1. Exponential Functions and Their Graphs, Logs and Natural Logs, Rational Functions and their Graphs

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.
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3. The Graph of f(x) = ax, a > 1y
Range: (0, )
(0, 1) x
Horizontal Asymptote y = 0
Domain: (–, )
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4. The Graph of f(x) = ax, 0 < a <1y
Range: (0, )
Horizontal Asymptote y = 0
(0, 1) x
Domain: (–, )
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5. Example: Sketch the graph of f(x) = 2x. x f(x) (x, f(x)) -2 ¼ (-2, ¼) y x
f(x)
(x, f(x)) -2
(-2, ¼) -1
(-1, ½) 1
(0, 1) 2
(1, 2) 4
(2, 4) 4 2 x –2 2
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6. Example: Sketch the graph of g(x) = 2x – 1. State the domain and range.y
f(x) = 2x
The graph of this function is a vertical translation of the graph of f(x) = 2x down one unit . 4 2
Domain: (–, ) x
y = –1
Range: (–1, )
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7. Example: Sketch the graph of g(x) = 2-x. State the domain and range.y
f(x) = 2x
The graph of this function is a reflection the graph of f(x) = 2x in the y-axis. 4
Domain: (–, ) x –2 2
Range: (0, )
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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
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9. The Graph of f(x) = ex x f(x) -2 0.14 -1 0.38 1 2.72 2 7.39 y x 6 4 21
2.72 2
7.39 6 4 2 x –2 2
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10. Properties of Logarithmic Functions
If b, M, and N are positive real numbers, b  1, and p and x are real numbers, then:
Log15 1 = 0
150 = 1
Log10 10 = 1
101 = 10
Log5 5x = x
5x = 5x
3log x = x 3

11. Graph and find the domain of the following functions.
y = ln x
x y -2 -1 1 2 3 4 .5
cannot take
the ln of a (-)
number or 0
ln 2 = .693
ln 3 = 1.098
ln 4 = 1.386
D: x > 0
ln .5 = -.693

12. Graph y = 2x
y = x
x y -2 -1 1 2
2-2 =
2-1 = 1 2 4
The graph of y = log2 x
is the inverse of y = 2x.

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

14. f (-x) = f (x): y-axis symmetry f (-x) = -f (x): origin symmetry
3.6: Rational Functions and Their Graphs
Strategy for Graphing a Rational Function
Suppose 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) = Find any vertical asymptote(s) by solving the equation q (x) = 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.

15. EXAMPLE: Graphing a Rational Function
3.6: Rational Functions and Their Graphs
EXAMPLE: Graphing a Rational Function
Solution
Step 1 Determine symmetry: f (-x) = = = f (x):
Symmetric with respect to the y-axis.
Step 2 Find the y-intercept: f (0) = = 0: y-intercept is 0.
Step 3 Find the x-intercept: 3x2 = 0, so x = 0: x-intercept is 0.
Step 4 Find the vertical asymptotes: Set q(x) = 0.
x2 - 4 = Set the denominator equal to zero.
x2 = 4
x = 2
Vertical asymptotes: x = -2 and x = 2.
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16. EXAMPLE: Graphing a Rational Function
3.6: Rational Functions and Their Graphs
EXAMPLE: Graphing a Rational Function
Solution
Step 5 Find the horizontal asymptote: y = 3/1 = 3.
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. -5 -4 -3 -2 -1 1 2 3 4 5 7 6
Vertical asymptote: x = 2
Vertical asymptote: x = -2
Horizontal asymptote: y = 3
x-intercept and y-intercept x -3 -1 1 3 4
f(x) =
The figure shows these points, the y-intercept, the x-intercept, and the asymptotes.
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