• Graph of polynomials. The leading item (the item with highest degree) totally
the behavior of the graph at two ends (lxl is very big). The roots of polynomial
to the x-intercept of the graph. The multiplicity of each root (if multiplicity
is odd, then
the graph crosses the root, if multiplicity is even, then the graph touches the
• Asymptotes and holes of rational functions.
, g(x) and h(x) are polynomials.
Solve for h(x) = 0. If the root (say, a) of h(x) = 0 satisfies g(a) ≠ 0, then x
= a is the
vertical asymptote. If a (the root of h(x) = 0) also satisfies g(a) = 0, and the
(x-a) can be canceled from the numerator and denominator, then x = a is a hole
of the graph of . As for horizontal
asymptote, if g(x) and h(x) have the same
degree, then the horizontal asymptote is . If
the degree of
g(x) is lower than the degree of h(x), then the horizontal asymptote is just y =
x-axis). For example,, then x = 0 is a hole
(because x can be canceled
from the numerator and denominator), x = 1 is a vertical asymptote, and y = 1 is
• Exponential functions . Domain and range of exponential functions. Graph of ex-
ponential functions (y-intercept). Word problems of exponential functions
increase and exponential decrease. What's the beginning in the exponential in-
• Logarithm functions (Read handouts for Sec5.3-5.4!). Domain of logarithm
functions. Logarithm statement and exponential statement.
• Power law , product law and quotient law of logarithm functions. Solve
tions by trying to write the left side and right side of the equation into one
For example: log (x - 1)+1 = -log x. How to solve
? (Multiply log (x - 1)
on both sides first!) Don't forget to check your answer as the input for
must be positive numbers !
• Solving equations for a variable in an exponent . For example,
just write 4 as
22, we get
(with the same base!), then x = 2x + 4. If the base is different,
just take natural logarithm on both sides and apply the power law for
logarithm, we get x ln 2 = (x + 1) ln 3, which is a linear equation !
•Word Problems. These problems are always interesting but not so
the problems in your homework.
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