Question

the graph of a quadratic function with vertex (3, 2) is shown in the figure below. find the range and the domain. write your answers as inequalities, using x or y as appropriate. or, you may instead click on empty set or all reals as the answer. (a) range: (b) domain:

Explanation:

Step1: Analyze the Domain of Quadratic Function

For any quadratic function (a parabola), the domain is all real numbers because there are no restrictions on the values of \( x \) that we can plug into the function. Mathematically, this means the domain is \( (-\infty, \infty) \) or we can say all real numbers for \( x \).

Step2: Analyze the Range of Quadratic Function

The vertex of the parabola is \( (3, 2) \) and the parabola opens upwards (since the graph has a minimum at the vertex, as seen from the U - shape opening up). For a parabola that opens upwards, the range consists of all real numbers \( y \) such that \( y \) is greater than or equal to the \( y \) - coordinate of the vertex. So the range is \( y\geq2 \).

Answer:

(A) range: \( y\geq2 \)
(B) domain: All real numbers (or \( (-\infty, \infty) \) or \( x\in\mathbb{R} \))