**Quadratic Functions**

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
.

Go to Solutions.

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