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Moving on to Sec. 2.2… HW: p. 189-190 1-21 odd, 41,43PowerPoint Presentation

Moving on to Sec. 2.2… HW: p. 189-190 1-21 odd, 41,43

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### Moving on to Sec. 2.2…HW: p. 189-190 1-21 odd, 41,43

Power Functions

Definition: Power Function

Any function that can be written in the form

where k and a are nonzero constants,

is a power function.

The constant a is the power, and k is the constant of

variation, or constant of proportion.

We say that f (x) varies as the a-th power of x, or f (x) is

proportional to the a-th power of x.

Which of our twelve basic functions are power functions???

Identity Function, Squaring Function, Cubing Function,

Reciprocal Function, Square Root Function

Some Examples…

Power = 5/3, Constant = 9

Power = 0, Constant = 13

Power = 5, Constant = k/2

Power = 3, Constant = 4 /3

More Definitions

Statements of direct variation – power function formulas with

positive powers.

Statements of inverse variation – power function formulas with

negative powers.

Circumference

Power = 1, Constant =

The circumference of a circle varies directly as its radius.

Force of gravity

Power = –2, Constant =

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.

More Definitions

Statements of direct variation – power function formulas with

positive powers.

Statements of inverse variation – power function formulas with

negative powers.

Ex. from physics: The period of time (T) for the full swing of a

pendulum varies as the square root of the pendulum’s length (L).

Express this relationship as a power function.

Analyzing Power Functions

State the power and constant of variation for the given function,

graph it, and analyze it.

Domain:

All reals

Extrema:

None

Range:

All reals

Asymptotes:

None

Continuity:

Continuous

End Behavior:

Inc/Dec:

Inc. for all x

Symmetry:

Origin (odd func.)

Boundedness:

Not Bounded

Analyzing Power Functions

State the power and constant of variation for the given function,

graph it, and analyze it.

Boundedness:

Below

Domain:

Extrema:

None

Range:

Asymptotes:

H.A.: y = 0

V.A.: x = 0

Continuity:

Continuous on its

domain (discont. at

x = 0)

End Behavior:

Inc/Dec:

Inc. on

Dec. on

Symmetry:

y-axis (even func.)

Guided Practice

Write the given statement as a power function equation. Use k

for the constant of variation if one is not given.

The volume V of a circular cylinder with fixed height is

proportional to the square of its radius r.

Charles’s Law states the volume V of an enclosed ideal

gas at a constant pressure varies directly as the

absolute temperature T.

Guided Practice

Write the given statement as a power function equation. Use k

for the constant of variation if one is not given.

The speed p of a free-falling object that has been

dropped from rest varies as the square root of the

distance traveled d, with a constant of variation

Definition: Monomial Function

Any function that can be written as

or

where k is a constant and n is a positive integer,

is a monomial function.

…an “Exploration”:

is even if n is even and odd if n is odd

The graphs of , for n = 1, 2,…, 6

(1, 1)

(0, 0)

[0, 1] by [0, 1]

Guided Practice

Describe how to obtain the graph of the given function from the graph of with the same power n. Sketch the graph by hand and support your answer with a grapher.

Vertically stretch the graph of by a factor of 2.

Both functions are odd.

Guided Practice

Describe how to obtain the graph of the given function from the graph of with the same power n. Sketch the graph by hand and support your answer with a grapher.

Vertically shrink the graph of by a factor of 2/3, and reflect it across the x-axis. Both functions are even.

More Truths About Graphs of Power Functions

The general shapes that are possible for power functions

Of the form for positive x-values. In all

cases, the graph of f contains the point (1, k).

a < 0

a > 1

k < 0

a = 1

(1, k)

0 < a < 1

0 < a < 1

(1, k)

a = 1

a > 1

a < 0

k > 0

More Guided Practice

Observe the values of the constants k and a. Describe the

portion of the graph that lies in Quadrant I or IV. Determine

whether f is even, odd, or undefined for x < 0. Describe the

rest of the curve, if any. Graph the function to see whether

it matches the description.

Because k is positive and a is negative, the graph

passes through (1, 2) and is asymptotic to both axes.

The graph is decreasing in the first quadrant. The

function is odd because f (–x) = –f (x).

More Guided Practice

Observe the values of the constants k and a. Describe the

portion of the graph that lies in Quadrant I or IV. Determine

whether f is even, odd, or undefined for x < 0. Describe the

rest of the curve, if any. Graph the function to see whether

it matches the description.

Because k is negative and a > 1, the graph contains

(0, 0) and passes through (1, – 0.4). In the fourth

quadrant, it is decreasing. The function is undefined

for x < 0 (Why???).

More Guided Practice

Observe the values of the constants k and a. Describe the

portion of the graph that lies in Quadrant I or IV. Determine

whether f is even, odd, or undefined for x < 0. Describe the

rest of the curve, if any. Graph the function to see whether

it matches the description.

Because k is negative and 0 < a < 1, the graph contains

(0, 0) and passes through (1, –1). In the fourth

quadrant, it is decreasing. The function is even

because f (–x) = f (x).

Modeling with power Functions: Average distances

and orbit periods for the six innermost planets.

Average Distance

from Sun (Gm)

Period of

Orbit (days)

Planet

Mercury 57.9 88

Venus 108.2 225

Earth 149.6 365.2

Mars 227.9 687

Jupiter 778.3 4332

Saturn 1427 10760

Use the data from the previous slide to obtain a power function model

for orbital period as a function of average distance from the Sun.

Then use the model to predict the orbital period for Neptune,

which is 4497 Gm from the Sun on average.

Use your calculator to find power regression for the data…

How well does this model fit the data?

For Neptune:

It takes Neptune about 60,313 days to orbit the

Sun, or about 165 years.

More Quality Practice Problems function model

Charles’s Law. The volume of an enclosed gas (at a constant

pressure) varies directly as the absolute temperature. If the

pressure of a 3.46-L sample of neon gas at 302 degrees Kelvin

is 0.926 atm, what would the volume be at a temperature of

338 degrees Kelvin if the pressure does not change?

First, write the general equation:

Next, solve for k:

More Quality Practice Problems function model

Charles’s Law. The volume of an enclosed gas (at a constant

pressure) varies directly as the absolute temperature. If the

pressure of a 3.46-L sample of neon gas at 302 degrees Kelvin

is 0.926 atm, what would the volume be at a temperature of

338 degrees Kelvin if the pressure does not change?

Now, use our equation to solve for V at the new T:

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