Which equation describes a parabola that opens up or down and whose vertex is at the point $(h, v)$?
A. $x=a(y-h)^2+v$
B. $x=a(y-v)^2+h$
C. $y=a(x-v)^2+h$
D. $y=a(x-h)^2+v$
Answer (1)
The equation of a parabola opening up or down is in the form y = a ( x − h ) 2 + v .
The vertex of the parabola is ( h , v ) .
Comparing the given options to the standard form, we find that option D matches.
The correct equation is y = a ( x − h ) 2 + v .
Explanation
Problem Analysis We are looking for the equation of a parabola that opens either upwards or downwards and has its vertex at the point ( h , v ) . The options are: A. x = a ( y − h ) 2 + v B. x = a ( y − v ) 2 + h C. y = a ( x − v ) 2 + h D. y = a ( x − h ) 2 + v
Recalling the Standard Form The general form of a parabola that opens upwards or downwards is given by
y = a ( x − h ) 2 + v
where:
( h , v ) is the vertex of the parabola.
a determines the direction and width of the parabola.
If 0"> a > 0 , the parabola opens upwards.
If a < 0 , the parabola opens downwards.
Comparing Options Now, let's compare the given options with the standard form:
A. x = a ( y − h ) 2 + v - This equation represents a parabola that opens left or right, not up or down. B. x = a ( y − v ) 2 + h - This equation also represents a parabola that opens left or right. C. y = a ( x − v ) 2 + h - In this equation, the vertex would be ( v , h ) , not ( h , v ) .
D. y = a ( x − h ) 2 + v - This equation matches the standard form of a parabola that opens up or down with vertex ( h , v ) .
Conclusion Therefore, the correct equation is:
y = a ( x − h ) 2 + v
Examples
Parabolas are commonly seen in the real world, such as the trajectory of a ball thrown in the air or the shape of a satellite dish. Understanding the equation of a parabola allows us to model and predict these phenomena. For example, if you know the vertex of a parabolic arch and another point on the arch, you can determine the equation of the parabola and use it to calculate the height of the arch at any given point. Similarly, engineers use parabolic shapes in designing bridges and antennas to optimize their performance.
