1. CHAPTER 4 QUADRATIC FUNCTIONSAND EQUATIONS4.1 Quadratic Functions and Transformations
2. 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
3. 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.
4. 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
5. 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
6. 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
7. 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
8. EXAMPLE: GRAPH EACH FUNCTION. DESCRIBEHOW IT WAS TRANSLATED FROM
9. EXAMPLE: GRAPH EACH FUNCTION. DESCRIBEHOW IT WAS TRANSLATED FROM
10. EXAMPLE: GRAPH EACH FUNCTION. DESCRIBEHOW IT WAS TRANSLATED FROM
11. EXAMPLE: GRAPH EACH FUNCTION. DESCRIBEHOW IT WAS TRANSLATED FROM
12. 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
13. EXAMPLE: WRITE A QUADRATIC FUNCTION TOMODEL EACH GRAPH
14. EXAMPLE: WRITE A QUADRATIC FUNCTION TOMODEL EACH GRAPH