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Graph ( x + 2 ) 2 = −20 ( y 3 ) . Identify and label the vertex, axis of symmetry, focus, directrix, and endpoints of the latus rectum.

Vertex: ( 2 , 3 ) ; Axis of symmetry: x = −2 ; Focus: ( 2 , 2 ) ; Directrix: y = 8 ; Endpoints of the latus rectum: ( 12 , 2 ) and ( 8 , 2 ) .

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Solving applied problems involving parabolas

As we mentioned at the beginning of the section, parabolas are used to design many objects we use every day, such as telescopes, suspension bridges, microphones, and radar equipment. Parabolic mirrors, such as the one used to light the Olympic torch, have a very unique reflecting property. When rays of light parallel to the parabola’s axis of symmetry    are directed toward any surface of the mirror, the light is reflected directly to the focus. See [link] . This is why the Olympic torch is ignited when it is held at the focus of the parabolic mirror.

Reflecting property of parabolas

Parabolic mirrors have the ability to focus the sun’s energy to a single point, raising the temperature hundreds of degrees in a matter of seconds. Thus, parabolic mirrors are featured in many low-cost, energy efficient solar products, such as solar cookers, solar heaters, and even travel-sized fire starters.

Solving applied problems involving parabolas

A cross-section of a design for a travel-sized solar fire starter is shown in [link] . The sun’s rays reflect off the parabolic mirror toward an object attached to the igniter. Because the igniter is located at the focus of the parabola, the reflected rays cause the object to burn in just seconds.

  1. Find the equation of the parabola that models the fire starter. Assume that the vertex of the parabolic mirror is the origin of the coordinate plane.
  2. Use the equation found in part (a) to find the depth of the fire starter.
Cross-section of a travel-sized solar fire starter
  1. The vertex of the dish is the origin of the coordinate plane, so the parabola will take the standard form x 2 = 4 p y , where p > 0. The igniter, which is the focus, is 1.7 inches above the vertex of the dish. Thus we have p = 1.7.
    x 2 = 4 p y Standard form of upward-facing parabola with vertex (0,0) x 2 = 4 ( 1.7 ) y Substitute 1 .7 for  p . x 2 = 6.8 y Multiply .
  2. The dish extends 4.5 2 = 2.25 inches on either side of the origin. We can substitute 2.25 for x in the equation from part (a) to find the depth of the dish.
             x 2 = 6.8 y Equation found in part (a) . ( 2.25 ) 2 = 6.8 y Substitute 2 .25 for  x .            y 0.74   Solve for  y .

    The dish is about 0.74 inches deep.

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Balcony-sized solar cookers have been designed for families living in India. The top of a dish has a diameter of 1600 mm. The sun’s rays reflect off the parabolic mirror toward the “cooker,” which is placed 320 mm from the base.

  1. Find an equation that models a cross-section of the solar cooker. Assume that the vertex of the parabolic mirror is the origin of the coordinate plane, and that the parabola opens to the right (i.e., has the x -axis as its axis of symmetry).
  2. Use the equation found in part (a) to find the depth of the cooker.
  1. y 2 = 1280 x
  2. The depth of the cooker is 500 mm
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Access these online resources for additional instruction and practice with parabolas.

Questions & Answers

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SLIMANE
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Jenica
Hey I am new to precalculus, and wanted clarification please on what sine is as I am floored by the terms in this app? I don't mean to sound stupid but I have only completed up to college algebra.
rachel Reply
I don't know if you are looking for a deeper answer or not, but the sine of an angle in a right triangle is the length of the opposite side to the angle in question divided by the length of the hypotenuse of said triangle.
Marco
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Jenica
if you remember sine, cosine, and tangent from geometry, all the relationships are the same but they use x y and r instead (x is adjacent, y is opposite, and r is hypotenuse).
Natalie
it is better to use unit circle than triangle .triangle is only used for acute angles but you can begin with. Download any application named"unit circle" you find in it all you need. unit circle is a circle centred at origine (0;0) with radius r= 1.
SLIMANE
What is domain
johnphilip
the standard equation of the ellipse that has vertices (0,-4)&(0,4) and foci (0, -15)&(0,15) it's standard equation is x^2 + y^2/16 =1 tell my why is it only x^2? why is there no a^2?
Reena Reply
what is foci?
Reena Reply
This term is plural for a focus, it is used for conic sections. For more detail or other math questions. I recommend researching on "Khan academy" or watching "The Organic Chemistry Tutor" YouTube channel.
Chris
how to determine the vertex,focus,directrix and axis of symmetry of the parabola by equations
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Den Reply
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Jeffrey
Confunction Identity
Barcenas
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meena
For each year t, the population of a forest of trees is represented by the function A(t) = 117(1.029)t. In a neighboring forest, the population of the same type of tree is represented by the function B(t) = 86(1.025)t.
Shakeena Reply
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Shakeena
32.243
Kenard
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Rhudy Reply
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Drew Reply
It's just like any other number. The important thing to know is that they exist and can be used in computations like any number.
Steve
I would like to add that they are used in AC signal analysis for one thing
Scott
Good call Scott. Also radar signals I believe.
Steve
They are used in any profession where the phase of a waveform has to be accounted for in the calculations. Imagine two electrical signals in a wire that are out of phase by 90°. At some times they will interfere constructively, others destructively. Complex numbers simplify those equations
Tim
Practice Key Terms 4

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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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