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Today in Pre-Calculus

Today in Pre-Calculus. Notes: Power Functions Go over homework Go over test Homework. Power Functions. Definition: Any function that can be written in the form f(x)=k ∙x a , where k and a are nonzero constants.

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Today in Pre-Calculus

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  1. Today in Pre-Calculus • Notes: • Power Functions • Go over homework • Go over test • Homework

  2. Power Functions Definition: Any function that can be written in the form f(x)=k∙xa, where k and a are nonzero constants. The constant a is the power, and k is the constant of variation or constant of proportion. • f(x) varies as the ath power of x or is proportional to the ath power of x. • If the power is positive, it’s direct variation • If the power is negative, it’s inverse variation.

  3. Power Functions Which five of the ten basic functions are power functions? Many familiar formulas from geometry and science are power functions: Ex: C = 2πr power is 1, constant of variation is 2π Ex: A = s2 power is 2, constant of variation is 1

  4. Power Functions Which of the following are power functions? State the power and the constant of variation. If it isn‘t a power function, explain why a) b) g(x) = 4∙3x c) A = πr2 Power function Power: -4 Constant of variation: 2 Not a power function, power isn’t constant Power function Independent variable: r Power: 2 Constant of variation: π

  5. Writing Power Functions Express the following as power function equations: a) In physics, Hooke’s Law for a spring states that the distance a spring is stretched (or compressed) varies directly as the force on the spring. (let d = distance spring is stretched, F = force, k = constant) b) The distance a ball rolls down an inclined plane is directly proportional to the square of the time it rolls. (let d = distance, t = time, and k= constant) d = kF d = kt2

  6. Graph x2, x4, x6 and compare/contrast the graphs. All have shape similar to x2. Same domain and range All have the same end behavior. All are bounded below b=0. All have even symmetry. All increase/decrease on same interval. All go through (-1,1), (0,0), (1,1) Even Functions

  7. Graph x, x3, x5 and compare/contrast the graphs. All have the same end behavior. All are unbounded All have odd symmetry. All increase on (-∞,∞) Domain and Range all reals All go through (-1,-1), (0,0), (1,1) Odd Functions

  8. Graph x, x2, x3,x4, x5, x6 with window [0,1] by[0,1] Lower the power the higher the graph with x values between 0 and 1. All functions

  9. Graph x, x2, x3,x4, x5, x6 with window [0,2] by[0,2] The higher the power the higher the function with x values greater than 1. All functions

  10. We learned a power function has the form f(x)=kxa, so how does k change these graphs? If k > 1, there is a vertical stretch of k If k< 1, there is a vertical shrink of k If k is negative there is a reflection over the x-axis. All functions

  11. Graph f(x) = 4x6 and f(x) = Graphing

  12. Homework • Pg. 196: 1-10, 17-22, 31-42

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