# 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. The Olympic torch concludes its journey around the world when it is used to light the Olympic cauldron during the opening ceremony. (credit: Ken Hackman, U.S. Air Force)

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    .

explain and give four Example hyperbolic function
The denominator of a certain fraction is 9 more than the numerator. If 6 is added to both terms of the fraction, the value of the fraction becomes 2/3. Find the original fraction. 2. The sum of the least and greatest of 3 consecutive integers is 60. What are the valu
1. x + 6 2 -------------- = _ x + 9 + 6 3 x + 6 3 ----------- x -- (cross multiply) x + 15 2 3(x + 6) = 2(x + 15) 3x + 18 = 2x + 30 (-2x from both) x + 18 = 30 (-18 from both) x = 12 Test: 12 + 6 18 2 -------------- = --- = --- 12 + 9 + 6 27 3
Pawel
2. (x) + (x + 2) = 60 2x + 2 = 60 2x = 58 x = 29 29, 30, & 31
Pawel
Mark and Don are planning to sell each of their marble collections at a garage sale. If Don has 1 more than 3 times the number of marbles Mark has, how many does each boy have to sell if the total number of marbles is 113?
Mark = x,. Don = 3x + 1 x + 3x + 1 = 113 4x = 112, x = 28 Mark = 28, Don = 85, 28 + 85 = 113
Pawel
how do I set up the problem?
what is a solution set?
Harshika
find the subring of gaussian integers?
Rofiqul
hello, I am happy to help!
Abdullahi
hi mam
Mark
find the value of 2x=32
divide by 2 on each side of the equal sign to solve for x
corri
X=16
Michael
Want to review on complex number 1.What are complex number 2.How to solve complex number problems.
Beyan
yes i wantt to review
Mark
use the y -intercept and slope to sketch the graph of the equation y=6x
how do we prove the quadratic formular
Darius
hello, if you have a question about Algebra 2. I may be able to help. I am an Algebra 2 Teacher
thank you help me with how to prove the quadratic equation
Seidu
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Opoku
what is math number
4
Trista
x-2y+3z=-3 2x-y+z=7 -x+3y-z=6
can you teacch how to solve that🙏
Mark
Solve for the first variable in one of the equations, then substitute the result into the other equation. Point For: (6111,4111,−411)(6111,4111,-411) Equation Form: x=6111,y=4111,z=−411x=6111,y=4111,z=-411
Brenna
(61/11,41/11,−4/11)
Brenna
x=61/11 y=41/11 z=−4/11 x=61/11 y=41/11 z=-4/11
Brenna
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 By By OpenStax By Hannah Sheth By Sean WiffleBoy By JavaChamp Team By By Joli Julianna By Vanessa Soledad By Tod McGrath By David Corey By OpenStax