538 I Chapter 6 Systems of Equations
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" PR DJ E 0 T5
1. FINDING ZEROS OF A POLYNOMIAL One zero of c. What is the slope of the line betweenO and P?
P(x) = x3 + 2x2 + Cx 6 is the sum of the other two . , . _ .
zeros of Pa). Find C and the three zeros of
Conic Sections
Standard Forms of the Equation of a Parabola
Equation
Vertex
Focus
Directrix
Axis of Symmetry
Opens
x 2 = 4 py y 2 = 4 px ( x - h) 2 = 4 p ( y - k ) ( y - k ) 2 = 4 p ( x - h)
(0, 0) (0, 0) (h, k )
(0, p) ( p, 0) (h, k + p )
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6.1: Conic Sections
PARABOLA
A parabola is the set of points in the plane that are equidistant from a fixed line (the directrix) and a fixed point (the focus) not on the directrix.
The midpoint of the line segment between the focus and the d
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3.6: Exponential Growth and Decay
The Compound Interest Formula nt A = P 1 + r n
where
A = balance P = principal r = interest rate (expressed as a decimal) t = time n = the number of times compounded
Continuous Compounding Interest Formula
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3.5: Exponential and Logarithmic Equations
SOLVE EXPONENTIAL EQUATIONS
Equality of Exponents Theorem
x y If b = b , then x = y, provided that b > 0 and b 1.
Example: Solve for x algebraically.
3
x - 2
= 81
Solution:
3
x - 2 x
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3.4: Logarithms and Logarithmic Scales
Properties of Logarithms In the following properties, b, M, and N are positive real numbers (b 1). Product Property logb (MN) = logb M + logb N Quoti
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3.3: Logarithmic Functions
If x > 0 and b is a positive constant (b 1), then y = logb x if and only if by = x.
The exponential form of y = logb x is by = x. The logarithmic form of by =
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3.2: Exponential Functions
The exponential function with base b is defined by f(x) = bx where b > 0, b 1, and x is a real number. 1. Evaluate f(x) = 3 x at x = 4, x = -3, a
3.3: Zeros of Polynomial Functions
A polynomial function P of degree n has at most n zeros, where each zero of multiplicity k is counted k times.
The Rational Zero Theorem
If P(x) = anx n + an-1x n-1 + . . . + a1x + a0 has integer coefficients (an 0
2.5 - Transformations of Graphs
Every College Algebra student must be able to identify the following six parent graphs.
f(x) = x
y
f(x) = x2
y
x
x
Domain = _ Range = _
Domain = __ Range = _
f(x) = x
3
y
f(x) = |x|
y
x
x
Domain = _ Range =
3.1: The Remainder Theorem and Factor Theorem
Consider the polynomial function P(x)
= x3 - 7x - 6. Notice,
P(-1) = (-1)3 - 7(-1) - 6 = -1 + 7 - 6 =0
So, -1 is called a zero of the function P. We will be interested in finding the zeros of many polyn
Steps To Graph Rational Functions
1. Make sure the numerator and denominator of the function are arranged in the correct descending order of power. 2. Find the Domain a. Factor the denominator of the function completely. b. Find the real zeros of the
Steps To Graph Polynomial Functions
1. Make sure the function is arranged in the correct descending order of power.
f(x) = anx + an-1x
n
n-1
+ . . . + a1x + a0 , where the leading coefficient an 0
2. Determine the far-left and far-right behavio
2.4 - Graphing Quadratic Functions
A quadratic function of x is of the form
f(x) = ax2 + bx + c
where a, b, and c are real numbers and a 0. This equation is called the expanded form of the function, and its graph is called a parabola.
y axis of sym
3.2: Polynomial Functions of Higher Degree
So far, we have only graphed two kinds of polynomial functions this semester: lines and parabolas. In this section we will graph polynomials of degree 3 or higher. All polynomial functions have graphs that a
2.3 - Linear Functions
Definition of a Linear Function A linear function of x is of the form
f(x) = mx + b, m 0
where m and b are real numbers.
The slope of the line is the steepness, or rate of change between any two points on the line.
y y2 P2
P1 - Summary of Interval and Set Notation
If a and b are real numbers such that a < b, then
Interval Notation Type of Interval Set-builder Notation Graph
(a, b) [a, b] (a, b] [a, b) (b, ) (-, a) [b, ) (-, a] (-, )
open interval
{x|a<x<b} {x|axb} {
3.5: Graphs of Rational Functions
Definition: If P(x) and Q(x) are polynomials, then the function F given by F(x) = P(x) Q(x)
is called a rational function. The domain of F is the set of all real numbers except those for which Q(x) = 0.
Example:
F
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MATH 2412: Algebra Review Packet
oe oe oe oe This packet is worth 10% of your overall course grade. Show all intermediate steps (on these pages) to receive full credit. Circle or box your final answer, when appropriate. This packet is due on
A Quick Guide to Cancellation
Students are often required to simplify, or reduce, rational expressions in Algebra. When simplifying such expressions, remember that you are only allowed to cancel common factors in a fraction and not common terms. See