How to Find the Domain Range of a Parabola Parameter Change

Find the domain and range of a parabola.
••• calculatrice image by Noé Rouxel from

A parabola is a conic section, or a graph in the shape of a U that opens either upward or downward. A parabola opens from the vertex, which is the lowest point on a parabola that opens up, or the lowest point on one that opens down -- and is symmetrical. The graph corresponds to a quadratic equation in the form "y=x^2." The domain and range of that graph are all the x and y coordinates through which the function passes. When teachers speak of changing the parameter of a parabola, they refer to the values that can be added or changed in the former equation. The full equation is -- ax^2+bx+c -- where a, b and c are the parameters that are variable.

    Determine the domain of the function. The domain is defined as all values of x that can be input into the equation and produce a corresponding y. Work with the equation: y=2x^2-5x+6. In this case, any real number can be entered into the equation and produce a y value, so the domain is all real numbers.

    Decide if the parabola opens up or down. If the a value is positive, the graph will open up, and if the a value is negative, the graph will open down. This will let you know if the vertex represents the minimum or maximum value of the parabola.

    Use the formula "-b/2a" to determine the X value of the vertex. Using the formula: y=2x^2-5x+6: x= -(-5)/2(2) = 5/4.

    Plug the X value back into the originally equation and solve for y: y = 2(5/4)^2-5(5/4)+6 = 2.875

    So the vertex -- and in this case minimum value of the parabola since the parabola opens up -- is (1.25, 2.875).

    Determine the range of the function. If the minimum y value of the parabola is 2.875, then the range is all points greater than or equal to that minimum value, or "y>=2.875."


    • Plug equations in the form "y= ax^2+bx+c" with different parameters into your graphing calculator and observe how each parameter changes the graph.

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