Lecture Notes: Representations and Functions

A set of question-and-answer flashcards covering graphical, numerical, analytical, verbal representations and key function concepts including limits, behavior, and types of functions.

What is graphical representation?

A visual representation of data results using graphs, plots, and charts.

What is numerical representation?

Representation using numbers.

What is analytical representation?

Represented by an algebraic equation.

What is verbal representation?

A means of describing the relationship between inputs and outputs with words.

What is the rate of change?

Defines how one quantity changes in relation to another quantity.

What is a linear function?

Has one independent and one dependent variable; graph is a straight line; form f(x)=a + bx.

What is a quadratic function?

Form f(x)= a x^2 + b x + c; graph is a parabola.

What is a polynomial function?

An expression that may contain variables of varying degrees, coefficients, positive exponents, and constants.

What is a complex zero?

A complex number that makes the polynomial evaluate to zero.

What is end behavior?

How the function behaves as the input grows without bound or shrinks without bound.

What is a rational function?

The ratio of two polynomial functions where the denominator is not zero.

What is a vertical asymptote?

A vertical line where the function becomes unbounded (y tends to infinity or negative infinity) and does not touch or cross the curve.

What is a hole in a graph?

A point on the graph where the function is not defined.

What is a transformation?

A change to a graph's size or position, including translation, rotation, reflection, or dilation.

What is function modeling?

The process of creating a mathematical representation of a real-world scenario to predict or gain insight.

What is construction in functions?

Creating new functions using transformations, compositions, inverses, and regressions.

What is an assumption?

The conditions that must be met for the model to be valid.

What is application in functions?

The act of applying a function to an input from its domain to obtain the corresponding value from its range.

What is an inverse function?

A function that undoes another function; the domain and range swap places.

What is regression?

A tool used to model the relationship between two or more variables.

What is a conclusion?

The final decision or judgment reached after analyzing the results of a mathematical problem or experiment.

What is an inequality?

A statement indicating that two or more quantities are not equal, using symbols >, <, or ≠.

What is the composition of functions?

The function f(g(x)) formed by first applying g and then applying f.

What is the domain?

The set of input values; also called the independent variable.

What is the range?

The set of output values; also called the dependent variable.

What is an interval?

A set of real numbers between two endpoints, finite or infinite.

What is concave up?

A graph where the rate of change is increasing.

What is concave down?

A graph where the rate of change is decreasing.

What is a secant line?

The line through two points on a curve whose slope equals the average rate of change over the interval.

What is a relative maximum?

A point where the function changes from increasing to decreasing.

What is a relative minimum?

A point where the graph changes from decreasing to increasing.

What is a point of inflection?

A point where concavity changes (increasing to decreasing or vice versa) or the rate of change changes.

What is multiplicity in polynomials?

The exponent of each factor, or how many times a factor appears.

What is a conjugate?

The binomial formed by changing the sign between the two terms (e.g., a+b and a−b).

What is a piecewise function?

A function defined by different expressions on different subintervals of its domain.