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( y + 4 ) 2 = 16 ( x + 4 )

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y 2 + 12 x 6 y + 21 = 0

( y 3 ) 2 = −12 ( x + 1 ) , V : ( 1 , 3 ) ; F : ( 4 , 3 ) ; d : x = 2

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x 2 4 x 24 y + 28 = 0

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5 x 2 50 x 4 y + 113 = 0

( x 5 ) 2 = 4 5 ( y + 3 ) , V : ( 5 , 3 ) ; F : ( 5 , 14 5 ) ; d : y = 16 5

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y 2 24 x + 4 y 68 = 0

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x 2 4 x + 2 y 6 = 0

( x 2 ) 2 = −2 ( y 5 ) , V : ( 2 , 5 ) ; F : ( 2 , 9 2 ) ; d : y = 11 2

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y 2 6 y + 12 x 3 = 0

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3 y 2 4 x 6 y + 23 = 0

( y 1 ) 2 = 4 3 ( x 5 ) , V : ( 5 , 1 ) ; F : ( 16 3 , 1 ) ; d : x = 14 3

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x 2 + 4 x + 8 y 4 = 0

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Graphical

For the following exercises, graph the parabola, labeling the focus and the directrix.

( y 2 ) 2 = 4 3 ( x + 2 )

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−5 ( x + 5 ) 2 = 4 ( y + 5 )

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−6 ( y + 5 ) 2 = 4 ( x 4 )

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y 2 6 y 8 x + 1 = 0

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x 2 + 8 x + 4 y + 20 = 0

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3 x 2 + 30 x 4 y + 95 = 0

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y 2 8 x + 10 y + 9 = 0

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y 2 + 2 y 12 x + 61 = 0

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2 x 2 + 8 x 4 y 24 = 0

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For the following exercises, find the equation of the parabola given information about its graph.

Vertex is ( 0 , 0 ) ; directrix is y = 4 , focus is ( 0 , −4 ) .

x 2 = −16 y

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Vertex is ( 0 , 0 ) ; directrix is x = 4 , focus is ( −4 , 0 ) .

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Vertex is ( 2 , 2 ) ; directrix is x = 2 2 , focus is ( 2 + 2 , 2 ) .

( y 2 ) 2 = 4 2 ( x 2 )

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Vertex is ( −2 , 3 ) ; directrix is x = 7 2 , focus is ( 1 2 , 3 ) .

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Vertex is ( 2 , 3 ) ; directrix is x = 2 2 , focus is ( 0 , 3 ) .

( y + 3 ) 2 = −4 2 ( x 2 )

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Vertex is ( 1 , 2 ) ; directrix is y = 11 3 , focus is ( 1 , 1 3 ) .

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For the following exercises, determine the equation for the parabola from its graph.

( y 2 ) 2 = 1 4 ( x + 2 )

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( y 3 ) 2 = 4 5 ( x + 2 )

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Extensions

For the following exercises, the vertex and endpoints of the latus rectum of a parabola are given. Find the equation.

V ( 0 , 0 ) , Endpoints  ( 2 , 1 ) , ( −2 , 1 )

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V ( 0 , 0 ) , Endpoints  ( −2 , 4 ) , ( −2 , −4 )

y 2 = −8 x

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V ( 1 , 2 ) , Endpoints  ( −5 , 5 ) , ( 7 , 5 )

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V ( −3 , −1 ) , Endpoints  ( 0 , 5 ) , ( 0 , −7 )

( y + 1 ) 2 = 12 ( x + 3 )

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V ( 4 , −3 ) , Endpoints  ( 5 , 7 2 ) , ( 3 , 7 2 )

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Real-world applications

The mirror in an automobile headlight has a parabolic cross-section with the light bulb at the focus. On a schematic, the equation of the parabola is given as x 2 = 4 y . At what coordinates should you place the light bulb?

( 0 , 1 )

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If we want to construct the mirror from the previous exercise such that the focus is located at ( 0 , 0.25 ) , what should the equation of the parabola be?

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A satellite dish is shaped like a paraboloid of revolution. This means that it can be formed by rotating a parabola around its axis of symmetry. The receiver is to be located at the focus. If the dish is 12 feet across at its opening and 4 feet deep at its center, where should the receiver be placed?

At the point 2.25 feet above the vertex.

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Consider the satellite dish from the previous exercise. If the dish is 8 feet across at the opening and 2 feet deep, where should we place the receiver?

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A searchlight is shaped like a paraboloid of revolution. A light source is located 1 foot from the base along the axis of symmetry. If the opening of the searchlight is 3 feet across, find the depth.

0.5625 feet

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If the searchlight from the previous exercise has the light source located 6 inches from the base along the axis of symmetry and the opening is 4 feet, find the depth.

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An arch is in the shape of a parabola. It has a span of 100 feet and a maximum height of 20 feet. Find the equation of the parabola, and determine the height of the arch 40 feet from the center.

x 2 = −125 ( y 20 ) , height is 7.2 feet

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If the arch from the previous exercise has a span of 160 feet and a maximum height of 40 feet, find the equation of the parabola, and determine the distance from the center at which the height is 20 feet.

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An object is projected so as to follow a parabolic path given by y = x 2 + 96 x , where x is the horizontal distance traveled in feet and y is the height. Determine the maximum height the object reaches.

2304 feet

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For the object from the previous exercise, assume the path followed is given by y = −0.5 x 2 + 80 x . Determine how far along the horizontal the object traveled to reach maximum height.

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Questions & Answers

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
Aislinn Reply
cm
tijani
what is titration
John Reply
what is physics
Siyaka Reply
A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
Jude Reply
Can you compute that for me. Ty
Jude
what is the dimension formula of energy?
David Reply
what is viscosity?
David
what is inorganic
emma Reply
what is chemistry
Youesf Reply
what is inorganic
emma
Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
Adjei
please, I'm a physics student and I need help in physics
Adjanou
chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.
Pedro
A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
Krampah Reply
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
Sahid Reply
you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
Samuel Reply
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Joseph Reply
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
Ryan
what's motion
Maurice Reply
what are the types of wave
Maurice
answer
Magreth
progressive wave
Magreth
hello friend how are you
Muhammad Reply
fine, how about you?
Mohammed
hi
Mujahid
A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
yasuo Reply
Who can show me the full solution in this problem?
Reofrir Reply
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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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