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4.1 quadratic functions and transformations

This document discusses quadratic functions and their transformations. It defines key terms like parabola, vertex, and axis of symmetry. The vertex form of a quadratic function makes it easy to identify transformations - the a value determines stretching or compression, the h value shifts the graph horizontally, and the k value shifts it vertically. For any quadratic function, the minimum or maximum value will occur at the vertex. Graphing involves plotting the vertex and axis of symmetry, then using a table of values. Writing quadratic functions starts with identifying the vertex coordinates, then using another point to solve for the a value.

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CHAPTER 4 QUADRATIC FUNCTIONS AND EQUATIONS 4.1 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 symmetry 2. 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