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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. - PowerPoint PPT Presentation
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Today in Precalculus
• Need a calculator• Notes:
– Power Functions
• Go over homework• Homework
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