Question

The graph of a quadratic function has a domain of (-∞, ∞) and range of [4, ∞). In two or more complete sentences, explain how the given range of the function can help you to determine whether the graph opens up or down.

Answer 1

A parabola is represented by a polynomial of degree 2:

$f(x) = ax^2+bx+c,\ a,b,c \in \mathbb{R}, a\neq 0$

Depending on the sign of the leading coefficient $a$, the parabola open up or down.

More specifically, if $a>0$, the parabola opens up, which means that the vertex is a minimum, and the parabola grows indefinitely. Conversely, if $a$, the parabola opens down, which means that the vertex is a maximum, and the parabola decreases indefinitely.

The range is the set of the point reached by the graph, and we can see that it is unbounded to the right, which means that the range has a minimum, and grows indefinitely.

This corresponds to the case $a>0$, which means that the parabola opens up.