**Problem : **
Given the following quadratic function *f* (*x*) = 3*x*^{2} - 12*x* + 13, decide
whether the graph opens upward or downward, find the vertex and axis of
the graph, and find any real roots of the
function.

The graph opens upward. The vertex is at

(2, 1) and the axis is the line

*x* = 2. It has no real roots.

**Problem : **
Given the following quadratic function *f* (*x*) = - 3*x*^{2} - 6*x* - 3, decide whether
the graph opens upward or downward, find the vertex and axis of the graph, and
find any real roots of the function.

The graph opens downward. The vertex is at

(- 1, 0) and the axis is the line

*x* = - 1. It has one real root at

*x* = - 1.

**Problem : **
Given the following quadratic function *f* (*x*) = *x*^{2} - 8*x* + 19, decide whether
the graph opens upward or downward, find the vertex and axis of the graph, and
find any real roots of the function.

The graph opens upward. The vertex is at

(4, 3) and the axis is the line

*x* = 4. It has no real roots.

**Problem : **
Given the following quadratic function *f* (*x*) = *x*^{2}, decide whether the graph
opens upward or downward, find the vertex and axis of the graph, and find any
real roots of the function.

The graph opens upward. The vertex is at

(0, 0) and the axis is the line

*x* = 0. It has one real root at

*x* = 0.

**Problem : **
Given the following quadratic function *f* (*x*) = *x*^{2} - 2*x*, decide whether the
graph opens upward or downward, find the vertex and axis of the graph, and find
any real roots of the function.

The graph opens upward. The vertex is at

(1, - 1) and the axis is the line

*x* = 1. It has two real roots, at

*x* = {0, 2}.