Precalculus Chapter 2 Section 2 Power Functions with Modeling

Power Functions and Variation Definition: Any function that can be written in the form f(x) = k(xa), where k and a are nonzero constants, is a power function. a is the power k is the constant of variation, or constant of proportion We say that the function f(x) varies as the ath power of x.

Power Functions and Variation Many formula’s from Geometry and Science are power functions. NAME FORMULA POWER CONSTANT OF VARIATION Circumference C = 2πr 1 2π Area of a Circle A = πr2 2 π Force of Gravity F = k/d2 -2 k Boyle’s Law V = k/P -1

Power Functions and Variation Find the Area of a Circle for the following radii: r = 1, 2, 3, 4 Describe the variation between the increase in the radius to the change in the area. Use Boyle’s Law to find Volume for the given Pressure: P = 1, 2, 3, 4 and k = 1 Describe the variation between the increase in the pressure to the change in the volume.

Power Functions and Variation You have just described what it means to have Direct Variation and Inverse Variation. A power function with positive powers have direct variation. A power function with negative powers have inverse variation. You need to be able to identify this from a function. You also need to be able to analyze a power function. Analyze: 𝑓 𝑥 = 3 𝑥 and 𝑔 𝑥 = 1 𝑥 2

Monomial Functions A 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. Examples: y = x3, y = 2x5 You need to able to graph a monomial function on a calculator and do an analysis.

Graphs of Power Functions Graphs of Power Functions have 4 basic shapes. See graphs at end of notes. For 𝑥≥0, the graph contains point (1, k). Those with positive powers also pass through (0, 0). For 𝑥<0, one of three following things happen: f is undefined for x < 0, i.e. 𝑓 𝑥 = 𝑥 1 2 f is an even function, i.e. 𝑓 𝑥 = 𝑥 −2 f is an odd function, i.e. 𝑓 𝑥 = 𝑥 −1

Graphs of Power Functions State the values of the constants k and a. Describe the portion of the curve that lies in Quadrant I or IV. Determine whether f(x) is even, odd, or undefined for x < 0. 𝑓 𝑥 =2 𝑥 −3 𝑓 𝑥 =−0.4 𝑥 1.5 𝑓 𝑥 =− 𝑥 0.4

Modeling with Power Functions Let’s see how we can use a graphing calculator in order to solve power models. We will do Example 5 & 6 on page 194 - 195 together.

Homework Exercises # 3 – 36 by multiples of 3, 37 – 48 all, 51 – 57 odds on pages 196 – 198.