1. Radical Functions and Equations L. Waihman 2002 radical radicand index

2. A radical function is a function that has a variable in the radicand. Horizontal shift left one Vertical stretch two. Vertical shift down three You can apply the same transformations to the graphs of radical functions as you can to polynomial functions.

3. Radical Parent functions As x , f(x)  As x  0 +, f(x)  0 As x , f(x)  As x  – , f(x)  –  As x , f(x)  As x  0 +, f(x)  0 Click on each graph above to link to interactive page!

4. Transformations Vertical stretch of 3 Vertical shift up 2 Horizontal shift right 3 We apply the transformations on these functions in the same manner as we did with the polynomial functions. Domain: Range Domain: Range: Domain: Range:

5. Applying the transformations Vertical shrink of ½ Vertical stretch 3 Horizontal shift right 2 Vertical shift up 4 Horizontal shift left 1 Vertical shift down 1 Domain:Range: As x x , f(x)  ;; As x x -1 +, f(x)  As x x , f(x)  - ;; As x x 2 +, f(x)  4 Domain:Range: Reflect @ x-axis

6. Applying the transformations Vertical stretch of 3 Horizontal shift left 2 Vertical shift down 4 Vertical shrink of ½ Horizontal shift right 1 Vertical shift up 2 Domain:Range: As x x , f(x)   ; As x x - , f(x)  -  Domain:Range: As x x , f(x)  -  ; As x x - , f(x)   Reflect @ x-axis

7. To solve a radical equation that has only one variable in the radicand, isolate that term on one side of the equation. If the index is 2, then square both sides of the equation. Be careful! The new equation you created when you Squared both sides might have extraneous solutions! Isolate the radical. Square both sides. Simplify and set equal to zero. Factor. Solve. Radical Equations Given:

8. These solutions may not be solutions to the original equation. Check your solutions ! The graph below illustrates that the derived equation may be different from the original equation. √ Solving Radical Equations

9. Try: Check: Try: Check

10. A radical equation may contain two radical expressions with an index of 2. To solve these, rewrite the equation with one of the radicals isolated on one side of the equals sign. Then, square both sides. If a variable remains in a radicand, you must repeat the squaring process. More Solving Radical Equations

11. Check: Try: More Solving Radical Equations Isolate one radical on each side of equals sign. Square both sides. Collect like terms on each side of equals sign. Simplify. Square both sides again.

12. More Solving Radical Equations Check: √ Try:

13. Radical equations with indexes greater than 2 can be solved using similar techniques. Check: After isolating the term containing the radical, raise each side of the equation to the power equal to the index of the radical. √

14. Bonus Questions! One type of transformation was not covered in this PowerPoint. Identify it and give and example of an equation with this type of transformation and then sketch its graph to illustrate the transformation. Identify the domain and range as well. Solve the equation: click