In the equation above, \( a \) is a positive constant and the graph of the equation in the \( x y \)-plane is a parabola. Which of the following is an equivalent form of the equation? A) \( y=(x+a)(x-a) \) B) \( y=(x+\sqrt{a})(x-\sqrt{a}) \) C) \( y=\left(x+\frac{a}{2}\right)\left(x-\frac{a}{2}\right) \) D) \( y=(x+a)^{2} \)

To determine an equivalent form of the equation that represents a parabola, we can analyze the given options by looking at their structures. Option A, \( y=(x+a)(x-a) \), can be simplified using the difference of squares: \( y = x^2 - a^2 \), which indeed is a parabola. Option B, \( y=(x+\sqrt{a})(x-\sqrt{a}) \), can also be expressed as \( y = x^2 - a \), which is another parabolic form. Thus, both options A and B present legitimate forms of a parabola, but they would represent different constants related to the graph's shape and position. If we break down more about the options, option C \( y=\left(x+\frac{a}{2}\right)\left(x-\frac{a}{2}\right) \) simplifies to \( y= x^2 - \frac{a^2}{4} \), which is still a parabolic shape but with a different constant. Lastly, option D \( y=(x+a)^{2} \) expands to a perfect square, \( y=x^2 + 2ax + a^2 \), which also represents a parabola but is not equivalent to the parabolic forms presented in options A, B, or C if they start from standard quadratic expressions. All in all, the question implies recognizing forms structurally matching parabolas. Enjoy the journey through parabolas!