What is the standard form of the equation of a parabola that opens up or down?
A. $x=a(y-y)^2+h$
B. $x=a y^2+b y+c$
C. $y=a(x-h)^2+v$
D. $y=a x^2+b x+c$
Answer (1)
The problem requires identifying the standard form of a parabola that opens up or down.
Recall that the standard forms are y = a ( x − h ) 2 + k and y = a x 2 + b x + c .
Analyze each option and eliminate those that do not represent a parabola opening up or down.
The correct answer is y = a x 2 + b x + c .
Explanation
Understanding the Problem The question asks us to identify the standard form of a parabola equation that opens upwards or downwards from a list of options.
Recalling Standard Forms Let's recall the standard forms of parabola equations. A parabola that opens up or down has the form y = a ( x − h ) 2 + k or y = a x 2 + b x + c , where a, b, c, h, and k are constants.
Analyzing the Options Now, let's examine the given options:
Option A: x = a ( y − y ) 2 + h simplifies to x = a ( 0 ) 2 + h , which means x = h . This is a vertical line, not a parabola.
Option B: x = a y 2 + b y + c represents a parabola that opens left or right, not up or down.
Option C: y = a ( x − h ) 2 + v is a parabola that opens up or down, where (h,v) is the vertex. This is a valid standard form.
Option D: y = a x 2 + b x + c is also a parabola that opens up or down. This is another valid standard form.
Choosing the Best Option Both options C and D are valid standard forms for a parabola that opens up or down. However, option D, y = a x 2 + b x + c , is generally considered the more common or general standard form.
Final Answer Therefore, the standard form of the equation of a parabola that opens up or down is y = a x 2 + b x + c .
Examples
Understanding the standard form of a parabola is crucial in various real-world applications, such as designing parabolic mirrors for telescopes or satellite dishes. The shape of a parabola allows these devices to focus incoming signals or light to a single point, maximizing efficiency. Similarly, architects use parabolic curves in bridge design to distribute weight evenly and ensure structural stability. Knowing the equation of a parabola helps engineers and designers predict and control these shapes accurately.
