Flashcards in Ch.5 - 8 Vocabulary Deck (80)
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.
The coefficient of the first term of a polynomial in standard form.
A continuous function that can be described by a polynomial in one variable.
The simplest form of a polynomia function in the form of f(x)= ax^b, where a and b are nonzero real numbers.
A polynomial function with the degree of four.
A polynomial function with the degree of five.
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.
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.
A point with no other nearby points that have a greater y-coordinante.
A point with no other nearby points that have a lesser y-coordinate.
The maximum and minimum values of a function.
A point that is a relative maximum or minimum of the graph in which the graph changes direction.
A polynomial that cannot be factored.
au^2+bu+c, which a polynomial in x could be rewritten as.
Sum of two cubes (formula)
a^3 + b^3 = (a+b)(a^2-ab+b^2)
Difference of two tubes (formula)
a^2-b^3 = (a-b) (a^2+ab+b^2)
Applying the Remainder Theorem using synthetic division to evaluate a function.
The quotient after dividing a polynomial by a binomial, which would have a degree one less than the original polynomial.
If a polynomial is divided by x-r, the remainder isa constant P(r).
If the binomial x-r is a factor of the polynomial P(x) if and only if P(r)=0.
The fundamental theorem of algebra
Every polynomial equation with degree greater than zero has at least one root in the set of complex numbers.
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.
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).
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.
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.
Composition of functions
The results of one function are used to evaluate a second function.
The set of ordered pair obtained by exchanging the coordinates of each ordered pair.
The inverse function of f(x) is written as f-1(x), f(a) = b if and only if f-1(b) = a.
Square root function
A function that contains the square root of a variable.
A function in which the variable is under the radical sign.sign.
Square root inequality
An inequality involving square roots of a variable.
The number under the radical sign
For nth root (the inverse of raising a number to the nth power), the index is n.
The nonnegative root when there is more than one real root and the index is even.
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.
Like radical expressions
Radicals with the same index and radicand.
a√b + c √d and a√b - c √d are conjugates if a,b,c,d are all rational numbers.
An equation that include radical expressions, can be solved by raising each side of the equation to a power.
The result of an equation that does not satisfy the original equation.
An inequality that has a variable in the radicand.
A function where the base is a constant and the exponent is the independent variable.
A function of the form f(x)=b^x, where b > 1.
A line that the graph of the function approaches.
The base of the exponential function, 1+r.
A function of the form f(x) = b^x, where 0
The base of the exponential function, 1-r.
An equation in which variables occur as exponents.
Interest paid on the principal of an investment and any previously earned interest.
An inequality involving exponential functions.
X=b^y, the variable y is the logarithm of x.
A function in the form y=logbx, where b is not 1.
Contains one or more logarithms.
An inequality that involves logarithms.
Product property of logarithms
Logx(ab) = logx(a)+logx(b)
Quotient property of logarithms
Logx(a/b) = logx(a)-logx(b)
Power property of logarithms
Logb(m^p) = p logb(m)
Base 10 logarithms.
Change of base formula
Loga(n) = logb(n) / logb(a)
An irrational number with the value of 2.71828
Natural base exponential function
An exponential function with base e.
The inverse of a natural base exponential function, often abbreviated as ln.
Rate of continuous growth
Represented as k in an exponential growth function,
f(t) = ae^kt
Rate of continuous decay
Represented as k in an exponential decay function,
f(t) = ae^-kt
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
A ratio of two polynomial expressions.
A rational expression with a numerator and/or denominator that is also a rational expression.
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.
The shape of the graph of a reciprocal 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.
b(x) = 0.
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).
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.
Can be expressed in the form y=kx.
Constant of variation
The value k in the equation y=kx.
When one quantity varies directly as the product of the two or more other quantities, y=kxz.
When the product of two quantities is equal to a constant k, xy=k, y=k/x.
When one quantity varies directly and/or inversely as two or more other quantities.
Equations that contain one or more rational expressions.
The method for finding the mean of a set of numbers in which some elements carry more importance.