1. Power Functions and Radical Equations Lesson 4.8

2. Properties of Exponents Given m and m positive integers r, b and p real numbers

3. Power Function Definition Where k and p are constants Power functions are seen when dealing with areas and volumes Power functions also show up in gravitation (falling bodies)

4. Special Power Functions Parabolay = x 2 Cubic functiony = x 3 Hyperbolay = x -1

5. Special Power Functions y = x -2 Text calls them "root" functions

6. Special Power Functions Most power functions are similar to one of these six x p with even powers of p are similar to x 2 x p with negative odd powers of p are similar to x -1 x p with negative even powers of p are similar to x -2 Which of the functions have symmetry? What kind of symmetry?

7. Variations for Different Powers of p For large x, large powers of x dominate x5x5 x4x4 x3x3 x2x2 x

8. Variations for Different Powers of p For 0 < x < 1, small powers of x dominate x5x5 x4x4 x3x3 x2x2 x

9. Variations for Different Powers of p Note asymptotic behavior of y = x -3 is more extreme y = x -3 approaches x-axis more rapidly 0.5 10 20 y = x -3 climbs faster near the y-axis

10. Think About It… Given y = x –p for p a positive integer What is the domain/range of the function? Does it make a difference if p is odd or even? What symmetries are exhibited? What happens when x approaches 0 What happens for large positive/negative values of x?

11. Equations with Radicals Consider Strategy is to square both sides to solve May require this step twice Make sure to check for extraneous roots The squaring process can produce results that do not satisfy the original equation

12. Assignment Lesson 4.7 Page 350 Exercises 1 – 65 EOO 89, 91, 93