Properties of Graphs of Quadratic FunctionsParabola: the curved graph of a quadratic function

Vertex: the point on a parabola where a minimum or maximum y-value occurs.

Axis of symmetry: a line in which a parabola is reflected onto itself.

Vertical stretch: a ratio that compares the change in y-values of a quadratic function with the corresponding y-values of y=x2

Quadratics can be expressed in different forms:

1. Transformational

2. Standard

3. General

Transformational form:

21hxky

a

a = vertical stretch

k = vertical translation

h = horizontal translation

Standard form: khxay 2)(

General form: cbxaxy 2

Review Squaring Binomials and Factoring

2

2

2

2

2

2

1

3

5

2

3

x

x

x

x

x

Factor:

2510

12

3612

168

2510

2

2

2

2

2

xx

xx

xx

xx

xx

Finding the Maximum and Minimum Value The vertex gives you the maximum or

minimum value. Putting quadratics in transformational form

makes finding the vertex easy

Creating the Transformational form of a Quadratic:Completing the Square

•Divide all terms by ‘a’

•Move ‘c’ to the other side

•Add half of ‘b’ squared to both sides.

•Factor both sides

382 xxy

484 2 xxy

Determining Quadratic Functions from Parabolas

If the vertex and at least one other point of a parabola are known, the transformational form of the quadratic function can be found.

−3 −2 −1 1 2 3 4 5

−3

−2

−1

1

2

3

4

5

6

x

y

Roots of Quadratic Equations

Finding the roots of a quadratic means solving the equation.

Roots, zeros, solutions The value of x that makes the equation

equal to zero.

Method 1: Graphing

Let equation equal zero Use TI-Calculator Enter equation into y= CALC:zeros TABLE

Method 2: Factoring by Decomposition

01625 2 xx

Method 3: Completing the Square

01625 2 xx

Quadratic Formula

There is another way to determine the roots that will always work.

Quadratic Formula:

It is used when the quadratic is in general form:

a

acbbx

2

42

0,2 acbxaxy

Imaginary numbers:

What is the square root of -4???

Can’t find the square root of a negative number, so the answer is imaginary.

A complex number is made up of a real number and an imaginary number: a+bi

Some quadratics have no real roots. Therefore the roots are imaginary.

The Number of Roots of a Quadratic Equation The expression b2-4ac in the quadratic formula

is called the discriminant. The discriminant is used to determine the type of

roots a quadratic will have. If the discriminant is larger than zero, the

quadratic has 2 distinct real roots. If the discriminant is zero, the quadratic has one

root, or two equal real roots If the discriminant is less than zero, the

quadratic has imaginary roots.