Exponential Functions and Their Graphs, Logs and Natural Logs, Rational Functions and their Graphs.
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Transcript of Exponential Functions and Their Graphs, Logs and Natural Logs, Rational Functions and their Graphs.
Exponential Functions and Their Graphs, Logs and Natural Logs, Rational
Functions and their Graphs
Copyright © by Houghton Mifflin Company, Inc. All rights reserved 2
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.
Copyright © by Houghton Mifflin Company, Inc. All rights reserved 3
The Graph of f(x) = ax, a > 1
y
x(0, 1)
Domain: (–, )
Range: (0, )
Horizontal Asymptote y = 0
Copyright © by Houghton Mifflin Company, Inc. All rights reserved 4
The Graph of f(x) = ax, 0 < a <1
y
x
(0, 1)
Domain: (–, )
Range: (0, )
Horizontal Asymptote y = 0
Copyright © by Houghton Mifflin Company, Inc. All rights reserved 5
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
Copyright © by Houghton Mifflin Company, Inc. All rights reserved 6
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
Copyright © by Houghton Mifflin Company, Inc. All rights reserved 7
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
Copyright © by Houghton Mifflin Company, Inc. All rights reserved 8
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
Copyright © by Houghton Mifflin Company, Inc. All rights reserved 9
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