Now we extend the discussion to include other key features of the parabola. See [link] . Notice that the axis of symmetry passes through the focus and vertex and is perpendicular to the directrix. The vertex is the midpoint between the directrix and the focus.

The line segment that passes through the focus and is parallel to the directrix is called the latus rectum    . The endpoints of the latus rectum lie on the curve. By definition, the distance $\text{\hspace{0.17em}}d\text{\hspace{0.17em}}$ from the focus to any point $\text{\hspace{0.17em}}P\text{\hspace{0.17em}}$ on the parabola is equal to the distance from $\text{\hspace{0.17em}}P\text{\hspace{0.17em}}$ to the directrix.

To work with parabolas in the coordinate plane , we consider two cases: those with a vertex at the origin and those with a vertex    at a point other than the origin. We begin with the former.

Let $\text{\hspace{0.17em}}\left(x,y\right)\text{\hspace{0.17em}}$ be a point on the parabola with vertex $\text{\hspace{0.17em}}\left(0,0\right),$ focus $\text{\hspace{0.17em}}\left(0,p\right),$ and directrix $\text{\hspace{0.17em}}y= -p\text{\hspace{0.17em}}$ as shown in [link] . The distance $\text{\hspace{0.17em}}d\text{\hspace{0.17em}}$ from point $\text{\hspace{0.17em}}\left(x,y\right)\text{\hspace{0.17em}}$ to point $\text{\hspace{0.17em}}\left(x,-p\right)\text{\hspace{0.17em}}$ on the directrix is the difference of the y -values: $\text{\hspace{0.17em}}d=y+p.\text{\hspace{0.17em}}$ The distance from the focus $\text{\hspace{0.17em}}\left(0,p\right)\text{\hspace{0.17em}}$ to the point $\text{\hspace{0.17em}}\left(x,y\right)\text{\hspace{0.17em}}$ is also equal to $\text{\hspace{0.17em}}d\text{\hspace{0.17em}}$ and can be expressed using the distance formula    .

what is math number
x-2y+3z=-3 2x-y+z=7 -x+3y-z=6
Need help solving this problem (2/7)^-2
x+2y-z=7
Sidiki
what is the coefficient of -4×
-1
Shedrak
the operation * is x * y =x + y/ 1+(x × y) show if the operation is commutative if x × y is not equal to -1
An investment account was opened with an initial deposit of \$9,600 and earns 7.4% interest, compounded continuously. How much will the account be worth after 15 years?
lim x to infinity e^1-e^-1/log(1+x)
given eccentricity and a point find the equiation
12, 17, 22.... 25th term
12, 17, 22.... 25th term
Akash
College algebra is really hard?
Absolutely, for me. My problems with math started in First grade...involving a nun Sister Anastasia, bad vision, talking & getting expelled from Catholic school. When it comes to math I just can't focus and all I can hear is our family silverware banging and clanging on the pink Formica table.
Carole
I'm 13 and I understand it great
AJ
I am 1 year old but I can do it! 1+1=2 proof very hard for me though.
Atone
hi
Not really they are just easy concepts which can be understood if you have great basics. I am 14 I understood them easily.
Vedant
find the 15th term of the geometric sequince whose first is 18 and last term of 387
I know this work
salma
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
hmm well what is the answer
Abhi
If f(x) = x-2 then, f(3) when 5f(x+1) 5((3-2)+1) 5(1+1) 5(2) 10
Augustine
how do they get the third part x = (32)5/4
make 5/4 into a mixed number, make that a decimal, and then multiply 32 by the decimal 5/4 turns out to be
AJ
how
Sheref
can someone help me with some logarithmic and exponential equations.
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
hmm
Abhi
is it a question of log
Abhi
🤔.
Abhi
I rally confuse this number And equations too I need exactly help
salma
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salma
Commplementary angles
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
hii
Uday
hi
salma
hi
Ayuba
Hello
opoku
hi
Ali
greetings from Iran
Ali
salut. from Algeria
Bach
hi
Nharnhar