Standards for Radical Functions

1. Standards for Radical Functions • MM1A2a. Simplify algebraic and numeric expressions involving square root. • MM1A2b. Perform operations with square roots. • MM1A3b. Solve equations involving radicals such as , using algebraic techniques.

2. Radical Functions • Essential questions: • What is a radical function? • What does the graph look like and how does it move? • How are they used in real life applications?

3. Real Life Applications • Pythagorean Theorem • Distance Formula • Solving any equation that includes a variable with an exponent, such as:

4. Radical Expressions • Index Radical Sign Radicand

5. General Radical Equation Vertical translation k units up for k > 0 and |k| units down for k < 0 Vertical stretch or compression by a factor of |a|; for a < 0, the graph is a reflections across the x-axis Horizontal translation h units to the right for h > 0 and |h| units to the left if h < 0. (h = 0 for this course) Horizontal stretch or compression by a factor of |1/b|; for b < 0, the graph is a reflection across the y-axis (b = 1 or -1 for this course)

6. Radical Functions • Make a (some) table(s), graph the following functionsand describe the transformations for x = 0, 1, 4, 9, 16 & 25. • What transformation Rules do you see from Your graphs?

7. That’s our smallest value in our t-chart. What value for x gives us a zero under the radical? 0

8. That’s our smallest value in our t-chart. What value for x gives us a zero under the radical? 0

9. That’s our smallest value in our t-chart. What value for x gives us a zero under the radical? 0

10. That’s our smallest value in our t-chart. What value for x gives us a zero under the radical? 0

11. Radical Functions • State an equation that would make the square root function shrink vertically by a factor of ½ and translate up 4 units. • How would we reflect the above equation across the y-axis? • Make the “x” negative

12. Domain & Range: Radical Functions • State the domain, range, and intervals of increasing and decreasing for each function.

13. Graphing Radical Functions Summary • Transformations for radical functions are the same as polynomial functions. • The domain of the parent function is limited to {x | x  0} (the set of all x such that x  0) • The range of the parent function is limited to {y | y  0}(the set of all y such that y  0) • The domain and range may change as a result of transformations. • The parent radical function continuously increases from the origin.

14. Simplifying Radical Expressions • Square and square root are inverse functions, but the square root has to be positive

15. Simplifying Radical Expressions • or • “Simplify” a radical means to: • Take all the perfect squares out of the radicand. • Simplify:

16. Simplifying Radical Expressions • “Simplify” a radical means to: • Combine terms with like radicands • Must have the same radicand to be able to add or subtract radials • Simplify:

17. Simplifying Radical Expressions  Quotient Property of Square Roots: If a ≥ 0 and b > 0: • “Simplify” a radical means to: • Do not leave a radical in the denominator • Simplify:

18. Simplify Expressions via Conjugates • Remember: (a+ b) is the conjugate of (a – b) • We get conjugates when we factor a perfect square minus a perfect square. • We also get conjugates other times. • Simplify:

19. Simplify Expressions via Conjugates Simplify:

20. Summary Important Operations • Square and square root are inverse functions, but the square root has to be positive • Product Property of Square Roots: If a ≥ 0 and b ≥ 0: • Quotient Property of Square Roots: If a ≥ 0 and b > 0:

21. Summary Simplification Rules • To “simplify” a radical means to: • Take all the perfect squares out of the radicand. • Combine terms with like radicands • Do not leave a radical in the denominator

22. Simplifying Radical Expressions • Homework page 144, # 3 – 24 by 3’s and 25 & 26

23. Simplify 1. 2. Solve. 3. 4. Warm-up

24. Standards for Radical Functions • MM1A2a. Simplify algebraic and numeric expressions involving square root. • MM1A2b. Perform operations with square roots. • MM1A3b. Solve equations involving radicals such as , using algebraic techniques.

25. Radical Functions • Today’s essential questions: • How do we find the solution of a radical function? • How are they used in real life applications?

26. 12.3 Solving Radical Equations • Get the radical on one side. 2. Square both sides of the equals sign. 3. Solve for the variable. 4. Check your answer. IF the answer doesn’t check, then “no solution.”

27. EXAMPLE 3: CHECK

28. EXAMPLE 1: CHECK No solution

29. Your Turn – Solve with your neighbor: CHECK

30. What if > One Radical? • The process is the same: • Get a radical on one side. • Square both sides of the equal sign. • Solve for the variable. • Repeat as necessary • Check your answer. IF the answer does not check, then there is NO SOLUTION for that answer

31. Example 3: CHECK • ? • ?

32. Example 3: CHECK • ? • ? • ? • ?

33. Your Turn – Solve Individually: CHECK • ? • ? • ? • ?

34. Practice with Tic-Tac-Toe • The object is to get three in a row. • Work together in designated pairs. • Notice: Different problems have different points. • Your score will be the three in a row you solve with the most points

35. Practice • Page 148, # 3 – 30 by 3’s and 31 & 32