lesson 6 video 4 Higher Order Polynomials and Piecewise Functions Flashcards

Vocabulary terms covering higher order polynomial characteristics, end behavior, extrema rules, and the definition and application of piecewise functions.

Linear Polynomial

A polynomial where the highest power of $x$ is 1 ($mx + b$), characterized by zero extrema and end behavior where one arm points up and one arm points down.

Quadratic Polynomial

A polynomial with a highest power of 2 ($ax^2 + bx + c$) that has exactly one extrema (a maximum or minimum) and both arms pointing in the same direction.

Cubic Polynomial

A polynomial with a highest power of 3 ($ax^3 + bx^2 + cx + d$) that has at most two extrema and end behavior with arms pointing in opposite directions.

Quartic Polynomial

A polynomial with a highest power of 4 ($ax^4 + bx^3 + cx^2 + d$) that has at most three extrema and both arms pointing in the same direction (both up or both down).

End behavior

The behavior of the 'arms' of a graph as it begins and ends; determined by whether the highest degree of the polynomial is even or odd.

Even power polynomial

A polynomial where the highest degree is even, resulting in end behavior where both arms point in the same direction (both up or both down).

Odd power polynomial

A polynomial where the highest degree is odd, resulting in end behavior where the arms point in opposite directions (one up and one down).

Extrema

The peaks and valleys (maximums and minimums) of a function; for any polynomial of degree $n$, there are at most $n-1$ extrema.

Piecewise Function

A function that is defined in pieces, using different formulas for different intervals of the input variable $x$.

Excelsior Electric membership corporation example

An application of a piecewise function used to calculate monthly charges based on kilowatt hours ($x$) used, where rates change at thresholds of $100$ and $500$ kilowatt hours.

Slope (in cost piecewise functions)

In the context of the Excelsior Electric example, the slope represents the specific cost per kilowatt hour within a given usage interval (e.g., $0.127$, $0.113$, or $0.119$).