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Algebra and Trigonometry (MindTap ...

4th Edition
James Stewart + 2 others
ISBN: 9781305071742

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BuyFindarrow_forward

Algebra and Trigonometry (MindTap ...

4th Edition
James Stewart + 2 others
ISBN: 9781305071742
Textbook Problem

SKILLS

59-60 ■Families of Parabolas (a) Find equations for the family of parabolas with the given description. (b) Draw the graphs. What do you conclude?

The family of parabolas with vertex at the origin and with directrixes y = 1 2 , y = 1 , y = 4 ,   and y = 8 .

To determine

(a)

To find:

The equations for the family of parabolas with vertex at the origin and with directrixes y=12,y=1,y=4,y=8.

Explanation

Given:

The directrix is y=12,y=1,y=4,y=8 and the vertex is V(0,0).

Approach:

The general equation for a parabola with a horizontal axis having the vertex at the origin and the focus F(0,p) is as follows.

x2=4py …… (1)

Here, p is the distance from the origin.

The focus is F(0,p) and the directrix line is y=p

The parabola opens to the right, if p>0 or to the left if p<0.

Calculation:

The directrix of a parabola is y=12

It is in the form of the directrix line is y=p.

Compare the value of x above two expressions. Then,

p=12p=12

Thus, the focus is F(0,12)

Substitute 12 for p in equation (1).

x2=4(12)y=2y

Therefore, the required equation of a parabola is x2=2y.

Similarly, The directrix of a parabola is y=1

It is in the form of the directrix line is y=p.

Compare the value of x above two expressions. Then,

p=1p=1

Thus, the focus is F(0,1)

Substitute 1 for p in equation (1).

x2=4(1)y=4y

Therefore, the required equation of a parabola is x2=4y

To determine

(b)

To draw:

The graph of the equation for the parabola x2=2y,x2=4y,x2=16y,x2=32y.

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