Properties of Graphs of Quadratic Functions Parabola: 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: Transformational Standard General Transformational form: a = vertical stretch k = vertical translation h = horizontal translation Standard form: General form:

Review Squaring Binomials and Factoring

Factor:

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

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.

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

Method 3: Completing the Square

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:

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.