Question

How do you know if a parabola is positive or negative?

Answer 1

Hint In this question we have to find when the parabola is positive and when the parabola is negative. It depends on the open side of the parabola.

Complete step by step solution:
There are two types of patterns of the parabola. The first type is a horizontal parabola. The second type of parabola is a vertical parabola.
In the horizontal parabola, the parabola is opened to either the right side or the left side.
In the vertical parabola, the parabola is opened either upward or downward direction.
The standard equation of a horizontal parabola is \[x = a{\left( {y - k} \right)^2} + h\] where \[\left( {h,k} \right)\]is the vertex of the parabola and the focus is \[\left( {h,k + \dfrac{1}{{4a}}} \right)\].
The standard equation of a vertical parabola is \[y = a{\left( {x - h} \right)^2} + k\] where \[\left( {h,k} \right)\]is the vertex of the parabola and the focus is \[\left( {h + \dfrac{1}{{4a}},k} \right)\].
The positive and negative of a parabola depends on the value of \[a\].
Case I:
The diagram of the parabola whose equation is \[x = a{\left( {y - k} \right)^2} + h\] where \[a > 0\]

Image: Horizontal parabola
The parabola is positive if it is opened to the right side.
Case II:
The diagram of the parabola whose equation is \[x = a{\left( {y - k} \right)^2} + h\] where \[a < 0\]

Image: Horizontal parabola
The horizontal parabola is negative if it is opened to the left side.
Case III:
The diagram of the parabola whose equation is \[y = a{\left( {x - h} \right)^2} + k\] where \[a > 0\]
Image: Vertical parabola
The parabola is positive if it is opened upward direction.
Case IV:
The diagram of the parabola whose equation is \[y = a{\left( {x - h} \right)^2} + k\] where \[a < 0\].

When the parabola is opened in a downward direction, then the parabola is negative.

Hence the parabola is negative when it opens to either the left direction or downward direction.

Note: Sometimes students do not consider the horizontal parabola. This gives an incomplete solution. There are 4 types of parabolas. There are 2 horizontal parabolas and 2 vertical parabolas. When the parabola opens to the right side or upward direction, then the parabola is known as a positive parabola.