**Examples**

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

*x*^{2}= 6*y*-
The equation is in the form (

*x*−*h*)^{2}= 4*p*(*y*−*k*), so the parabola will open up or down. We have . Now we know three things—that the parabola opens up (because*p*is positive), that the focus is ), and the directrix is (from The graph that behaves this way is in Figure 12.5 . *y*^{2}= 6*x*-
The equation is in the form (

*y*−*k*)^{2}= 4*p*(*x*−*h*), so the parabola opens to the left or to the right. We have . Now we know that the parabola opens to the right, that the focus is , and that the directrix is . The graph for this equation is in Figure 12.3 . *y*^{2}= –6*x*-
The equation is in the form (

*y*−*k*)^{2}= 4*p*(*x*−*h*), so the parabola opens to the left or to the right. We have . The parabola opens to the left, the focus is , and the directrix is . The graph for this equation is in Figure 12.4. *x*^{2}= –6*y*-
The equation is in the form (

*x*−*h*)^{2}= 4*p*(*y*−*k*), so the parabola opens up or down. We have . The parabola opens down, the focus is . The directrix is (from . The graph for this equation is in Figure 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.

( |
( |

The vertex is ( |
The vertex is ( |

The parabola opens up if |
The parabola opens to the right if |

The focus is ( |
The focus is ( |

The directrix is |
The directrix is |

The axis of symmetry is |
The axis of symmetry is |

**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 (

*x*−*h*)^{2}= 4*p*(*y*−*k*). 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*=*k*−*p*= 2 − 1 = 1). - (
*y*+ 1)^{2}= 8(*x*− 3) -
The equation is in the form (

*y*−*k*)^{2}= 4*p*(*x*−*h*). 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*=*h*−*p*= 3−2 = 1).

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