Explanation Given information: The parabola of an equation x 2 = 12 y and ( x − 3 ) 2 = 12 ( y − 1 ) . Definition used: A parabola with vertex at ( 0 , 0 ) and focus at ( 0 , p ) faces upwards if p > 0 and the focus is above the vertex or faces downwards if p < 0 and the focus is below the vertex. The directrix of the parabola is y = − p . The parabola equation in the standard form is, x 2 = 4 p y .............(1) A parabola with vertex at ( 0 , 0 ) and focus at ( p , 0 ) faces right if p > 0 and the focus is on the right side of the vertex or faces left if p < 0 and the focus is left side of the vertex. The directrix of the parabola is y = − p and The parabola equation in the standard form is, y 2 = 4 p x .............(2) A parabola with vertex at ( h , k ) and focus at ( h , k + p ) faces upwards if p > 0 or faces downwards if p < 0 . The parabola equation in the standard form is, ( x − h ) 2 = 4 p ( y − k ) .............(3). Here the directrix is y = − p + k and the parabola is vertically symmetric to the line x = h . A parabola with vertex at ( h , k ) and focus at ( h + p , k ) faces right if p > 0 or faces left if p < 0 . The parabola equation in the standard form is, ( y − k ) 2 = 4 p ( x − h ) .............(4). Here the directrix is x = − p + h and the parabola is horizontally symmetric to the line y = k . Calculation: The given equation of a parabola is x 2 = 12 y . Rewrite the equation of the parabola x 2 = 12 y as, x 2 = 4 p y x 2 = 12 y x 2 = 4 ( 3 ) y Compare the equation of parabola x 2 = 4 ( 3 ) y with the general equation of parabola x 2 = 4 p y to obtain the value of p. p = 3 Substitute 3 for p in the expression ( 0 , p ) to obtain the focus of the parabola as, focus = ( 0 , p ) = ( 0 , 3 ) Substitute 3 for p in the expression y = − p to obtain the directrix of the parabola as, y = − p y = − 3 . Substitute 0 for h and 0 for k in the expression ( h , k ) to obtain the vertex of the parabola as, vertex = ( h , k ) = ( 0 , 0 ) Therefore, the focus of the parabola is ( 0 , 3 ) , directrix y = − 3 and vertex is ( 0 , 0 ) is shown in figure 1

7th Edition

ISBN: 9781305115545

Author: James Stewart, Lothar Redlin, Saleem Watson

Publisher: Cengage Learning

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Chapter 7.4, Problem 2E

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The the focus, directrix and vertex of each parabola x2=12y and (x−3)2=12(y−1).

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Chapter 7.4, Problem 1E

Chapter 7.4, Problem 3E

Chapter 7 Solutions

College Algebra

Ch. 7.1 - A parabola is the set of all points in the plane...Ch. 7.1 - The graph of the equation x2=4py is a parabola...Ch. 7.1 - The graph of the equation y2=4px is a parabola...Ch. 7.1 - Label the focus, directrix, and vertex on the...Ch. 7.1 - Prob. 5E Ch. 7.1 - Prob. 6E Ch. 7.1 - Graphs of Parabolas Match the equation with the...Ch. 7.1 - Prob. 8E Ch. 7.1 - Prob. 9E Ch. 7.1 - Prob. 10E

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The the focus, directrix and vertex of each parabola x 2 = 12 y and ( x − 3 ) 2 = 12 ( y − 1 ) .

Solution Summary: The author explains the focus, directrix, and vertex of each parabola of an equation x2=12y

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To find:

The the focus, directrix and vertex of each parabola x2=12y and (x−3)2=12(y−1).

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Chapter 7 Solutions

The the focus, directrix and vertex of each parabola x 2 = 12 y and ( x − 3 ) 2 = 12 ( y − 1 ) .

Solution Summary: The author explains the focus, directrix, and vertex of each parabola of an equation x2=12y

Concept explainers

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To find: The the focus, directrix and vertex of each parabola x2=12y and (x−3)2=12(y−1).

Want to see the full answer?

Chapter 7 Solutions

To find:

The the focus, directrix and vertex of each parabola x2=12y and (x−3)2=12(y−1).