Exponential Functions and Their Graphs, Logs and Natural Logs, Rational

Functions and their Graphs

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The exponential function f with base a is defined by

f(x) = ax

where a > 0, a 1, and x is any real number.

For instance,

f(x) = 3x and g(x) = 0.5x

are exponential functions.

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The Graph of f(x) = ax, a > 1

y

x(0, 1)

Domain: (–, )

Range: (0, )

Horizontal Asymptote y = 0

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The Graph of f(x) = ax, 0 < a <1

y

x

(0, 1)

Domain: (–, )

Range: (0, )

Horizontal Asymptote y = 0

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Example: Sketch the graph of f(x) = 2x.

x

x f(x) (x, f(x))

-2 ¼ (-2, ¼)

-1 ½ (-1, ½)

0 1 (0, 1)

1 2 (1, 2)

2 4 (2, 4)

y

2–2

2

4

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Example: Sketch the graph of g(x) = 2x – 1. State the domain and range.

x

yThe graph of this function is a vertical translation of the graph of f(x) = 2x

down one unit .

f(x) = 2x

y = –1 Domain: (–, )

Range: (–1, )

2

4

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Example: Sketch the graph of g(x) = 2-x. State the domain and range.

x

yThe graph of this function is a reflection the graph of f(x) = 2x in the y-axis.

f(x) = 2x

Domain: (–, )

Range: (0, ) 2–2

4

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The irrational number e, where

e 2.718281828…

is used in applications involving growth and decay.

Using techniques of calculus, it can be shown that

ne

n

n

as 1

1

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The Graph of f(x) = ex

y

x2 –2

2

4

6

x f(x)

-2 0.14

-1 0.38

0 1

1 2.72

2 7.39

Properties of Logarithmic FunctionsIf b, M, and N are positive real numbers, b 1, and p and x are real numbers, then: Log15 1 = 0

Log10 10 = 1

Log5 5x = x

3log x = x 3

150 = 1

101 = 10

5x = 5x

Graph and find the domain of the following functions.

y = ln x

x y

-2-101234

.5

cannot takethe ln of a (-) number or 0

0ln 2 = .693ln 3 = 1.098ln 4 = 1.386

ln .5 = -.693

D: x > 0

Graph y = 2x

x y

-2-1012

2-2 =4

1

2-1 =2

1

124

The graph of y = log2 x is the inverse of y = 2x.

y = x

The domain of y = b +/- loga (bx + c), a > 1 consistsof all x such that bx + c > 0, and the V.A. occurs whenbx + c = 0. The x-intercept occurs when bx + c = 1.

Ex. Find all of the above for y = log3 (x – 2). Sketch.

D: x – 2 > 0

D: x > 2

V.A. @ x = 2

x-int. x – 2 = 1

x = 3

(3,0)

3.6: Rational Functions and Their Graphs

Strategy for Graphing a Rational FunctionStrategy for Graphing a Rational FunctionSuppose that

where p(x) and q(x) are polynomial functions with no common factors.

1. Determine whether the graph of f has symmetry. f (x) f (x): y-axis symmetry f (x) f (x): origin symmetry

2. Find the y-intercept (if there is one) by evaluating f (0).3. Find the x-intercepts (if there are any) by solving the equation p(x) 0. 4. Find any vertical asymptote(s) by solving the equation q (x) 0.5. Find the horizontal asymptote (if there is one) using the rule for determining the

horizontal asymptote of a rational function.6. Plot at least one point between and beyond each x-intercept and vertical asymptote.7. Use the information obtained previously to graph the function between and beyond

the vertical asymptotes.

Strategy for Graphing a Rational FunctionStrategy for Graphing a Rational FunctionSuppose that

where p(x) and q(x) are polynomial functions with no common factors.

1. Determine whether the graph of f has symmetry. f (x) f (x): y-axis symmetry f (x) f (x): origin symmetry

2. Find the y-intercept (if there is one) by evaluating f (0).3. Find the x-intercepts (if there are any) by solving the equation p(x) 0. 4. Find any vertical asymptote(s) by solving the equation q (x) 0.5. Find the horizontal asymptote (if there is one) using the rule for determining the

horizontal asymptote of a rational function.6. Plot at least one point between and beyond each x-intercept and vertical asymptote.7. Use the information obtained previously to graph the function between and beyond

the vertical asymptotes.

( )( )

( )p x

f xq x

EXAMPLE: Graphing a Rational Function

Step 4 Find the vertical asymptotes: Set q(x) 0.x2 4 0 Set the denominator equal to zero.

x2 4 x 2Vertical asymptotes: x 2 and x 2.

Solution

moremore

2

2

3Graph: ( ) .4

xf xx

Step 3 Find the x-intercept: 3x2 0, so x 0: x-intercept is 0.

Step 1 Determine symmetry: f (x) f

(x):

Symmetric with respect to the y-axis.

2 2

2 2

3 344

x xxx

Step 2 Find the y-intercept: f (0) 0: y-intercept is 0.

2

2

3 0 00 4 4

3.6: Rational Functions and Their Graphs

EXAMPLE: Graphing a Rational Function

Solution

The figure shows these points, the y-intercept, the x-intercept, and the asymptotes.

x 3 1 1 3 4

f(x) 1 1 4

moremore

2

2

3Graph: ( ) .4

xf xx

Step 6 Plot points between and beyond the x-intercept and the vertical asymptotes. With an x-intercept at 0 and vertical asymptotes at x 2 and x 2, we evaluate the function at 3, 1, 1, 3, and 4.

Step 5 Find the horizontal asymptote: y 3/1 3.

2

2

34

xx

275

275

-5 -4 -3 -2 -1 1 2 3 4 5

7

6

5

4

3

1

2

-1

-2

-3

Vertical asymptote: x = 2

Vertical asymptote: x = 2

Vertical asymptote: x = -2

Vertical asymptote: x = -2

Horizontal asymptote: y = 3

Horizontal asymptote: y = 3

x-intercept and y-intercept

x-intercept and y-intercept

3.6: Rational Functions and Their Graphs