Power Functions with Modeling. Any function that can be written in the form f(x) = k ·x ⁿ, where k and n are nonzero constants is a power function. The.

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Presentation transcript:

Power Functions with Modeling

Any function that can be written in the form f(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 variation and power function formulas with negative powers are statements of inverse variation.

Circumference formula C = 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 -2 Constant 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

Five Basic Power Functions

Identity Function f(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 Function The domain of this function is all NON-ZERO real numbers. The range is all NON-ZERO real numbers.

Square Root Function The domain of this function is NON-NEGATIVE real numbers. The range is NON-NEGATIVE real numbers

Cube Function f(x) = x 3 The domain of this function is all real numbers. The range is all real numbers

Square Function f(x) = x 2 The domain of this function is all real numbers. The range is all NON-NEGATIVE real numbers