A parabola is a symmetrical curve with a vertex that represents its minimum or maximum. The two mirroring sides of the parabola change in opposite ways: one side increases as you move from left to right whereas the other side decreases. Once you have located the vertex of the parabola, you can use interval notation to describe the values over which your parabola is either increasing or decreasing.
Write the equation of your parabola in the form y=ax^2 + bx + c, where a, b and c equal the coefficients of your equation. For example, y=5 + 3x^2 + 12x - 9x^2 would be rewritten as y=-6x^2 + 12x + 5. In this case, a=-6, b=12 and c=5.
Substitute your coefficients into the fraction -b/2a. This is the x-coordinate of the parabola's vertex. For y=-6x^2 + 12x + 5, -b/2a = -12/(2(-6)) = -12/-12 = 1. In this case, the x-coordinate of the vertex is 1. The parabola exhibits one trend between -∞ and the x-coordinate of the vertex and it exhibits the opposite trend between the x-coordinate of the vertex and ∞.
Write the intervals between -∞ and the x-coordinate and the x-coordinate and ∞ in interval notation. For example, write (-∞, 1) and (1, ∞). The parentheses indicate that these intervals do not include their endpoints. This is the case because neither -∞ nor ∞ are actual points. Furthermore, the function is neither increasing nor decreasing at the vertex.
Observe the sign of "a" in your quadratic equation to determine the behavior of the parabola. For example, if "a" is positive, the parabola opens up. If "a" is negative, the parabola opens down. In this case, a=-6. Therefore, the parabola opens down.
Write the behavior of the parabola next to each interval. If the parabola opens up, the graph decreases from -∞ to the vertex and increases from the vertex to ∞. If the parabola opens down, the graph increases from -∞ to the vertex and decreases from the vertex to ∞. In the case of y=-6x^2 + 12x + 5, the parabola increases over (-∞, 1) and decreases over (1, ∞).
- Interval notation always describes graph trends from left to right across the x-axis, from -∞ towards ∞.
- Square brackets in interval notation denote inclusive boundaries. Neither infinity nor the vertex should be included in parabola behavior interval notations. Therefore, do not use square brackets.