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. The constant a is the power, and k is the constant of variation or constant of proportion. -f(x) varies as the a th power of x or is proportional to the a th power of x. -If the power is positive, it’s direct variation -If the power is negative, it’s inverse variation.

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 = s 2 power is 2, constant of variation is 1

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∙3 x c) A = πr 2 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

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 = kt 2

Even Functions Graph x 2, x 4, x 6 and compare/contrast the graphs. All have shape similar to x 2. 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)

Odd Functions Graph x, x 3, x 5 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)

All functions Graph x, x 2, x 3,x 4, x 5, x 6 with window [0,1] by[0,1] Lower the power the higher the graph with x values between 0 and 1.

All functions Graph x, x 2, x 3,x 4, x 5, x 6 with window [0,2] by[0,2] The higher the power the higher the function with x values greater than 1.

All functions We learned a power function has the form f(x)=kx a, 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.

Graphing Graph f(x) = 4x 6 and f(x) =

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