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Math 150, Fall 2008, Benjamin
Aurispa
Chapter 3, continued
3.4 Complex Numbers
This section involves the imaginary number i.
i=
√
−1
This means that i2 = −1
A complex number is an expression of the form a + bi, where a and b are real numbers. a is called the
real part and b is called the imaginary part.
If we have a complex number such as 3i (where a = 0), then the number is called a pure imaginary
number.
Also, recognize that every real number is also a complex number. For example, the number 4 is complex.
You can express it as 4 + 0i. In other words, the imaginary part is just 0.
To add/subtract complex numbers, add/subtract the real parts and add/subtract the imaginary parts.
Multiply complex numbers just like binomials (FOIL method), remembering that i2 = −1.
Examples
• i3
• i4
• i19
• (−6 + 6i) + (9 − i)
• (3 − 4i)(5 − 12i)
The complex conjugate of a complex number z = a + bi is z = a − bi.
Find the complex conjugates of the following.
−2 + 5i
3 − 6i
4i
Important Property: zz = (a + bi)(a − bi) = a2 + b2
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Math 150, Fall 2008, Benjamin
Aurispa
This property allows us to simplify a fraction with complex numbers by multiplying the numerator and
denominator by the conjugate of the denominator.
Example: Simplify and write in the form a + bi.
5−i
3 + 4i
If −r is a negative number, then the principal square root of −r is
√
√
−r = i r
√
(Note that there are two square roots of −r, ±i r, but only the positive one is the principal square root).
Examples
•
√
−25
•
√
−19
Note: When multiplying any radicals with negative numbers, you MUST first write them using i.
√
√
√
√
• ( 3 − −4)( 6 − −8)
We can now solve quadratic equations using the quadratic formula even if the discriminant is negative.
• 9x2 + 4 = 0
• x2 − 3x + 3 = 0
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Math 150, Fall 2008, Benjamin
Aurispa
3.6 Rational Functions
A rational function is a fraction where the numerator and denominator are both polynomials.
Since there are no radicals in a polynomial, the domain of a rational function is all x except for where the
denominator is 0.
Introductory Example: Consider r(x) = x1 . What happens as x → ∞, x → −∞, x → 0+ , x → 0− ?
Note:
x → a+ means “as x approaches a from the right” (from the positive direction).
x → a− means “as x approaches a from the left” (from the negative direction).
In this example, the line x = 0 is called a vertical asymptote and the line y = 0 is called a horizontal
asymptote.
1. The line x = a is a vertical asymptote of the function y = f (x) if y approaches ±∞ as x approaches
a from the right or left.
2. The line y = b is a horizontal asymptote of the function y = f (x) if y approaches b as x approaches
±∞.
Note that the asymptotes are not actually a part of the graph of f (x). They are just tools in telling what
the graph looks like.
Finding Asymptotes
Suppose r(x) =
an xn + . . . + a1 x + a0
g(x)
=
where the degree of g(x) is n and the degree of h(x) is m.
h(x)
bm xm + . . . + b1 x + b0
1. Vertical Asymptotes: A vertical asymptote exists wherever the denominator is 0 (after cancelling any
common factors from the numerator and denominator).
2. Horizontal Asymptotes:
(a) If n < m, then r(x) has horizontal asymptote y = 0.
an
(b) If n = m, then r(x) has horizontal asymptote y =
, the ratio of the leading coefficients.
bm
(c) If n > m, then there is no horizontal asymptote.
3. Holes: If a factor cancels from the numerator and denominator, there is a “hole” in the graph of the
function at this point, NOT a vertical asymptote.
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Math 150, Fall 2008, Benjamin
Aurispa
Examples: Describe any asymptotes and/or holes for the following rational functions.
1. r(x) =
4x2 + 1
(x + 7)(x2 − 6x + 8)
2. r(x) =
(3x − 2)(x2 + x − 12)
(x − 3)(2x + 5)
3. r(x) =
(x − 2)(3x + 4)
(4x − 1)(2x + 7)
Graphing Rational Functions
1. Factor the numerator and denominator. Cancel any common factors.
2. Determine the x and y intercepts.
(a) x-intercepts are where the function is 0. This occurs where the NUMERATOR is 0 (since the
denominator cannot be 0).
(b) y-intercepts occur where x = 0.
3. Determine the vertical and horizontal asymptotes.
4. Make a sign chart to determine the behavior of the graph near the asymptotes.
5. Sketch the graph. Sometimes it is useful to plot additional points to see what the graph looks like.
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Math 150, Fall 2008, Benjamin
Aurispa
Examples. Sketch the graphs of the following rational functions.
• r(x) =
x2
x−1
−x−6
This example shows that a function CAN CROSS ITS HORIZONTAL ASYMPTOTE.
• r(x) =
2x(x + 3)2
(x + 1)2 (x + 5)
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Math 150, Fall 2008, Benjamin
Aurispa
• r(x) =
(6 − x)(x2 − 4)
x2 − 2x
We know that if the degree of the numerator is larger than the degree of the denominator, then there
is no horizontal asymptote. However, if the degree of the numerator is exactly one more than the
degree of the denominator, we have a slant asymptote (or oblique asymptote), as in the example
above.
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