Question

How to determine the direction a parabola opens

Answer 1

Upward or Downward

Explanation:

A parabola is the graph of a quadratic function y = ax2 + bx + c. The graphs below depict two typical parabolas:

For clarity, we indicate their x-intercepts with red dots, their y-intercepts with pink dots, and the vertex of each parabola with a green dot:

The first parabola (a U shape) opens vertically, whereas the second parabola opens downwards (is an upside down U shape).

Given the function y = ax2 + bx + c, write the following:

• If an is greater than zero (positive), the parabola widens upward.
• If the value is 0 (negative), the parabola expands downward.

Answer 2

• positive leading coefficient: upward
• negative leading coefficient: downward

Explanation:

A parabola is defined by a quadratic equation, which can be written generically as ...

y = ax² +bx +c

The graph of this equation will have a ∪ or ∩ shape. That is, it will open upward or downward.

Direction of opening

The direction the parabola opens depends on the sign of y when values of x get large. The term x² always has a positive sign, so the sign of y for large x will depend on the sign of the leading coefficient, a.

When a is positive, y-values are large positive values when x gets large, so the graph of the parabola opens upward.

When a is negative, y-values are large negative values when x gets large, so the graph of the parabola opens downward.

The parabola opens upward when the leading coefficient is positive; downward otherwise.

Additional comment

This dependence on the sign of the leading coefficient is true for the graph of any even-degree polynomial.