Glossary

Degree (of a polynomial)

Criticality: 3

The highest exponent of the variable in a polynomial function. It, along with the leading coefficient, determines the polynomial's end behavior pattern.

Example:

A polynomial modeling the volume of a cube with side length 'x' would be $V(x) = x^3$ , which has a degree of 3.

End behavior

Criticality: 3

Describes what happens to a function's output values ($y$-values) as the input values ($x$-values) approach positive or negative infinity. It illustrates the long-term trend of the function's graph.

Example:

When analyzing the flight path of a rocket, understanding its end behavior helps predict if it will continue to ascend indefinitely or eventually fall back to Earth.

Even Degree Polynomials

Criticality: 2

Polynomial functions where the highest exponent of the variable is an even number. Their end behaviors are the same on both the left and right sides (either both rise or both fall).

Example:

The trajectory of a ball thrown upwards, modeled by a quadratic function like $h(t) = -16t^2 + 64t$ , is an even degree polynomial whose graph opens downwards on both ends.

Leading coefficient

Criticality: 3

The numerical coefficient of the leading term in a polynomial function. Its sign is crucial for determining the direction (up or down) of the end behavior.

Example:

For the polynomial $P(x) = 2x^5 - 8x^3 + 1$ , the leading coefficient is 2, indicating the graph will rise to the right.

Leading term

Criticality: 3

The term in a polynomial function that has the highest degree. This single term dictates the overall end behavior of the entire polynomial.

Example:

In the function $f(x) = -7x^6 + 3x^4 - x + 10$ , the leading term is $-7x^6$ , which determines how the graph behaves at its far ends.

Limits (at infinity)

Criticality: 3

A mathematical concept used to describe the value a function approaches as its input variable tends towards positive or negative infinity. For polynomials, this describes their end behavior.

Example:

To describe the long-term behavior of a population model $P(t)$ , we might evaluate the limit as $t o \infty$ , written as $lim_{t o \infty} P(t)$ .

Negative Leading Coefficient

Criticality: 2

A characteristic where the numerical coefficient of the leading term is a negative number. This typically means the right side of the polynomial's graph will fall towards negative infinity.

Example:

A function modeling the temperature of a cooling object might be $T(t) = -0.5t^2 + 20t + 100$ , where the negative leading coefficient indicates the temperature will eventually decrease.

Odd Degree Polynomials

Criticality: 2

Polynomial functions where the highest exponent of the variable is an odd number. Their end behaviors are opposite on the left and right sides (one rises while the other falls).

Example:

A function describing the growth of a population that eventually stabilizes and then declines might be an odd degree polynomial, with one end rising and the other falling.

Polynomial functions

Criticality: 3

Functions that can be expressed as a sum of terms, where each term consists of a coefficient and a variable raised to a non-negative integer power. They are continuous and smooth, without breaks or sharp corners.

Example:

The cost of producing 'x' number of items might be modeled by a polynomial function like $C(x) = 0.5x^3 - 10x^2 + 500x + 1000$ , representing varying production efficiencies.

Positive Leading Coefficient

Criticality: 2

A characteristic where the numerical coefficient of the leading term is a positive number. This typically means the right side of the polynomial's graph will rise towards positive infinity.

Example:

If a company's profit function is $P(x) = 0.01x^4 - 50x + 1000$ , the positive leading coefficient of 0.01 suggests that profits will eventually increase significantly as production 'x' grows large.

Glossary

Degree (of a polynomial)

Criticality: 3

The highest exponent of the variable in a polynomial function. It, along with the leading coefficient, determines the polynomial's end behavior pattern.

Example:

A polynomial modeling the volume of a cube with side length 'x' would be $V(x) = x^3$ , which has a degree of 3.

End behavior

Criticality: 3

Describes what happens to a function's output values ($y$-values) as the input values ($x$-values) approach positive or negative infinity. It illustrates the long-term trend of the function's graph.

Example:

When analyzing the flight path of a rocket, understanding its end behavior helps predict if it will continue to ascend indefinitely or eventually fall back to Earth.

Even Degree Polynomials

Criticality: 2

Polynomial functions where the highest exponent of the variable is an even number. Their end behaviors are the same on both the left and right sides (either both rise or both fall).

Example:

The trajectory of a ball thrown upwards, modeled by a quadratic function like $h(t) = -16t^2 + 64t$ , is an even degree polynomial whose graph opens downwards on both ends.

Leading coefficient

Criticality: 3

The numerical coefficient of the leading term in a polynomial function. Its sign is crucial for determining the direction (up or down) of the end behavior.

Example:

For the polynomial $P(x) = 2x^5 - 8x^3 + 1$ , the leading coefficient is 2, indicating the graph will rise to the right.

Leading term

Criticality: 3

The term in a polynomial function that has the highest degree. This single term dictates the overall end behavior of the entire polynomial.

Example:

In the function $f(x) = -7x^6 + 3x^4 - x + 10$ , the leading term is $-7x^6$ , which determines how the graph behaves at its far ends.

Limits (at infinity)

Criticality: 3

A mathematical concept used to describe the value a function approaches as its input variable tends towards positive or negative infinity. For polynomials, this describes their end behavior.

Example:

To describe the long-term behavior of a population model $P(t)$ , we might evaluate the limit as $t o \infty$ , written as $lim_{t o \infty} P(t)$ .

Negative Leading Coefficient

Criticality: 2

A characteristic where the numerical coefficient of the leading term is a negative number. This typically means the right side of the polynomial's graph will fall towards negative infinity.

Example:

A function modeling the temperature of a cooling object might be $T(t) = -0.5t^2 + 20t + 100$ , where the negative leading coefficient indicates the temperature will eventually decrease.

Odd Degree Polynomials

Criticality: 2

Polynomial functions where the highest exponent of the variable is an odd number. Their end behaviors are opposite on the left and right sides (one rises while the other falls).

Example:

A function describing the growth of a population that eventually stabilizes and then declines might be an odd degree polynomial, with one end rising and the other falling.

Polynomial functions

Criticality: 3

Example:

The cost of producing 'x' number of items might be modeled by a polynomial function like $C(x) = 0.5x^3 - 10x^2 + 500x + 1000$ , representing varying production efficiencies.

Positive Leading Coefficient

Criticality: 2

A characteristic where the numerical coefficient of the leading term is a positive number. This typically means the right side of the polynomial's graph will rise towards positive infinity.

Example: