Chapter 3 - Quadratic Functions - 3.1 Introduction to the Family of Quadratic Functions - Exercises and Problems for Section 3.1 - Exercises and Problems - Page 121: 32

The graph of $f(x)$ is a parabola opening upward and is concave up.

If the graph of $f(x)$ opens downward and passes through the point $(1,1)$, then it must cross the $x$-axis at two points but we know that $f(x)$ has no zeros, this cannot happen. Therefore the graph of $f(x)$ is a parabola opening upward and is concave up.