How can I determine whether a horizontal parabola opens to the left or to the right?

Question

Vedantu Content Team · Accepted Answer

Hint: A parabola can be horizontal, vertical, or tilted, depending upon the orientation of its axis. In the above question, we have been asked about a horizontal parabola, which means that its axis must be horizontal, that is, parallel to the x-axis. This means that the equation of the parabola must be linear in x and must be of the type $${{y}^{2}}=kx$$. The direction of the opening of the parabola will depend on the sign of $k$.Complete step by step solution:Since the parabola given in the above question is horizontal, its axis must be parallel to the x-axis. This implies that the equation of the parabola must be linear in x. So we can consider the general equation of a horizontal parabola as$\Rightarrow {{y}^{2}}=kx$Now, the direction of the opening of the parabola will depend on the sign of k in the above equation. On the basis of the signs of k, we can have two cases:Case I: When k is positiveConsidering again the equation of the horizontal parabola, we have$\Rightarrow {{y}^{2}}=kx$Since the LHS is equal to the square of y, it must be positive. This implies that the RHS, equal to the product $kx$, must be positive. For this case, we have considered positive value for k. This means that $x$ must also be positive for the product $kx$ to be positive.Now, we know that the region $x>0$ lies to the right of the origin.This means that the parabola must open to the right for $k>0$, as shown below.\n \n \n \n \n Case II: When k is negativeThe parabolic equation is$\Rightarrow {{y}^{2}}=kx$In this case, the value of $k$ is negative, and since the product $kx$ must be positive, as shown in the above case, the value of $x$ must be negative.We know that the region $x<0$ lies to the left to the origin.This means that the parabola must open to the left for $k<0$, as shown below.\n \n \n \n \n Hence, the parabola will open to the left when $k<0$ and to the right when $k>0$.Note: We must note that before deciding the direction of opening, it is necessary to write the equation of the parabola in the standard form of ${{y}^{2}}=kx$. Then the sign of the coefficient of x will decide the direction of its opening, according to the caes discussed in the above solution.