1. Moving on to Sec. 2.2… HW: p. 189-190 1-21 odd, 41,43 Power Functions

2. Definition: Power Function Any function that can be written in the form where k and a are nonzero constants, is a power function. The constant a is the power, and k is the constant of variation, or constant of proportion. We say that f (x) varies as the a-th power of x, or f (x) is proportional to the a-th power of x. Which of our twelve basic functions are power functions??? Identity Function, Squaring Function, Cubing Function, Reciprocal Function, Square Root Function

3. Some Examples… Power = 5/3, Constant = 9 Power = 0, Constant = 13 Power = 5, Constant = k/2 Power = 3, Constant = 4 /3

4. More Definitions Statements of direct variation – power function formulas with positive powers. Statements of inverse variation – power function formulas with negative powers. CircumferencePower = 1, Constant = The circumference of a circle varies directly as its radius. Force of gravityPower = –2, Constant = The force of gravity acting on an object is inversely proportional to the square of the distance from the object to the center of the Earth.

5. More Definitions Statements of direct variation – power function formulas with positive powers. Statements of inverse variation – power function formulas with negative powers. Ex. from physics: The period of time (T) for the full swing of a pendulum varies as the square root of the pendulum’s length (L). Express this relationship as a power function.

6. Analyzing Power Functions State the power and constant of variation for the given function, graph it, and analyze it. All reals Domain: Range: Continuous Continuity: Inc. for all x Inc/Dec: Origin (odd func.) Symmetry: Not Bounded Boundedness: None Extrema: None Asymptotes: End Behavior:

7. Analyzing Power Functions State the power and constant of variation for the given function, graph it, and analyze it. Domain: Range: Continuous on its domain (discont. at x = 0) Continuity: Dec. on Inc/Dec: y-axis (even func.) Symmetry: Below Boundedness: None Extrema: H.A.: y = 0 V.A.: x = 0 Asymptotes: End Behavior: Inc. on

8. Guided Practice Write the given statement as a power function equation. Use k for the constant of variation if one is not given. The volume V of a circular cylinder with fixed height is proportional to the square of its radius r. Charles’s Law states the volume V of an enclosed ideal gas at a constant pressure varies directly as the absolute temperature T.

9. Guided Practice Write the given statement as a power function equation. Use k for the constant of variation if one is not given. The speed p of a free-falling object that has been dropped from rest varies as the square root of the distance traveled d, with a constant of variation

10. Definition: Monomial Function Any function that can be written as or where k is a constant and n is a positive integer, is a monomial function.

11. …an “Exploration”: The graphs of, for n = 1, 2,…, 6 [0, 1] by [0, 1] (0, 0) (1, 1) is even if n is even and odd if n is odd

12. Guided Practice Describe how to obtain the graph of the given function from the graph of with the same power n. Sketch the graph by hand and support your answer with a grapher. Vertically stretch the graph of by a factor of 2. Both functions are odd.

13. Guided Practice Describe how to obtain the graph of the given function from the graph of with the same power n. Sketch the graph by hand and support your answer with a grapher. Vertically shrink the graph of by a factor of 2/3, and reflect it across the x-axis. Both functions are even.

14. More Truths About Graphs of Power Functions The general shapes that are possible for power functions Of the form for positive x-values. In all cases, the graph of f contains the point (1, k). k > 0 (1, k) a < 0a > 1 a = 1 0 < a < 1 k < 0 (1, k) a < 0 a > 1 a = 1 0 < a < 1

15. More Guided Practice Observe the values of the constants k and a. Describe the portion of the graph that lies in Quadrant I or IV. Determine whether f is even, odd, or undefined for x < 0. Describe the rest of the curve, if any. Graph the function to see whether it matches the description. Because k is positive and a is negative, the graph passes through (1, 2) and is asymptotic to both axes. The graph is decreasing in the first quadrant. The function is odd because f (–x) = –f (x).

16. More Guided Practice Observe the values of the constants k and a. Describe the portion of the graph that lies in Quadrant I or IV. Determine whether f is even, odd, or undefined for x < 0. Describe the rest of the curve, if any. Graph the function to see whether it matches the description. Because k is negative and a > 1, the graph contains (0, 0) and passes through (1, – 0.4). In the fourth quadrant, it is decreasing. The function is undefined for x < 0 (Why???).

17. More Guided Practice Observe the values of the constants k and a. Describe the portion of the graph that lies in Quadrant I or IV. Determine whether f is even, odd, or undefined for x < 0. Describe the rest of the curve, if any. Graph the function to see whether it matches the description. Because k is negative and 0 < a < 1, the graph contains (0, 0) and passes through (1, –1). In the fourth quadrant, it is decreasing. The function is even because f (–x) = f (x).

18. Modeling with power Functions: Average distances and orbit periods for the six innermost planets. Planet Average Distance from Sun (Gm) Period of Orbit (days) Mercury57.988 Venus108.2225 Earth149.6365.2 Mars227.9687 Jupiter778.34332 Saturn142710760

19. Use the data from the previous slide 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. Use your calculator to find power regression for the data… How well does this model fit the data? For Neptune: It takes Neptune about 60,313 days to orbit the Sun, or about 165 years.

20. More Quality Practice Problems Charles’s Law. The volume of an enclosed gas (at a constant pressure) varies directly as the absolute temperature. If the pressure of a 3.46-L sample of neon gas at 302 degrees Kelvin is 0.926 atm, what would the volume be at a temperature of 338 degrees Kelvin if the pressure does not change? First, write the general equation: Next, solve for k:

21. More Quality Practice Problems Charles’s Law. The volume of an enclosed gas (at a constant pressure) varies directly as the absolute temperature. If the pressure of a 3.46-L sample of neon gas at 302 degrees Kelvin is 0.926 atm, what would the volume be at a temperature of 338 degrees Kelvin if the pressure does not change? Now, use our equation to solve for V at the new T: