Calculus
Calculus
7th Edition
ISBN: 9781524916817
Author: SMITH KARL J, STRAUSS MONTY J, TODA MAGDALENA DANIELE
Publisher: Kendall Hunt Publishing
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Question
Chapter F, Problem 1PS
To determine

Draw a graph of the parabola using given data.

Expert Solution & Answer
Check Mark

Answer to Problem 1PS

The equations of the parabola are

  x2=8yandy2=8x

Explanation of Solution

Given:

The distance between the focus and the directrix is the 4 .

Calculation:

A parabola is the set of all points in the plane equidistant from a given point (focus) and a given line (directrix).

The vertex, exactly between the focus and directrix.

Consider a parabola with focus (0,c) and directrix y=c , where c is any positive number. Vertex of the parabola is (0,0) .

Let (x,y) be any point on the parabola. Then from definition of the parabola

  ( x0)2+( yc)2=|y+c|

Squaring both sides.

  ( ( x0 ) 2 + ( yc ) 2 )2=(| y+c|)2(x0)2+(yc)2=(y+c)2x2+y2+c22yc=y2+c2+2ycx2=4cy

Similarly for negative c

  y2=4cx

and for positive c .

  x2=4cyy2=4cx

In the given problem,

The vertex, exactly between the focus and directrix.

The distance between vertex to directrix =42=2

Now,

Directrixis y=2andx=2 .

Equation of the parabolas are

  x2=42yx2=8yy2=42xy2=8x

Write the equation in the standard form of 4f(yk)=(xh)2

  4(2)(y0)=(x0)2

Direction it opens to the downward, because the value of ‘ a ’ is negative.

Vertex =(h,k)=(0,0)

Focus =(0,0+f)=(0,02)=(0,2) .

Directrix,

  y=0f=0(2)=2

Draw a table for the x and y values.

    x04488
    y=x2802288

Hence the graph of the equation is given below.

  Calculus, Chapter F, Problem 1PS , additional homework tip  1

Write the equation in the standard form of 4f(xh)=(yk)2

  4(2)(x0)=(y0)2

Direction it opens to the left, because the value of ‘ a ’ is negative.

Vertex =(h,k)=(0,0)

Focus =(0+f,0)=(02,0)=(2,0) .

Directrix,

  x=0f=0(2)=2

Draw a table for the x and y values.

    x02288
    x=y2804488

Hence the graph of the equation is given below.

  Calculus, Chapter F, Problem 1PS , additional homework tip  2

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