Download Chapter 2 Summary

Chapter 2 Summary
§2.1
Polynomials
Let n be a nonnegative integer and an , an−1 , . . . , a2 , a1 , a0 be real numbers with an 6= 0. A polynomial is a function
of the form
p(x) = an xn + an−1 xn−1 + . . . + a2 x2 + a1 x + a0
The highest power of x that occurs is called the degree. Above, since an 6= 0, we say p(x) is a polynomial of degree
n.
The numbers an , an−1 , . . . , a2 , a1 , a0 are called the coefficients of the polynomial. an is called the leading coefficient.
We typically don’t write down terms corresponding to coefficients that equal 0 (unless we’re doing polynomial
division).
Quadratic Functions
A quadratic function is a polynomial of degree 2 (the convention is to name the coefficients a, b, c instead of a2 , a1 , a0 ):
q(x) = ax2 + bx + c
The graph of a quadratic function has the shape of a parabola, and is symmetric about a line called its axis of
symmetry. The point where this line intercepts the parabola is called the vertex.
Given a quadratic in the form q(x) = ax2 + bx + c, the axis of symmetry will be the line
x=−
b
2a
and the vertex will be at the point
b
b
− ,q −
2a
2a
If a (the leading coefficient) is positive, then the parabola will open upward and the vertex will be the lowest point
on the graph (i.e., it will be at the minimum value of the function).
If a is negative, then the parabola will open downward and the vertex will be the highest point on the graph (i.e., it
will be at the maximum value of the function).
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Standard Form
The standard form of a quadratic function is
q(x) = a(x − h)2 + k
(this is the same a as above). To put a quadratic in standard form we may have to do some algebra (such as
completing the square), but once we have it in standard form we can easily identify the axis and vertex and sketch a
graph the function: The axis of symmetry is the line x = h and the vertex is the point (h, k). (As before, the graph
opens upward if a > 0 and downward if a < 0.)
§2.2
Power Functions
A polynomial of the form
p(x) = xn
is called a power function. If n is even, then the shape of the graph will be similar to y = x2 . If n is odd, then the
shape of the graph will be similar to y = x3 .
Leading Term Test
The leading term of a polynomial, an xn , can tell us about the end behavior of a graph as x moves without bound to
the left or the right:
• If n is even and an > 0, then the end behavior of the graph will be similar to that of y = x2 .
• If n is even and an < 0, then the end behavior of the graph will be similar to that of y = −x2 .
• If n is odd and an > 0, then the end behavior of the graph will be similar to that of y = x3 .
• If n is odd and an < 0, then the end behavior of the graph will be similar to that of y = −x3 .
Zeros of Polynomials
Given a polynomial p(x), the following are equivalent:
• x = a is a zero of p(x).
• x = a is a root of p(x).
• x = a is a solution of the equation p(x) = 0.
• (x − a) is a factor of p(x).
• (a, 0) is an x-intercept of the graph of p(x).
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Repeated Roots
If k is a positive integer and (x − a)k is a factor of a polynomial, then we say that x = a is a repeated root of
multiplicity k.
• If the multiplicity is odd, then the graph crosses the x-axis at x = a.
• If the multiplicity is even, then the graph touches (but doesn’t cross) the x-axis at x = a.
Intermediate Value Theorem (IVT)
“Let [a, b] be an interval and let p(x) be a polynomial. Then for any number d between p(a) and p(b), there exists a
number c in the interval [a, b] such that p(c) = d.”
Note: The IVT is true for any “continuous” function on an interval [a, b], not just polynomials.
§2.3
The Division Algorithm
If p(x) and d(x) are polynomials such that d(x) 6= 0 (d(x) could be zero for some values of x, but it can’t be the
constant function y = 0), and the degree of d is less than or equal to the degree of p, then there exist unique
polynomials q(x) and r(x) such that
p(x) = d(x)q(x) + r(x)
where the degree of r is strictly less than the degree of d. If r = 0 (the constant function), then we sat that d divides
[evenly into] p.
[Note: “q” is used a function name in this theorem because we’re talking about quotients (which may or may not be
quadratics).]
The Remainder Theorem
The remainder when a polynomial p(x) is divided by the linear factor x − k is
r = p(k)
The Factor Theorem
(x − k) is a factor of a polynomial p(x) if and only if p(k) = 0.
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§2.4
Complex Numbers, C
Let a and b be real numbers, and let i =
√
−1 (so i2 = −1. A complex number is a number of the form
z = a + bi
a is called the real part of z, and bi is called the imaginary part of z. The set of all complex numbers is denoted by
C (in older literature you may see =).
