Which statements are true for the equation [tex]$x^2=-4 y$[/tex]? Check all that apply.
The axis of symmetry is [tex]$x=0$[/tex].
The focus is at [tex]$(0,-1)$[/tex].
The parabola opens up.
The parabola opens right.
The value of [tex]$p=-1$[/tex]
The equation for the directrix is [tex]$y=0$[/tex].
Answer (1)
The equation is in the form x 2 = 4 p y , so we find p = − 1 .
Since p < 0 , the parabola opens downwards.
The axis of symmetry is x = 0 , and the focus is at ( 0 , p ) = ( 0 , − 1 ) .
The directrix is y = − p = 1 . Therefore, the true statements are: axis of symmetry is x = 0 , the focus is at ( 0 , − 1 ) , and the value of p = − 1 .
Explanation
Problem Analysis We are given the equation x 2 = − 4 y and need to determine which of the provided statements about the parabola are true. Let's analyze the equation and its properties.
Finding the Value of p The standard form of a parabola that opens either upwards or downwards is x 2 = 4 p y , where p is the distance from the vertex to the focus and from the vertex to the directrix. Comparing x 2 = − 4 y with x 2 = 4 p y , we have 4 p = − 4 , which gives us p = − 1 .
Determining the Direction of Opening Since p = − 1 is negative, the parabola opens downwards, not upwards or to the right.
Finding the Axis of Symmetry For a parabola in the form x 2 = 4 p y , the axis of symmetry is the y-axis, which is described by the equation x = 0 .
Finding the Focus The focus of a parabola in the form x 2 = 4 p y is at the point ( 0 , p ) . Since p = − 1 , the focus is at ( 0 , − 1 ) .
Finding the Directrix The directrix of a parabola in the form x 2 = 4 p y is the line y = − p . Since p = − 1 , the directrix is y = − ( − 1 ) , which simplifies to y = 1 .
Comparing with Given Statements Now, let's compare our findings with the given statements:
The axis of symmetry is x = 0 . - True
The focus is at ( 0 , − 1 ) . - True
The parabola opens up. - False
The parabola opens right. - False
The value of p = − 1 - True
The equation for the directrix is y = 0 . - False
Examples
Understanding parabolas is crucial in various fields like physics and engineering. For instance, satellite dishes and radio telescopes use parabolic reflectors to focus signals at a single point. The properties of the parabola, such as its focus and directrix, are essential in designing these devices to efficiently capture and transmit signals. Similarly, the trajectory of a projectile, like a ball thrown in the air, approximately follows a parabolic path, which can be analyzed using the equation of a parabola to predict its range and maximum height.
