CHAPTER 4 QUADRATIC FUNCTIONS AND EQUATIONS4.1 Quadratic Functions and Transformations

DEFINITIONS

A parabola is the graph of a quadratic function. A parabola is a “U” shaped graph

The parent Quadratic Function is

Vertex at (0, 0)Axis of Symmetry at x = 0

DEFINITIONS

The vertex form of a quadratic function makes it easy to identify the transformations

The axis of symmetry is a line that divides the parabola into two mirror images (x = h)

The vertex of the parabola is (h, k) and it represents the intersection of the parabola and the axis of symmetry.

REFLECTION, STRETCH, AND COMPRESSION Working with functions of the form

The determines the “width” of the parabola If the the graph is vertically stretched (makes

the “U” narrow) If the graph is vertically compressed

(makes the “U” wide) If a is negative, the graph is reflected over the

x – axis

MINIMUM AND MAXIMUM VALUES

The minimum value of a function is the least y – value of the function; it is the y – coordinate of the lowest point on the graph.

The maximum value of a function is the greatest y – value of the function; it is the y – coordinate of the highest point on the graph.

For quadratic functions the minimum or maximum point is always the vertex, thus the minimum or maximum value is always the y – coordinate of the vertex

TRANSFORMATIONS – USING VERTEX FORM

Remember transformations from Chapter 2 The vertex form makes identifying

transformations easy

a gives you information about stretch, compression, and reflection over the x – axis

h gives you information about the horizontal shift k gives you information about the vertical shift

The vertex is at (h, k) The axis of symmetry is at x = h

TRANSFORMATIONS – USING VERTEX FORM

Graphing Quadratic Functions:1. Identify and Plot the vertex and axis of

symmetry2. Set up a Table of Values. Choose x – values

to the right and left of the vertex and find the corresponding y – values

3. Plot the points and sketch the parabola

EXAMPLE: GRAPH EACH FUNCTION. DESCRIBE HOW IT WAS TRANSLATED FROM

EXAMPLE: GRAPH EACH FUNCTION. DESCRIBE HOW IT WAS TRANSLATED FROM

EXAMPLE: GRAPH EACH FUNCTION. DESCRIBE HOW IT WAS TRANSLATED FROM

EXAMPLE: GRAPH EACH FUNCTION. DESCRIBE HOW IT WAS TRANSLATED FROM

TRANSFORMATIONS – USING VERTEX FORM

Writing the equations of Quadratic Functions:1. Identify the vertex (h, k)2. Choose another point on the graph (x, y)3. Plug h, k, x, and y into

and solve for a 4. Use h, k, and a to write the vertex form of

the quadratic function

EXAMPLE: WRITE A QUADRATIC FUNCTION TO MODEL EACH GRAPH

EXAMPLE: WRITE A QUADRATIC FUNCTION TO MODEL EACH GRAPH