Section+2.2

//(Pages 188-199)//
 * __Section 2.2 Power Functions with Modeling__**

A __power function__ is any function that can be written in the form of f(x) = k(xa) where ‘a’ is the power and ‘k’ is called the constant of variation or constant of proportion.


 * Two Types of Power Functions**

Power functions in the form of f(x) = k(xa) are direct variations or are said to be directly proportional to. Power functions in the form of f(x) = k/xa are inverse variations or are said to be inversely proportional to.

Some power functions that we have seen before are:

Circumference … C = 2πr … Power =1 Constant = 2π Area of a circle … A= πr2 … Power = 2 Constant = π Boyle’s Law … V = k/P … Power = -1 Constant = k

Power function formulas can also be written as statements. Formulas with positive powers are written as statements of direct variation while formulas with negative powers are written as statements of inverse variation. Some examples are:

The circumference of a circle varies as its radius. The area enclosed by a circle is directly proportional to the square of its radius. Boyle’s Law states that the volume of an enclosed gas varies inversely as the applied pressure.


 * Graphing/Modeling Power Functions**

//EVERY// power function goes through the point (1, k)

When graphing power functions there are four basic shapes used. The four possibilities are determined if... __a < 0__, __a > 1__ , __a = 1__ , or __0 < a < 1__. K also determines which quadrant the power function lies in (__k > 0__ or __k < 0__). (Figure 2.14 in the book on page 192)

Power functions have even, odd, or undefined (for x < 0) symmetry that is found the same way as we’ve done before by solving for f(-x) and comparing to the original function.


 * Examples**

f(x) = 2x-3 Power of -3 and Constant of 2 f(-x) = 2(-x)-3 = 2/(-x)3 =-2/x3 = -2x-3 = -f(x) Odd symmetry and we know that it goes through the point (1,2).

f(x) = -.4x1.5 = -.4x3/2 = -.4( √x)3 Power of 1.5 Constant of -.4 Undefined symmetry because x has to be greater than 0 (can’t take square root of a negative) and goes through the point (1,-.4)

Page 196 #s 1-35 odd and 55 Page 197 #s 32, 34, 36, 37-48
 * Homework**