HPC Unit 1B Lecture Notes Review: Rational Functions, Transformations, Binomial Expansion, and Modeling

Flashcards covering key vocabulary from HPC Unit 1B lecture notes, including rational functions, their characteristics (zeros, asymptotes, holes, end behavior), function transformations (translations, reflections, dilations), binomial expansion, and mathematical modeling techniques.

Rational Function

A function that can be expressed as the ratio of two polynomials, f(x) = P(x)/Q(x), where Q(x) is not the zero polynomial.

Real Zeros of Rational Functions

The x-values where a rational function equals zero, which occur when the numerator is zero and the x-value is within the function's domain.

Domain of a Rational Function

All real numbers except those that make the denominator of the function equal to zero, as these values would make the function undefined.

End Behavior (Rational Functions)

Describes how the graph of a rational function behaves as the input x approaches positive or negative infinity.

Holes (Rational Functions)

Removable discontinuities in the graph of a rational function that occur when a common factor exists in both the numerator and the denominator, which cancels out when simplifying.

Vertical Asymptote

A vertical line (x=a) that the graph of a rational function approaches but never crosses, occurring at x-values where the (simplified) denominator is zero.

Limit Notation

A mathematical notation used to describe the behavior of a function as its input approaches a certain value or infinity, often used for asymptotes and holes.

Multiplicity (in Rational Function denominators)

The number of times a factor appears in the denominator of a rational function, which can affect the behavior of the graph around a vertical asymptote.

Slant Asymptote (Oblique Asymptote)

An asymptote that is a non-horizontal, non-vertical line, occurring in a rational function when the degree of the numerator is exactly one greater than the degree of the denominator.

Horizontal Asymptote

A horizontal line that the graph of a rational function approaches as x approaches positive or negative infinity. Determined by comparing the degrees of the numerator and denominator.

Translation (Functions)

A type of transformation that shifts the graph of a function horizontally (input change) or vertically (output change) without changing its shape or orientation.

Reflection (Functions)

A type of transformation that flips a function's graph across an axis, either the x-axis (output change) or the y-axis (input change).

Dilation (Functions)

A type of transformation that stretches or compresses a function's graph either horizontally (input change) or vertically (output change).

Input Changes (Function Transformations)

Transformations that affect the x-values of a function, such as horizontal shifts, reflections across the y-axis, or horizontal dilations (e.g., f(x+c), f(-x), f(cx)).

Output Changes (Function Transformations)

Transformations that affect the y-values of a function, such as vertical shifts, reflections across the x-axis, or vertical dilations (e.g., f(x)+c, -f(x), cf(x)).

Factored Form (Polynomials)

A way of writing a polynomial as a product of its factors (e.g., f(x) = (x+2)^2(x-3)).

Standard (or General) Form (Polynomials)

A way of writing a polynomial by collecting and combining like terms and arranging them in descending order of degree (e.g., f(x) = x^3 + x^2 - 8x - 12).

Binomial Expansion

The process of expanding a binomial (an algebraic expression with two terms) raised to an integer power, such as (a+b)^n.

Pascal's Triangle

A triangular array of numbers that serves as a useful tool for finding the coefficients in binomial expansions.

Binomial Theorem

A formula that provides a systematic way to expand any binomial raised to a non-negative integer power, (a+b)^n = Σ [nCk * a^(n-k) * b^k].

Mathematical Model

A mathematical representation of a real-world situation, phenomenon, or data set, used to understand, analyze, or predict outcomes.

Model Domain Restrictions

Limitations on the input values (domain) of a function due to the real-world context of a mathematical model, ensuring the model remains realistic.

Model Range Restrictions

Limitations on the output values (range) of a function due to the real-world context of a mathematical model, ensuring the output makes sense within the application.

Average Rate of Change (AROC)

The change in the output of a function divided by the change in its input over a specific interval, representing the slope of the secant line between two points.

Regression Analysis

A powerful statistical technique used to find a function (model) that best fits a set of data points, often by minimizing the distance between the data and the curve.

Inversely Proportional

A relationship between two quantities where an increase in one quantity results in a proportional decrease in the other, typically expressed as y = k/x^n for some constant k and power n.

Rational Expression

An algebraic expression that can be written as a fraction where both the numerator and the denominator are polynomials.

Lowest Terms (Rational Expressions)

A rational expression is in lowest terms when the numerator and denominator have no common factors other than 1.

Excluded Values (Rational Expressions)

Values of the variable that would make the denominator of a rational expression equal to zero, rendering the expression undefined and thus excluded from the domain.