A quadratic function is a function of the form

I've already discussed quadratic functions a little; you know that you can use the graph of a quadratic function is a parabola. The parabola opens upward if and opens downward if . Knowing which way the parabola opens and where the roots are gives a pretty good picture of the graph.

The vertex of a parabola is the "tip" of the graph --- the lowest point on a parabola that opens up and the highest point on a parabola that opens down.

• The vertex is halfway between the roots (if there are real roots).

• The x-coordinate of the vertex is .

The formula is easy to remember, since it's the first "piece" of the quadratic formula:

Example. Find the coordinates of the vertex of .

The roots are and . The vertex is halfway between the roots, which is at

The y-coordinate of the vertex is

Therefore, the vertex is .

Example. Find the coordinates of the vertex of .

Since it would be too hard to find the roots by factoring, I'll use the formula instead. and , so the x-coordinate of the vertex is

The y-coordinate of the vertex is

The vertex is .

If the vertex of a parabola is , the parabola's equation has the form

This checks, since plugging in gives

is called the standard or vertex form of the equation of a parabola.

Example. Find the standard/vertex equation of the parabola whose equation is . What are the coordinates of the vertex?

The idea is to complete the square in x. , and . Therefore, I write the equation as

The vertex is .

Example. The vertex of a parabola is . The parabola passes through the point . Find the equation of the parabola.

Since the vertex is , the equation is

Plug in , :

The equation is .

Example. Find the dimensions of the rectangle with the largest area that has a perimeter of 20 feet.

Let x be the width of the rectangle and let y be the height. The area is

The perimeter is 20, so

Then

Substitute into :

The graph of A is a parabola opening downward. The roots are and . The vertex is at , and it's the highest point on the graph. Therefore, this value gives the largest value of A. When , .

Thus, the rectangle of largest area is 5 feet wide and 5 feet tall --- a square.

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