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Parabola Help (page 2)

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By — McGraw-Hill Professional
Updated on Oct 4, 2011

Examples

Match the graphs in Figures 12.3 through 12.6 with their equations.

Conic Sections Parabolas

Fig. 12.3

Conic Sections Parabolas

Fig. 12.4

Conic Sections Parabolas

Fig. 12.5

Conic Sections Parabolas

Fig. 12.6

  • x 2 = 6 y
  • The equation is in the form ( xh ) 2 = 4 p ( yk ), so the parabola will open up or down. We have Conic Sections Parabolas . Now we know three things—that the parabola opens up (because p is positive), that the focus is Conic Sections Parabolas ), and the directrix is Conic Sections Parabolas (from Conic Sections Parabolas The graph that behaves this way is in Figure 12.5 .

  • Conic Sections Parabolas

    Fig. 12.5

  • y 2 = 6 x
  • The equation is in the form ( yk ) 2 = 4 p ( xh ), so the parabola opens to the left or to the right. We have Conic Sections Parabolas . Now we know that the parabola opens to the right, that the focus is Conic Sections Parabolas , and that the directrix is Conic Sections Parabolas . The graph for this equation is in Figure 12.3 .

  • Conic Sections Parabolas

    Fig. 12.3

  • y 2 = –6 x
  • The equation is in the form ( yk ) 2 = 4 p ( xh ), so the parabola opens to the left or to the right. We have Conic Sections Parabolas . The parabola opens to the left, the focus is Conic Sections Parabolas , and the directrix is Conic Sections Parabolas . The graph for this equation is in Figure 12.4.

  • Conic Sections Parabolas

    Fig. 12.4

  • x 2 = –6 y
  • The equation is in the form ( xh ) 2 = 4 p ( yk ), so the parabola opens up or down. We have Conic Sections Parabolas . The parabola opens down, the focus is Conic Sections Parabolas . The directrix is Conic Sections Parabolas (from Conic Sections Parabolas . The graph for this equation is in Figure 12.6.

  • Conic Sections Parabolas

    Fig. 12.6

Finding the Vertex, Focus, Directrix, and Direction of Parabola Examples

Using the information in Table 12.1, we can find the vertex, focus, directrix, and whether the parabola opens up, down, to the left, or to the right by looking at its equation.

Table 12.1

( xh ) 2 = 4 p ( yk )

( yk ) 2 = 4 p ( xh )

The vertex is ( h, k ).

The vertex is ( h, k ).

The parabola opens up if p is positive and down if p is negative.

The parabola opens to the right if p is positive and to the left if p is negative.

The focus is ( h, k + p ).

The focus is ( h + p, k ).

The directrix is y = k − p .

The directrix is x = hp .

The axis of symmetry is x = h.

The axis of symmetry is y = k.

Examples

Find the vertex, focus, and directrix. Determine if the parabola opens up, down, to the left, or to the right.

  • ( x − 3) 2 = 4( y − 2)
  • This equation is in the form ( xh ) 2 = 4 p ( yk ). The vertex is (3, 2). Once we have found p , we can find the focus and directrix and how the parabola opens. p = 1 (from 4 = 4 p ). The parabola opens up because p is positive; the focus is ( h , k + p ) = (3, 2 + 1) = (3, 3); and the directrix is y = 1 (from y = kp = 2 − 1 = 1).

  • ( y + 1) 2 = 8( x − 3)
  • The equation is in the form ( yk ) 2 = 4 p ( xh ). The vertex is (3, −1), p = 2 (from 8 = 4 p ); the parabola opens to the right; the focus is ( h + p , k ) = (3 + 2, −1) = (5, −1); and the directrix is x = 1 (from x = hp = 3−2 = 1).

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