<< Chapter < Page Chapter >> Page >

Graph y 2 = −16 x . Identify and label the focus, directrix, and endpoints of the latus rectum.

Focus: ( 4 , 0 ) ; Directrix: x = 4 ; Endpoints of the latus rectum: ( 4 , ± 8 )

Got questions? Get instant answers now!

Graphing a parabola with vertex (0, 0) and the y -axis as the axis of symmetry

Graph x 2 = −6 y . Identify and label the focus , directrix    , and endpoints of the latus rectum    .

The standard form that applies to the given equation is x 2 = 4 p y . Thus, the axis of symmetry is the y -axis. It follows that:

  • 6 = 4 p , so p = 3 2 . Since p < 0 , the parabola opens down.
  • the coordinates of the focus are ( 0 , p ) = ( 0 , 3 2 )
  • the equation of the directrix is y = p = 3 2
  • the endpoints of the latus rectum can be found by substituting   y = 3 2   into the original equation, ( ± 3 , 3 2 )

Next we plot the focus, directrix, and latus rectum, and draw a smooth curve to form the parabola    .

Got questions? Get instant answers now!
Got questions? Get instant answers now!

Graph x 2 = 8 y . Identify and label the focus, directrix, and endpoints of the latus rectum.

Focus: ( 0 , 2 ) ; Directrix: y = −2 ; Endpoints of the latus rectum: ( ± 4 , 2 ) .

Got questions? Get instant answers now!

Writing equations of parabolas in standard form

In the previous examples, we used the standard form equation of a parabola to calculate the locations of its key features. We can also use the calculations in reverse to write an equation for a parabola when given its key features.

Given its focus and directrix, write the equation for a parabola in standard form.

  1. Determine whether the axis of symmetry is the x - or y -axis.
    1. If the given coordinates of the focus have the form ( p , 0 ) , then the axis of symmetry is the x -axis. Use the standard form y 2 = 4 p x .
    2. If the given coordinates of the focus have the form ( 0 , p ) , then the axis of symmetry is the y -axis. Use the standard form x 2 = 4 p y .
  2. Multiply 4 p .
  3. Substitute the value from Step 2 into the equation determined in Step 1.

Writing the equation of a parabola in standard form given its focus and directrix

What is the equation for the parabola    with focus ( 1 2 , 0 ) and directrix     x = 1 2 ?

The focus has the form ( p , 0 ) , so the equation will have the form y 2 = 4 p x .

  • Multiplying 4 p , we have 4 p = 4 ( 1 2 ) = −2.
  • Substituting for 4 p , we have y 2 = 4 p x = −2 x .

Therefore, the equation for the parabola is y 2 = −2 x .

Got questions? Get instant answers now!
Got questions? Get instant answers now!

What is the equation for the parabola with focus ( 0 , 7 2 ) and directrix y = 7 2 ?

x 2 = 14 y .

Got questions? Get instant answers now!

Graphing parabolas with vertices not at the origin

Like other graphs we’ve worked with, the graph of a parabola can be translated. If a parabola is translated h units horizontally and k units vertically, the vertex will be ( h , k ) . This translation results in the standard form of the equation we saw previously with x replaced by ( x h ) and y replaced by ( y k ) .

To graph parabolas with a vertex ( h , k ) other than the origin, we use the standard form ( y k ) 2 = 4 p ( x h ) for parabolas that have an axis of symmetry parallel to the x -axis, and ( x h ) 2 = 4 p ( y k ) for parabolas that have an axis of symmetry parallel to the y -axis. These standard forms are given below, along with their general graphs and key features.

Standard forms of parabolas with vertex ( h , k )

[link] and [link] summarize the standard features of parabolas with a vertex at a point ( h , k ) .

Axis of Symmetry Equation Focus Directrix Endpoints of Latus Rectum
y = k ( y k ) 2 = 4 p ( x h ) ( h + p ,   k ) x = h p ( h + p ,   k ± 2 p )
x = h ( x h ) 2 = 4 p ( y k ) ( h ,   k + p ) y = k p ( h ± 2 p ,   k + p )
(a) When p > 0 , the parabola opens right. (b) When p < 0 , the parabola opens left. (c) When p > 0 , the parabola opens up. (d) When p < 0 , the parabola opens down.

Questions & Answers

The sequence is {1,-1,1-1.....} has
amit Reply
circular region of radious
Kainat Reply
how can we solve this problem
Joel Reply
Sin(A+B) = sinBcosA+cosBsinA
Eseka Reply
Prove it
Eseka
Please prove it
Eseka
hi
Joel
June needs 45 gallons of punch. 2 different coolers. Bigger cooler is 5 times as large as smaller cooler. How many gallons in each cooler?
Arleathia Reply
find the sum of 28th term of the AP 3+10+17+---------
Prince Reply
I think you should say "28 terms" instead of "28th term"
Vedant
if sequence sn is a such that sn>0 for all n and lim sn=0than prove that lim (s1 s2............ sn) ke hole power n =n
SANDESH Reply
write down the polynomial function with root 1/3,2,-3 with solution
Gift Reply
if A and B are subspaces of V prove that (A+B)/B=A/(A-B)
Pream Reply
write down the value of each of the following in surd form a)cos(-65°) b)sin(-180°)c)tan(225°)d)tan(135°)
Oroke Reply
Prove that (sinA/1-cosA - 1-cosA/sinA) (cosA/1-sinA - 1-sinA/cosA) = 4
kiruba Reply
what is the answer to dividing negative index
Morosi Reply
In a triangle ABC prove that. (b+c)cosA+(c+a)cosB+(a+b)cisC=a+b+c.
Shivam Reply
give me the waec 2019 questions
Aaron Reply
the polar co-ordinate of the point (-1, -1)
Sumit Reply
Practice Key Terms 4

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Algebra and trigonometry' conversation and receive update notifications?

Ask