Power and Polynomial Functions: Understanding Change and Behavior, Exams of Calculus

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POWER FUNCTIONS

Functions Modeling Change:A Preparation for Calculus,

Proportionality and Power Functions

• Example 1

The area, A, of a circle is directly proportional to

the square of its radius, r: A = k • r^2.

Functions Modeling Change: A Preparation for Calculus,

• Example 2

The weight, w, of an object is inversely

proportional to the square of the object’s

distance, d, from the earth’s center:

w = k/d^2 or w = kd −.

product

division

Proportionality and Power Functions

• Example 3

Which of the following functions are power functions? For each power function, state the value of the constants k and p in the formula y = k xp. (a) f (x) =

(b) g (x) = 2 ( x + 5)^3

(c) u (x) =

(d) v (x) = 6 ・ 3 x

Solution: The functions f ( k =13, p =1/3) and u ( k =5, p =-3/2) are power functions; the functions g and v are not power functions.

Functions Modeling Change: A Preparation for Calculus,

3

1 13 3 x  13 x

2

3 2

3

25 / x^3  5 / x  5 x 

The Effect of the Power p

• Graphs of the Special Cases y = x^0 and y = x^1

Functions Modeling Change: A Preparation for Calculus,

1 1 2 3 4 5 x 1

1

2

3

4

5^ y y = x^1

(1,1) ● y = x 0

• The power functions corresponding to p = 0 and p = 1 are both linear. The function y = x^0 = 1, except at x = 0. Its graph is a horizontal line with a hole at (0,1). The graph of y = x^1 = x is a line through the origin with slope +1. Both graphs contain the point (1,1).

Show where the graphs intersect.

Preparation for Calculus,^ Functions Modeling Change: 4th Edition,^ A 2011, Connally

Algebraically!

Preparation for Calculus,^ Functions Modeling Change: 4th Edition,^ A 2011, Connally

x  x

4 2

x  x 

2 2

x x  

2

x x  x  

0 , 1 , 1

, x  x   x  

so

There are three intersection points!

3 2 1 0 1 2 3^ x

1

2

3

4

5

6^ y

3 2 1 1 2 3 x

4

2

2

4

y

The Effect of the Power p

Negative Integer Powers

Functions Modeling Change: A Preparation for Calculus,

(1,1)

(-1,-1)

y = x -

(-1,1) (1,1)

Both graphs have a vertical asymptote of x = 0

Both graphs have a horizontal asymptote of y = 0

y = x -

x 0.1 0.05 0.01 0.001 0.0001 0 1/x 10 20 100 1000 10000 undefined 1/x^2 100 400 10,000 1,000,000 100,000,000 undefined

x 0 10 20 30 40 50 1/x undefined 0.1 0.05 0.033333 0.025 0. 1/x^2 undefined 0.01 0.0025 0.001111 0.000625 0.

The Effect of the Power p Positive Fractional

Even

Odd

Functions Modeling Change: A Preparation for Calculus,

Graph y = x1/2 y = x1/

Graph y = x1/3 y = x1/

POLYNOMIAL FUNCTIONS

Functions Modeling Change:A Preparation for Calculus,

A General Formula for the Family of

Polynomial Functions

The general formula for the family of polynomial

functions can be written as

p(x) = an xn^ + an−1 xn−1^ +... + a 1 x + a 0 ,

where n is a positive integer called the degree of

p(x) and where an ≠ 0.

Functions Modeling Change: A Preparation for Calculus,

Functions ModelingChange: A Preparation for Calculus, 4th

Like the power functions from which they are built, polynomials are defined for all values of x. Except for polynomials of degree zero (whose graphs are horizontal lines), the graphs of polynomials do not have horizontal or vertical asymptotes; they are smooth and unbroken. The shape of the graph depends on its degree; typical graphs are shown below.

Quadratic Cubic Quartic Quintic n = 2 n = 3 n = 4 n =

The Long-Run Behavior

of Polynomial Functions

When viewed on a large enough scale, the graph

of the polynomial p(x) = anxn^ + an−1xn−1^ + ・ ・ ・

+ a 1 x + a 0 looks like the graph of the power

function y = anxn. This behavior is called the long-

run behavior of the polynomial. Using limit

notation, we write:

Functions Modeling Change: A Preparation for Calculus,

n lim x  p ( x )  lim x  anx

n x lim  p^ ( x )^ x lim anx

THE SHORT-RUN BEHAVIOR

OF POLYNOMIALS

Functions Modeling Change:A Preparation for Calculus,

Visualizing Short-Run and Long-Run

Behaviors of a Polynomial

Example 1 Compare the graphs of the polynomials f , g , and h given by f(x) = x^4 − 4x^3 + 16x − 16, g(x) = x^4 − 4x^3 − 4x^2 + 16x, h(x) = x^4 + x^3 − 8x^2 − 12x.

Functions Modeling Change: A Preparation for Calculus,

4 2 2 4 x

30

20

10

10

20

30

y

10 5 5 10^ x

1000

2000

3000

4000^ y

A close-up look near zeros and turns A larger scale look resembling x^4