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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

show that the set of all natural number form semi group under the composition of addition
Nikhil Reply
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Dominic
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_3_2_1
felecia
⅗ ⅔½
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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
SABAL Reply
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
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Ifeanyi
on number 2 question How did you got 2x +2
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combine like terms. x + x + 2 is same as 2x + 2
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x*x=2
felecia
2+2x=
felecia
×/×+9+6/1
Debbie
Q2 x+(x+2)+(x+4)=60 3x+6=60 3x+6-6=60-6 3x=54 3x/3=54/3 x=18 :. The numbers are 18,20 and 22
Naagmenkoma
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?
mariel Reply
Mark = x,. Don = 3x + 1 x + 3x + 1 = 113 4x = 112, x = 28 Mark = 28, Don = 85, 28 + 85 = 113
Pawel
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what is a solution set?
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find the subring of gaussian integers?
Rofiqul
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find the value of 2x=32
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divide by 2 on each side of the equal sign to solve for x
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X=16
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Want to review on complex number 1.What are complex number 2.How to solve complex number problems.
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16
Makan
x=16
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use the y -intercept and slope to sketch the graph of the equation y=6x
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4
Trista
x-2y+3z=-3 2x-y+z=7 -x+3y-z=6
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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
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(61/11,41/11,−4/11)
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x=61/11 y=41/11 z=−4/11 x=61/11 y=41/11 z=-4/11
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Need help solving this problem (2/7)^-2
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x+2y-z=7
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what is the coefficient of -4×
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-1
Shedrak
Practice Key Terms 4

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Source:  OpenStax, College algebra. OpenStax CNX. Feb 06, 2015 Download for free at https://legacy.cnx.org/content/col11759/1.3
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