Problem 59 - College Algebra

Question

If you are given the standard form of the equation of a parabola with vertex at the origin, explain how to determine if the parabola opens to the right, left, upward, or downward.

Step-by-Step Solution

Answer

The direction in which a parabola opens is determined by its standard equation form and the coefficient 'a'. If 'a' is positive, the parabola opens to the right (for \(y^2 = 4ax\)) or upwards (for \(x^2 = 4ay\)). If 'a' is negative, the parabola opens to the left (for \(y^2 = 4ax\)) or downwards (for \(x^2 = 4ay\)).

Step 1: Identify the Standard Equation

The standard form of a parabolic equation with vertex at origin can be either \(y^2 = 4ax\) or \(x^2 = 4ay\). The form of the equation can help determine the general direction in which a parabola opens.

Step 2: Examine the Coefficient

In the standard form of parabolic equation, the coefficient 'a' plays an important role in determining the direction of the parabola. If 'a' is positive, the parabola opens to the right (for \(y^2 = 4ax\)) or upwards (for \(x^2 = 4ay\)), and if 'a' is negative, the parabola opens to the left (for \(y^2 = 4ax\)) or downwards (for \(x^2 = 4ay\)).

Step 3: Conclusion

If the equation is of the form \(y^2 = 4ax\), and 'a' is positive then the parabola opens to the right, if 'a' is negative, it opens to the left. If the equation is of the form \(x^2 = 4ay\), and 'a' is positive then the parabola opens upwards, if 'a' is negative, it opens downwards.