A parabola has a vertex at $(0,0)$. The equation for the directrix of the parabola is $x=-4$.
In which direction does the parabola open?
up
down
right
left
Answer (2)
The vertex of the parabola is at ( 0 , 0 ) .
The directrix is x = − 4 .
The parabola opens away from the directrix, so it opens either left or right.
Since the vertex is at ( 0 , 0 ) and the directrix is x = − 4 , the parabola opens to the right.
The final answer is r i g h t
Explanation
Problem Analysis The problem states that a parabola has a vertex at ( 0 , 0 ) and its directrix is given by the equation x = − 4 . We need to determine the direction in which the parabola opens.
Basic Concepts Recall that a parabola is defined as the set of all points equidistant from the focus and the directrix. The parabola opens in the direction away from the directrix and towards the focus. Since the directrix is a vertical line x = − 4 , the parabola must open either to the left or to the right.
Determining the Focus Since the vertex of the parabola is at ( 0 , 0 ) and the directrix is x = − 4 , the focus must be at ( 4 , 0 ) . This is because the vertex is the midpoint between the focus and the directrix. The parabola opens from the vertex towards the focus.
Determining the Direction Since the focus is at ( 4 , 0 ) , which lies to the right of the vertex ( 0 , 0 ) , the parabola opens to the right.
Examples
Parabolas are commonly found in the design of satellite dishes and reflective telescopes. The parabolic shape helps to focus incoming signals or light to a single point, the focus, where the receiver or detector is placed. Understanding the direction a parabola opens is crucial in these applications to ensure proper alignment and signal reception.
The parabola has its vertex at ( 0 , 0 ) and the directrix at x = − 4 . It opens toward the focus at ( 4 , 0 ) , which is located to the right of the vertex. Thus, the parabola opens to the right.
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