How do you determine the range of a quadratic function?

To determine the range of a quadratic function, you can follow these steps:

Identify the vertex: The vertex of a quadratic function, represented as (h, k), gives you the minimum or maximum point on the graph. The range will depend on whether the parabola opens upward or downward.

Determine the direction of the parabola: If the coefficient of the x^2 term (a) is positive, the parabola opens upward, and the range will be greater than or equal to the y-coordinate of the vertex (k). If a is negative, the parabola opens downward, and the range will be less than or equal to the y-coordinate of the vertex (k).

Check for the presence of a minimum or maximum value: If the quadratic function has a minimum value (for an upward-opening parabola) or a maximum value (for a downward-opening parabola), the range will be bounded. In this case, the range will be all real numbers greater than or equal to the minimum value (for an upward-opening parabola) or all real numbers less than or equal to the maximum value (for a downward-opening parabola).

If the parabola does not have a minimum or maximum value, the range will be either all real numbers or restricted based on other conditions (such as domain restrictions).

By considering these steps, you can determine the range of a quadratic function.