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Key equations

Parabola, vertex at origin, axis of symmetry on x -axis y 2 = 4 p x
Parabola, vertex at origin, axis of symmetry on y -axis x 2 = 4 p y
Parabola, vertex at ( h , k ) , axis of symmetry on x -axis ( y k ) 2 = 4 p ( x h )
Parabola, vertex at ( h , k ) , axis of symmetry on y -axis ( x h ) 2 = 4 p ( y k )

Key concepts

  • A parabola is the set of all points ( x , y ) 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.
  • The standard form of a parabola with vertex ( 0 , 0 ) and the x -axis as its axis of symmetry can be used to graph the parabola. If p > 0 , the parabola opens right. If p < 0 , the parabola opens left. See [link] .
  • The standard form of a parabola with vertex ( 0 , 0 ) and the y -axis as its axis of symmetry can be used to graph the parabola. If p > 0 , the parabola opens up. If p < 0 , the parabola opens down. See [link] .
  • When given the focus and directrix of a parabola, we can write its equation in standard form. See [link] .
  • The standard form of a parabola with vertex ( h , k ) and axis of symmetry parallel to the x -axis can be used to graph the parabola. If p > 0 , the parabola opens right. If p < 0 , the parabola opens left. See [link] .
  • The standard form of a parabola with vertex ( h , k ) and axis of symmetry parallel to the y -axis can be used to graph the parabola. If p > 0 , the parabola opens up. If p < 0 , the parabola opens down. See [link] .
  • Real-world situations can be modeled using the standard equations of parabolas. For instance, given the diameter and focus of a cross-section of a parabolic reflector, we can find an equation that models its sides. See [link] .

Section exercises

Verbal

Define a parabola in terms of its focus and directrix.

A parabola is the set of points in the plane that lie equidistant from a fixed point, the focus, and a fixed line, the directrix.

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If the equation of a parabola is written in standard form and p is positive and the directrix is a vertical line, then what can we conclude about its graph?

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If the equation of a parabola is written in standard form and p is negative and the directrix is a horizontal line, then what can we conclude about its graph?

The graph will open down.

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What is the effect on the graph of a parabola if its equation in standard form has increasing values of p ?

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As the graph of a parabola becomes wider, what will happen to the distance between the focus and directrix?

The distance between the focus and directrix will increase.

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Algebraic

For the following exercises, determine whether the given equation is a parabola. If so, rewrite the equation in standard form.

y = 4 x 2

yes y = 4 ( 1 ) x 2

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( y 3 ) 2 = 8 ( x 2 )

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

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y 2 + 12 x 6 y 51 = 0

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For the following exercises, rewrite the given equation in standard form, and then determine the vertex ( V ) , focus ( F ) , and directrix   ( d )   of the parabola.

x = 8 y 2

y 2 = 1 8 x , V : ( 0 , 0 ) ; F : ( 1 32 , 0 ) ; d : x = 1 32

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y = −4 x 2

x 2 = 1 4 y , V : ( 0 , 0 ) ; F : ( 0 , 1 16 ) ; d : y = 1 16

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x = 36 y 2

y 2 = 1 36 x , V : ( 0 , 0 ) ; F : ( 1 144 , 0 ) ; d : x = 1 144

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( x 1 ) 2 = 4 ( y 1 )

( x 1 ) 2 = 4 ( y 1 ) , V : ( 1 , 1 ) ; F : ( 1 , 2 ) ; d : y = 0

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

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

( y 4 ) 2 = 2 ( x + 3 ) , V : ( 3 , 4 ) ; F : ( 5 2 , 4 ) ; d : x = 7 2

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

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( x + 4 ) 2 = 24 ( y + 1 )

( x + 4 ) 2 = 24 ( y + 1 ) , V : ( 4 , 1 ) ; F : ( 4 , 5 ) ; d : y = −7

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Questions & Answers

How look for the general solution of a trig function
collins Reply
stock therom F=(x2+y2) i-2xy J jaha x=a y=o y=b
Saurabh Reply
root under 3-root under 2 by 5 y square
Himanshu Reply
The sum of the first n terms of a certain series is 2^n-1, Show that , this series is Geometric and Find the formula of the n^th
amani Reply
cosA\1+sinA=secA-tanA
Aasik Reply
why two x + seven is equal to nineteen.
Kingsley Reply
The numbers cannot be combined with the x
Othman
2x + 7 =19
humberto
2x +7=19. 2x=19 - 7 2x=12 x=6
Yvonne
because x is 6
SAIDI
what is the best practice that will address the issue on this topic? anyone who can help me. i'm working on my action research.
Melanie Reply
simplify each radical by removing as many factors as possible (a) √75
Jason Reply
how is infinity bidder from undefined?
Karl Reply
what is the value of x in 4x-2+3
Vishal Reply
give the complete question
Shanky
4x=3-2 4x=1 x=1+4 x=5 5x
Olaiya
hi can you give another equation I'd like to solve it
Daniel
what is the value of x in 4x-2+3
Olaiya
if 4x-2+3 = 0 then 4x = 2-3 4x = -1 x = -(1÷4) is the answer.
Jacob
4x-2+3 4x=-3+2 4×=-1 4×/4=-1/4
LUTHO
then x=-1/4
LUTHO
4x-2+3 4x=-3+2 4x=-1 4x÷4=-1÷4 x=-1÷4
LUTHO
A research student is working with a culture of bacteria that doubles in size every twenty minutes. The initial population count was  1350  bacteria. Rounding to five significant digits, write an exponential equation representing this situation. To the nearest whole number, what is the population size after  3  hours?
David Reply
v=lbh calculate the volume if i.l=5cm, b=2cm ,h=3cm
Haidar Reply
Need help with math
Peya
can you help me on this topic of Geometry if l help you
litshani
( cosec Q _ cot Q ) whole spuare = 1_cosQ / 1+cosQ
Aarav Reply
A guy wire for a suspension bridge runs from the ground diagonally to the top of the closest pylon to make a triangle. We can use the Pythagorean Theorem to find the length of guy wire needed. The square of the distance between the wire on the ground and the pylon on the ground is 90,000 feet. The square of the height of the pylon is 160,000 feet. So, the length of the guy wire can be found by evaluating √(90000+160000). What is the length of the guy wire?
Maxwell Reply
the indicated sum of a sequence is known as
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Practice Key Terms 4

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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