Ch.5 - 8 Vocabulary Flashcards Preview

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Flashcards in Ch.5 - 8 Vocabulary Deck (80)
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1

Polynomial in one variable

An expression of the form an x^n + an-1 x^n-1 + ... + a2 x^2 + a1 x + a0, where an is not 0, all the coefficients are real numbers, and n is a nonnegative integer.

2

Leading coefficient

The coefficient of the first term of a polynomial in standard form.

3

Polynomial function

A continuous function that can be described by a polynomial in one variable.

4

Power function

The simplest form of a polynomia function in the form of f(x)= ax^b, where a and b are nonzero real numbers.

5

Quartic function

A polynomial function with the degree of four.

6

Quintic function

A polynomial function with the degree of five.

7

End behavior

The behavior of the graph of f(x) as x approaches positive or negative infinity, determined by the degree and leading coefficient of the function.

8

Location principle

If the value of f(x) changes signs from one value of x to the next, then there is a zero between those two values.

9

Relative maximum

A point with no other nearby points that have a greater y-coordinante.

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Relative minimum

A point with no other nearby points that have a lesser y-coordinate.

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Extrema

The maximum and minimum values of a function.

12

Turning point

A point that is a relative maximum or minimum of the graph in which the graph changes direction.

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Prime polynomial

A polynomial that cannot be factored.

14

Quadratic form

au^2+bu+c, which a polynomial in x could be rewritten as.

15

Sum of two cubes (formula)

a^3 + b^3 = (a+b)(a^2-ab+b^2)

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Difference of two tubes (formula)

a^2-b^3 = (a-b) (a^2+ab+b^2)

17

Synthetic substitution

Applying the Remainder Theorem using synthetic division to evaluate a function.

18

Depressed polynomial

The quotient after dividing a polynomial by a binomial, which would have a degree one less than the original polynomial.

19

Remainder theorem

If a polynomial is divided by x-r, the remainder isa constant P(r).

20

Factor theorem

If the binomial x-r is a factor of the polynomial P(x) if and only if P(r)=0.

21

The fundamental theorem of algebra

Every polynomial equation with degree greater than zero has at least one root in the set of complex numbers.

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Corollary to the fundamental theorem of algebra

A polynomial equation of degree n has exactly n roots in the set of complex numbers, including repeated roots.

23

Descartes' rule of signs

The number of positive real zeros of P(x) is the same as the number of changes in sign of the coefficient of the terms, or is less than this by an even number.
The number of negative real zeros of P(x) is that in terms of P(-x).

24

Complex conjugates theorem

Let a and b be real numbers and b is not 0. If a+bi is a zero of a polynomial function with real coefficients, then a-bi is also a zero of the function.

25

Rational zero theorem

If P(x) is a polynomial function with integral coefficients, then every rational zero of P(x) = 0 is of the form p/q, a rational number in simplest form, where p is a factor of the constant term and q is a factor of the leading coefficient.

26

Composition of functions

The results of one function are used to evaluate a second function.

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Inverse relation

The set of ordered pair obtained by exchanging the coordinates of each ordered pair.

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Inverse function

The inverse function of f(x) is written as f-1(x), f(a) = b if and only if f-1(b) = a.

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Square root function

A function that contains the square root of a variable.

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Radical function

A function in which the variable is under the radical sign.sign.

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Square root inequality

An inequality involving square roots of a variable.

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Radicand

The number under the radical sign

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Index

For nth root (the inverse of raising a number to the nth power), the index is n.

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Primary root

The nonnegative root when there is more than one real root and the index is even.

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Rationalizing the denominator

Multiplying the numerator and denominator by a quantity so that the radicand of the denominator has an exact root to eliminate radicals from the denominator.

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Like radical expressions

Radicals with the same index and radicand.

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Conjugates

a√b + c √d and a√b - c √d are conjugates if a,b,c,d are all rational numbers.

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Radical equation

An equation that include radical expressions, can be solved by raising each side of the equation to a power.

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Extraneous solution

The result of an equation that does not satisfy the original equation.

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Radical inequality

An inequality that has a variable in the radicand.

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Exponentail function

A function where the base is a constant and the exponent is the independent variable.

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Exponential growth

A function of the form f(x)=b^x, where b > 1.

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Asymptote

A line that the graph of the function approaches.

44

Growth factor

The base of the exponential function, 1+r.

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Exponential decay

A function of the form f(x) = b^x, where 0

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Decay factor

The base of the exponential function, 1-r.

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Exponential equation

An equation in which variables occur as exponents.

48

Compound interest

Interest paid on the principal of an investment and any previously earned interest.

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Exponential inquality

An inequality involving exponential functions.

50

Logarithm

X=b^y, the variable y is the logarithm of x.

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Logarithmic function

A function in the form y=logbx, where b is not 1.

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Logarithmic equation

Contains one or more logarithms.

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Logarithmic inequality

An inequality that involves logarithms.

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Product property of logarithms

Logx(ab) = logx(a)+logx(b)

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Quotient property of logarithms

Logx(a/b) = logx(a)-logx(b)

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Power property of logarithms

Logb(m^p) = p logb(m)

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Common logarithms

Base 10 logarithms.

58

Change of base formula

Loga(n) = logb(n) / logb(a)

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Natural base

An irrational number with the value of 2.71828

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Natural base exponential function

An exponential function with base e.

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Natural logarithm

The inverse of a natural base exponential function, often abbreviated as ln.

62

Rate of continuous growth

Represented as k in an exponential growth function,
f(t) = ae^kt

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Rate of continuous decay

Represented as k in an exponential decay function,
f(t) = ae^-kt

64

Logistic growth model

Represents growth that has a limiting factor,
f(t) = c / (1+ae^-bt) with a,b,c all being positive constant and b

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Rational expression

A ratio of two polynomial expressions.

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Complex fraction

A rational expression with a numerator and/or denominator that is also a rational expression.

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Reciprocal function

A function with an equation of the form f(x) = 1/a(x), where a(x) is a linear function and does not equal to 0.

68

Hyperbola

The shape of the graph of a reciprocal function.

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Rational function

A function with an equation of the form f(x) = a(x)/b(x)m where a(x) and b(x) are both polynomial functions and b(x) does not equal to 0.

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Vertical asymptote

b(x) = 0.

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Horizontal asymptote

If the degree of a(x) is greater, no horizontal asymptote.
If the degree of a(x) is less, f(x) = 0.
If the degrees are equal, f(x) = leading coefficient of a(x) / leading coefficient of b(x).

72

Oblique asymptote

if the degree of a(x) minus that of b(x) is 1, f(x) = a(x) / b(x) with no remainder is the oblique asymptote.

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Direct variation

Can be expressed in the form y=kx.

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Constant of variation

The value k in the equation y=kx.

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Joint variation

When one quantity varies directly as the product of the two or more other quantities, y=kxz.

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Inverse variation

When the product of two quantities is equal to a constant k, xy=k, y=k/x.

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Combined variation

When one quantity varies directly and/or inversely as two or more other quantities.

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Rational equations

Equations that contain one or more rational expressions.

79

Weighted average

The method for finding the mean of a set of numbers in which some elements carry more importance.

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Rational inequalities

Inequalities that contain one more or rational expressions.