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Calc Unit 1.2

1) The document defines and discusses various types of functions including power functions, polynomial functions, rational functions, exponential functions, and logarithmic functions. 2) Power functions have the form y=x^n and their graphs are similar to y=x^2 and y=x^3. Polynomial functions are the sum of terms with variables raised to whole number powers. 3) Rational functions are the ratio of two polynomial functions. Exponential functions have the form y=a^x where a is a positive number not equal to 1. Logarithmic functions are the inverse of exponential functions.

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58 views3 pages

Calc Unit 1.2

1) The document defines and discusses various types of functions including power functions, polynomial functions, rational functions, exponential functions, and logarithmic functions. 2) Power functions have the form y=x^n and their graphs are similar to y=x^2 and y=x^3. Polynomial functions are the sum of terms with variables raised to whole number powers. 3) Rational functions are the ratio of two polynomial functions. Exponential functions have the form y=a^x where a is a positive number not equal to 1. Logarithmic functions are the inverse of exponential functions.

Uploaded by

DASitch
Copyright
© Attribution Non-Commercial (BY-NC)
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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wtw158 Definition (p 31) The function y f x

Unit 1.2 - Some important functions

First sem 2008

x a , with a a constant is called a power function. x n is similar to the graph of y x n is similar to the graph of y x2. x3.

6. If n is a positive rational number and 0 n 1 the form of the graph y x n is similar to the graph of y x. 7. If n is a positive rational number (not an integer) and n 1 the form of the graph y x n is similar to the graph of y x 1.5 .

x n , n even, 0

x n , n odd, 0

x 1.5

xn, n

2, 4, 6, . . .

xn, n

1, 3, 5, 7, . . .

5. If n

1, 3, 5, . . . the form of the graph y

4. If n

2, 4, . . . the form of the graph y

x n is similar to the graph of y

1 . x2 1. n x is similar to the graph of y x

xn, n

2, 4, 6, . . .

xn, n

 

 

1. y x 2. If n 2, 4, 6, . . . the form of the graph y 3. If n 3, 5, 7, . . . the form of the graph y

 

3, 5, 7, . . .

Remark A polynomial of degree n has at most n n 1 turning points.


1 0.8

1 turning points. Such a polynomial can have less than

10 8 6

0.6 0.4 -3 0.2 -2 -1

4 2 0 -2 -4 1 x 2 3

0.2 0.4 0.6 0.8

1 x

1.2 1.4 1.6 1.8

Definition(p 32) A Rational function is the ration of two polynomials. Px with P x and Q x polynomials. y Rx Qx We will later sketch these functions. Definition (p 34) The function y f x

ax, 0

ax, a

0, x

is called an exponential function.

ax, a

Both are polynomials of degree 4. The first one has one turning point at x three turning points.

x4

4x 3

6x 2

4x

x4

3x 2

x 1. The second has

Examples of polynomials: y ax 2 bx c (a quadratic polynomial if a polynomial if a 0).

0) and y

Definition (p 29) anxn an 1xn 1 . . . a1x The function y P x The degree of the polynomial is n (assuming that a n

a 0 is call a polynomial. 0. ax 3 bx 2 cx d (a cubic

ax, a

Definition (p 35) The function y f x log a x, with a a positive number is called a logarithmic function and is the inverse of an exponential function.

Example Sketch the function f x 2x 9 . The form of the graph of y x 9 is the same as y x 3 . The form of y 2x 9 is the same as y x 9 because you multiply each function value with a positive number 2 1. If you multiply the formula of the function with 1 you reflect the graph about the x-axis.

y x9 y 2x 9 y 2x 9 (y x is the dotted graph.) Example Sketch y x 2 and y x 4 on the same set of axis. The shapes are the same, but the graph with the higher power dominates. Dominates means that for large values of x (in this case large in more than one!) the positive function values of the dominating function is larger. The dotted graph is y x 4 .
9
5 4 3 2 1

-2

-1

1 x

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