Which equation describes a parabola that opens left or right and whose vertex is at the point ($h, v$)?
A. $x=a(y-h)^2+v$
B. $x=a(y-v)^2+h$
C. $y=a(x-h)^2+v$
D. $y=a(x-v)^2+h$
Answer (2)
The problem requires identifying the equation of a horizontal parabola with vertex ( h , v ) .
The general form of a horizontal parabola is x = a ( y − k ) 2 + h , where ( h , k ) is the vertex.
Substituting the given vertex ( h , v ) into the general form gives x = a ( y − v ) 2 + h .
Therefore, the correct equation is x = a ( y − v ) 2 + h .
Explanation
Understanding the Problem The question asks us to identify the equation of a parabola that opens either to the left or to the right, given that its vertex is at the point ( h , v ) . We need to recall the standard forms of parabolic equations and match the given conditions.
Recalling the General Form A parabola that opens to the left or right has the general form: x = a ( y − k ) 2 + h where ( h , k ) is the vertex of the parabola and a determines the direction and width of the opening. If 0"> a > 0 , the parabola opens to the right, and if a < 0 , it opens to the left.
Substituting the Vertex Coordinates We are given that the vertex of the parabola is ( h , v ) . Substituting v for k in the general equation, we get: x = a ( y − v ) 2 + h
Comparing with the Options Now, we compare this equation with the given options:
A. x = a ( y − h ) 2 + v B. x = a ( y − v ) 2 + h C. y = a ( x − h ) 2 + v D. y = a ( x − v ) 2 + h
Option B matches our derived equation.
Final Answer Therefore, the equation that describes a parabola that opens left or right and has a vertex at the point ( h , v ) is: x = a ( y − v ) 2 + h So, the correct answer is B.
Examples
Parabolas that open to the left or right can model various real-world scenarios, such as the shape of a satellite dish or a radar antenna. The vertex of the parabola represents the focal point where signals are concentrated. Understanding the equation of such parabolas allows engineers to design these devices efficiently, ensuring optimal signal reception or transmission. For example, if you want to build a satellite dish with the focal point at (5, 3), you can use the equation x = a ( y − 3 ) 2 + 5 to determine the shape of the dish.
The equation that describes a parabola opening left or right with a vertex at ( h , v ) is x = a ( y − v ) 2 + h . Hence, the correct choice is option B. This formulation helps clearly identify how the parabola behaves based on the vertex's position.
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