1. C HAPTER 2 2-1 Using transformations to graph quadratic equations

2. O BJECTIVES Students will be able to: Explore how changes in the parameters of a quadratic function affect its graph.

4. Q UADRATIC F UNCTION

5. Q UADRATIC FUNCTION TABLE The table shows the linear and quadratic parent functions Notice that the graph of the parent function f ( x ) = x 2 is a U-shaped curve called a parabola. As with other functions, you can graph a quadratic function by plotting points with coordinates that make the equation true.

6. E XAMPLE 1 Graph f ( x ) = x 2 – 4 x + 3 by using a table.

7. E XAMPLE 1 CONTINUE

8. E XAMPLE 2 Graph g ( x ) = – x 2 + 6 x – 8 by using a table

9. E XAMPLE 2 CONTINUE

10. S TUDENT P RACTICE Go to the guided practice and do problems 2 to 4

11. U SING TRANSFORMATIONS TO GRAPH QUADRATIC FUNCTIONS You can also graph quadratic functions by applying transformations to the parent function f(x) = x 2. Transforming quadratic functions is similar to transforming linear functions.

12. E XAMPLE 3 h k

13. E XAMPLE 3 CONTINUE

14. E XAMPLE 4 hk

15. E XAMPLE 4 CONTINUE

16. E XAMPLE 5 hk

17. E XAMPLE 5 CONTINUE

18. S TUDENT PRACTICE Do problem 20 through 22 from page 64.

19. R EMEMBER !!!!! functions can also be reflected, stretched, or compressed.

20. R EFLECTIONS, S TRETCHES AND C OMPRESSSIONS

21. E XAMPLE 6

22. E XAMPLE 6 CONTINUE

23. E XAMPLE 7

24. E XAMPLE 8

25. E XAMPLE 8 CONTINUE

26. E XAMPLE 9

28. PARABOLA If a parabola opens upward, it has a lowest point. If a parabola opens downward, it has a highest point. This lowest or highest point is the vertex of the parabola. The parent function f ( x ) = x 2 has its vertex at the origin. You can identify the vertex of other quadratic functions by analyzing the function in vertex form. The vertex form of a quadratic function is f ( x ) = a ( x – h )2 + k, where a, h, and k are constants.

29. V ERTEX FORM OF A QUADRATIC FUNCTION Because the vertex is translated h horizontal units and k vertical from the origin, the vertex of the parabola is at ( h, k ).

30. W RITING TRANSFORMED QUADRATIC EQUATIONS Example 10 Use the description to write the quadratic function in vertex form. Description: The parent function f ( x ) = x 2 is vertically stretched by a factor of and then translated 2 units left and 5 units down to create g.

31. E XAMPLE 10 SOLUTION

32. Step 3 graph Graph both functions on a graphing calculator. Enter f as Y 1, and g as Y 2. The graph indicates the identified transformations. f g

33. H OMEWORK Do problems 2-7,23,25,29 and 30 from page 64

34. C LOSURE Today we talked about how we can graph quadratic functions, use transformations in the quadratic equations and also how we can translated and write quadratic equations. Tomorrow we are going to continue with 2-2