power functions with modeling. any function that can be written in the form f(x) = k ·x ⁿ, where...
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Power Functions with Modeling
Any function that can be written in the formf(x) = k ·x ⁿ, where k and n are nonzero
constants
is a power function. The constant n is the power, and k is the constant of variation, or constant of proportion. We say f(x) varies as the nth power of x, or f(x) is proportional to the nth power of x.
DEFINITION: POWER FUNCTION
Power function models involve output-form-input relationships that can be expressed in the language of
variation and proportion:
The power function formulas with positive powers are statements of direct variationand power function formulas with negative powers are statements of inverse variation.
Circumference
formulaC = 2∏r
Circumference has a power of one
Constant of Varation 2∏
The circumference of a circle varies directly as its radius
Area of a circle A = ∏r²
Area of a circle has a power of 2
Constant f Varation ∏
The area enclosed by a circle is directly proportional to the
square of its radius
Force of gravity F = k/d²
Force of gravity power of -2Constant f Varation k
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
Boyle’s law V = k/p
Boyle’s law has a power of -1
Constant f Varation k
The volume of an enclosed gas varies inversely as the applied
pressure.
Writing a Power Function Formula
• From empirical evidence and the laws of physics it has been found that the period of time T for the full swing of a pendulum varies as the square root of the pendulum’s length l, provided that the swing is small relative to the length of the pendulum.
Solution:
• Because it does not state otherwise, the variation is direct. So the power is positive. The wording tells us that T is a function of I. Using k as the constant of variation gives us
lklT )(
Five Basic Power Functions
xxf )(2)( xxf
3)( xxf
xxf
1)( xxf )(
Identity Functionf(x) = x, slope 1, y-intercept = 0
The domain of this function is all real numbers.
The range is also all real numbers
f(x) = x
If you put any real number in this function, you get the same real number “back”.
Reciprocal FunctionThe domain of this function is all NON-ZERO real numbers.
The range is all NON-ZERO real numbers.
x
xf1
Square Root Function
The domain of this function is NON-NEGATIVE real numbers.
The range is NON-NEGATIVE real numbers
xxf
Cube Functionf(x) = x3
The domain of this function is all real numbers.
The range is all real numbers
Square Functionf(x) = x2
The domain of this function is all real numbers.
The range is all NON-NEGATIVE real numbers