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# 4.1 quadratic functions and transformations

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### 4.1 quadratic functions and transformations

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