Question

the graph of a quadratic function with vertex (-4, -2) is shown in the figure below. find the domain and the range. 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) domain: (b) range:

Explanation:

Step1: Determine the domain of the 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 input into the function. In other words, the graph extends infinitely to the left and right along the \( x \)-axis. So the domain is all real numbers, which can be written as \( -\infty < x < \infty \) or simply "All reals".

Step2: Determine the range of the quadratic function

The vertex of the parabola is given as \( (-4, -2) \), and the parabola opens upwards (since the graph is a "U - shape" opening up, we can tell from the direction of the arrows). For a parabola that opens upwards, the vertex is the minimum point. So the \( y \)-values of the function start from the \( y \)-coordinate of the vertex and go to positive infinity. The \( y \)-coordinate of the vertex is \( -2 \), so the range is \( y \geq - 2 \).

Answer:

(a) domain: All reals
(b) range: \( y \geq -2 \)