Conjugates
Let z = a + bi be a complex number. The conjugate of z is
z̄ = a − bi
Operations on Complex Numbers
Given two complex numbers z = a + bi and w = c + di, then we can combine them in the following ways:
Sum: When adding complex numbers, we simply add the real parts together and add the imaginary parts together:
z + w = (a + bi) + (c + di)
= (a + b) + (c + d)i
Difference: When subtracting complex numbers, we take the difference of the real parts and the difference of the
imaginary parts:
z − w = (a + bi) − (c + di)
= a + bi − c − di
= (a − b) + (c − d)i
Product: When multiplying complex numbers, we expand/FOIL and use the fact that i2 = −1:
z · w = (a + bi) · (c + di)
= ac + adi + bci + bdi2
= ac + adi + bci + bd(−1)
= (ac − bd) + (ad + bc)i
Note: If we multiply a complex number by its conjugate, we get:
z · z̄ = (a + bi) · (a − bi)
= a2 − abi + bai − b2 i2
= a2 − abi + abi − b2 (−1)
= a2 + b2
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Quotient: Dividing two complex numbers is trickier. To find
w:
z
w,
we must multiply and divide by the conjugate of
z
z · w̄
=
w
w · w̄
(a + bi)(c + di)
=
(c + di)(c − di)
(ac − bd) + (ad + bc)i
=
c2 + b2
ac − bd ad + bc
= 2
+ 2
i
c + b2
c + b2
Complex Numbers and Polynomials
Complex roots always come in conjugate pairs: Let p(x) be a polynomial with real coefficients. If z = a + bi is a
complex root of p, then so is its conjugate, z̄ = a − bi.
§2.5 [skipping]
§2.6
Rational Functions
A rational function is the quotient of two polynomials:
f (x) =
an xn + an−1 xn−1 + · · · + a1 x + a0
N (x)
=
D(x)
bm xm + bm−1 xm−1 + · · · + b1 x + b0
The domain of a rational function is the set of all real numbers except those for which D(x) = 0, i.e.,
Df = {x ∈ R | D(x) 6= 0}
Vertical Asymptotes
The vertical line
x=a
is a vertical asymptote of f if and only if
f (x) → ±∞
as x → a from either the left or the right (or both).
If f is a rational function with no common factors (i.e., we’ve canceled as many factors as possible from the
numerator and the denominator), then f will have vertical asymptotes at the x’s that make D(x) = 0.
Note: The graph of a function (rational or otherwise) cannot cross or touch a vertical asymptote!
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Horizontal Asymptotes
The horizontal line
y=b
is a horizontal asymptote of f if and only if
f (x) → b
as x → ∞ or x → −∞ (or both).
If
f (x) =
N (x)
an xn + an−1 xn−1 + · · · + a1 x + a0
=
D(x)
bm xm + bm−1 xm−1 + · · · + b1 x + b0
is a rational function, then we find the horizontal asymptote by comparing the degrees of N (x) and D(x) as follows:
• If the degree of the numerator is greater than the degree of the denominator (n > m), then we have no
horizontal asymptote.
• If the degree of the numerator is less than the degree of the denominator (n < m), then we have a horizontal
asymptote of y = 0.
• If the degree of the numerator is equal to the degree of the denominator (n = m), then we have a horizontal
asymptote of y = abnn (i.e., take the ratio of the leading coefficients).
Note: The graph of a function (rational or otherwise) can cross or touch a horizontal asymptote.
Graphing a Rational Function f (x) =
N (x)
D(x)
1. Simplify f , if possible (factor as much as possible and cancel any common factors).
2. Find and plot the y-intercept by evaluating f (0) (it’s possible to have no y-intercept).
3. Find and plot any x-intercepts by solving N (x) = 0.
4. Find and sketch any vertical asymptotes by solving D(x) = 0.
5. Find and sketch the horizontal asymptote (if there is one) as described above.
6. Plot at least one point between/beyond each of the x-intercepts and vertical asymptotes you’ve already sketched
(this extra information allows us to more accurately fill in the rest of the graph).
7. Fill in the rest of the graph with smooth curves. Show asymptotic behavior at any vertical or horizontal
asymptotes, and remember to not cross or touch vertical asymptotes.
A note about common factors:
When canceling common factors of N and D, we may loose some information about the graph. If (x − a) is a factor
of both N and D, then there will be either a hole in the graph at x = a or a vertical asymptote at x = a (we get an
asymptote if the multiplicity of the factor is greater in the denominator than it is in the numerator).
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