How to Write Interval Notations Using the Infinity Symbol on a Parabola Graph

••• AleksandarGeorgiev/E+/GettyImages

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.

Related Articles

Difference Between Parabola and Line Equation
How to Find The Slope of a Line Given Two Points
How to Find X and Y Intercepts of Quadratic Equations
How to Convert Graphs to Equations
How to Find Vertical & Horizontal Asymptotes
How to Calculate the Slope of Regression Line
How to Tell If Lines Are Parallel, Perpendicular or...
How to Express Your Answer in Interval Notation
How to Find the Line of Symmetry in a Quadratic Equation
How to Solve a Parabola
How to Find the Domain Range of a Parabola Parameter...
How to Find the X Intercept of a Function
How do I Calculate the Range in Algebraic Equations?
How to Write Functions in Math
How to Find Equations of Tangent Lines
How to Find the Angle of a Curve
How to Convert an Equation Into Vertex Form
How to Find the Slope of Linear Equations
How Do We Write the Equation of a Horizontal Line?
What is the Definition of Slope in Algebra?