Chapter 2: Polynomial, Power, and Rational Functions

1. Section 2-2: Power Functions With Modeling Chapter 2:Polynomial, Power, and Rational Functions

2. Objectives • You will learn about: • Power functions and variation • Monomial functions and their graphs • Graphs of power functions • Modeling with power functions • Why? • Power functions specify the proportional relationships of geometry, chemistry, and physics

3. Vocabulary • Power function • Power • Constant of variation (constant of proportion) • Varies as • Proportional to • Direct variation • Inverse variation • Monomial function

4. Power Function • Any function that can be written in the form: • f(x)=k∙xa where k and x are nonzero constants is a power function. • Parts of a power function: • a is the power • k is the constant of variation/constant of proportion • We can say: • f(x) varies as the a-th power of x. • f(x) is proportional to the a-th power of x

5. Examples of Power Functions

6. Example 1Writing a Power Function Formula • From empirical evidence and the laws of physics, it has been found that the period time T for the full swing of a pendulum varies as the square root of the pendulums length L, provided that the swing is small relative to the length of the pendulum. Express the relationship as a power function.

7. Example 2Analyzing Power Functions • State the power and constant of variation for the function, graph it, and analyze it.

8. Monomial Function • Any function that can be written as f(x)= k or f(x) = k∙xn where k is a constant and n is a positive integer is a monomial function

9. Graphing Monomial Functions • Describe how to obtain the graph of the given function from the graph of g(x)=xn with the same power n. Sketch the graph by hand.

10. Graphs of Power Functions (x ≥ 0) k > 0 • There are 4 basic shapes for general power functions of the form f(x)=kxa x ≥ 0. • In every case: • The graph contains the point (1, k) • If the power is positive, the graph contains the point (0, 0) • If the exponent is negative, there are asymptotes on both axes. k < 0

11. Graphs of Power Functions (x < 0) • In general, for power functions where x < 0, one of the three following things will happen: • f is undefined for x < 0 • f is an even function, so it is symmetric about the y-axis • f is an odd function, so it is symmetric about the origin.

12. Example 4Graphing Power Functions • State the values of the constants k and a. Describe the portion of the cure that lies in Quadrant I or Quadrant IV. Determine whether f is even, odd, or undefined for x < 0. Describe the rest of the curve, if any. Graph the function. • F(x)=2x-3 • F(x)=-0.4x1.5 • F(x)=-x0.4

13. Example 5Modeling Planetary Data with a Power Function • Use the table to obtain a power function model for orbital period as a function of average distance from the sun. Then use the model to predict the orbital period for Neptune, which is 4497 GM from the sun on average.

14. Example 6Modeling Free-Fall Speed Versus Distance • Use the data to obtain a power function model for speed p versus distance traveled d. Then use the model to predict the speed of the ball at impact given that impact occurs when d=1.80 m.