# Quadratic Function

**DEFINITION.**

A quadratic function is a polynomial of degree 2:

The graph is a parabola.

• If a > 0, the horns point up.

• If a < 0, the horns point down.

•If |a| > 1, the parabola is narrower than y
= x^{2}.

• If |a| < 1, the parabola is wider than y =
x^{2}.

Determine the shape of the following graphs. Pick the

shape below.

Given

Get the roots by factoring or using the quadratic formula:

. No roots if

**COMPLETING THE SQUARE THEOREM. **Every quadratic

function may be written in the form:

where is the vertex (or nose) of the
parabola.

Proof. Given

• Factor the a out of the ax^{2}+bx part.

• Divide the new coefficient of x by 2 and square.

Add this to complete the square.

• Anything which is added must also be subtracted to

preserve equality.

• Find the roots (the roots are the x-intercepts).

• Write in completed square form:

• Graph. On the graph list both coordinates of the vertex.

Find the vertex.

Find the graph with the correct shape and position.

Find the roots. x = ?, ?

Write equation in the form:

Find the vertex.

**WORD PROBLEMS**

• Draw the picture. Indicate the variables in the picture.

• Write the given equations which relate the variables.

• Solve for the wanted quantities.

The perimeter of a rectangle is 10 feet.

Express the area A in terms of the width x.

The area of a rectangle is 10 square feet.

Express the perimeter P in terms of the width x.

List the given.

Write the perimeter as a function of x

The corner of a
triangle lies on the line

Express the triangle’s area and perimeter in terms of the

base x.

The area of an isosceles triangle is 16.

Write the triangle’s height h in terms of its
width w.

Write the triangle’s width w in terms of its
height h.

The curved surface

area is the area of the

can’s side, excluding

the top and bottom

The height of a can
(right circular cylinder)

is three times the radius.

Express the radius as a function

of the curved surface area.

The height of a can (right circular cylinder)

is four times the radius.

Express the curved surface area as a function

of the radius.