Today in Pre-Calculus

• Notes:– Power Functions

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Power FunctionsDefinition: 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.

Power FunctionsWhich 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

2 3 1( ) ; ( ) ; ( ) ; ( ) ; ( )f x x f x x f x x f x f x x

x

Power FunctionsWhich 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

4

2( )f x

x

Power functionPower: -4Constant of variation: 2

Not a power function, power isn’t constant

Power functionIndependent variable: rPower: 2Constant of variation: π

Writing Power FunctionsExpress 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

Even FunctionsGraph x2, x4, x6 and compare/contrast the graphs.

All have shape similar to x2.Same domain and rangeAll 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)

x

y

Odd FunctionsGraph 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)

x

y

All functionsGraph 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.

x

y

All functionsGraph 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.

x

y

All functionsWe 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.

GraphingGraph f(x) = 4x6 and f(x) = 5

25 x

Homework

• Pg. 196: 1-10, 17-22, 31-42