A quadratic function is a 2nd-degree polynomial function:

, where .

The graph of a quadratic function is a parabola, which opens up if and opens down if .

The x-coordinate of the vertex of the parabola is given by , and the y-coordinate can be found by substituting this value for into .

If the vertex of the parabola has coordinates , then the standard equation of the parabola has the form .

The x-intercepts of the parabola, if there are any, are the solutions of the quadratic equation .

Ex 1 Find the vertex of the parabola .

Sol We have that , and then .

Ex 2 Find the minimum value of the function .

Sol The minimum value of this function is given by the y-coordinate of the vertex. Since the x-coordinate of the vertex is given by , the minimum value is given by .

Pr 1 Find the vertex of the parabola .

Pr 2 Find the maximum value of the function .

Pr 3 Find a quadratic function which has 5 and 1 as the x-intercepts of its graph and which has a minimum value of -12.

Pr 4 Find a parabola which has its vertex at the point and which passes through the point .

Pr 5 Find the minimum value of the function , and find the values of for which is a minimum.

Pr 6 Find the vertex of the parabola .

Pr 7 Find a quadratic function such that is its minimum value, and such that .

Pr 8 Find an equation of the non-vertical line which intersects the parabola only at the point .

Pr 9 Find the maximum value for the function .