Glossary

Concavity

Criticality: 2

Describes the direction in which the graph of a function opens; it can be concave up (like a cup) or concave down (like a frown).

Example:

A graph showing accelerating growth would exhibit upward concavity, while a graph showing decelerating growth would show downward concavity.

Constant function

Criticality: 1

A polynomial function where the highest power of the variable is zero, resulting in a horizontal line graph.

Example:

If the temperature in a perfectly insulated room is a constant function $T(t) = 22^\circ C$ , it means the temperature remains 22 degrees Celsius over time.

Critical points

Criticality: 3

Points in the domain of a function where the derivative is zero or undefined, often corresponding to local maxima or minima.

Example:

To find the exact time a ball reaches its maximum height, you would calculate the critical points of its height function.

Degree (of a polynomial)

Criticality: 3

The highest power of the variable in a polynomial function.

Example:

For the polynomial $P(x) = 7x^5 - 3x^2 + 1$ , the degree is 5.

Even degree (polynomials)

Criticality: 2

Polynomial functions where the highest power of the variable is an even number, causing their end behavior to go in the same direction.

Example:

A quadratic function like $f(x) = x^2$ is an even degree polynomial, and both ends of its graph point upwards.

Global maximum

Criticality: 2

The absolute highest point of the entire graph of a function over its entire domain.

Example:

The highest point on Mount Everest is the global maximum elevation on Earth.

Global minimum

Criticality: 2

The absolute lowest point of the entire graph of a function over its entire domain.

Example:

The deepest point in the Mariana Trench represents the global minimum elevation on Earth.

Increasing/Decreasing intervals

Criticality: 3

Intervals on the x-axis where the function's y-values are consistently rising (increasing) or falling (decreasing) as x increases.

Example:

If a company's stock price graph shows an increasing interval from 9 AM to 1 PM, it means the stock value was rising during those hours.

Inflection points

Criticality: 2

Points on the graph of a function where the concavity changes, transitioning from concave up to concave down or vice versa.

Example:

On a graph showing the spread of a rumor, an inflection point might indicate when the rate of new people hearing the rumor starts to slow down.

Leading coefficient

Criticality: 3

The coefficient of the term with the highest power of the variable in a polynomial function.

Example:

In the polynomial $g(x) = -0.5x^4 + 2x^3 - 7$ , the leading coefficient is -0.5, which tells us the graph opens downwards.

Local maximum

Criticality: 3

A point on the graph of a function where the function's value is greater than or equal to the values at all nearby points within a specific interval.

Example:

On a graph representing a day's temperature, the peak temperature reached in the afternoon is a local maximum.

Local minimum

Criticality: 3

A point on the graph of a function where the function's value is less than or equal to the values at all nearby points within a specific interval.

Example:

The lowest point a roller coaster reaches in a dip before climbing again is a local minimum.

Polynomial function

Criticality: 3

A function expressed as a sum of terms, where each term is a constant multiplied by a variable raised to a non-negative integer power.

Example:

The trajectory of a rocket launch can be modeled by a polynomial function like $h(t) = -4.9t^2 + 100t + 10$ , where $t$ is time and $h(t)$ is height.

Zeros (of a polynomial function)

Criticality: 3

The x-values where a polynomial function crosses the x-axis, meaning the function's output is zero ($f(x) = 0$).

Example:

If a company's profit is modeled by $P(x)$ , the zeros of $P(x)$ would indicate the break-even points where profit is zero.

Glossary

Concavity

Criticality: 2

Describes the direction in which the graph of a function opens; it can be concave up (like a cup) or concave down (like a frown).

Example:

A graph showing accelerating growth would exhibit upward concavity, while a graph showing decelerating growth would show downward concavity.

Constant function

Criticality: 1

A polynomial function where the highest power of the variable is zero, resulting in a horizontal line graph.

Example:

If the temperature in a perfectly insulated room is a constant function $T(t) = 22^\circ C$ , it means the temperature remains 22 degrees Celsius over time.

Critical points

Criticality: 3

Points in the domain of a function where the derivative is zero or undefined, often corresponding to local maxima or minima.

Example:

To find the exact time a ball reaches its maximum height, you would calculate the critical points of its height function.

Degree (of a polynomial)

Criticality: 3

The highest power of the variable in a polynomial function.

Example:

For the polynomial $P(x) = 7x^5 - 3x^2 + 1$ , the degree is 5.

Even degree (polynomials)

Criticality: 2

Polynomial functions where the highest power of the variable is an even number, causing their end behavior to go in the same direction.

Example:

A quadratic function like $f(x) = x^2$ is an even degree polynomial, and both ends of its graph point upwards.

Global maximum

Criticality: 2

The absolute highest point of the entire graph of a function over its entire domain.

Example:

The highest point on Mount Everest is the global maximum elevation on Earth.

Global minimum

Criticality: 2

The absolute lowest point of the entire graph of a function over its entire domain.

Example:

The deepest point in the Mariana Trench represents the global minimum elevation on Earth.

Increasing/Decreasing intervals

Criticality: 3

Intervals on the x-axis where the function's y-values are consistently rising (increasing) or falling (decreasing) as x increases.

Example:

If a company's stock price graph shows an increasing interval from 9 AM to 1 PM, it means the stock value was rising during those hours.

Inflection points

Criticality: 2

Points on the graph of a function where the concavity changes, transitioning from concave up to concave down or vice versa.

Example:

On a graph showing the spread of a rumor, an inflection point might indicate when the rate of new people hearing the rumor starts to slow down.

Leading coefficient

Criticality: 3

The coefficient of the term with the highest power of the variable in a polynomial function.

Example:

In the polynomial $g(x) = -0.5x^4 + 2x^3 - 7$ , the leading coefficient is -0.5, which tells us the graph opens downwards.

Local maximum

Criticality: 3

A point on the graph of a function where the function's value is greater than or equal to the values at all nearby points within a specific interval.

Example:

On a graph representing a day's temperature, the peak temperature reached in the afternoon is a local maximum.

Local minimum

Criticality: 3

A point on the graph of a function where the function's value is less than or equal to the values at all nearby points within a specific interval.

Example:

The lowest point a roller coaster reaches in a dip before climbing again is a local minimum.

Polynomial function

Criticality: 3

A function expressed as a sum of terms, where each term is a constant multiplied by a variable raised to a non-negative integer power.

Example:

The trajectory of a rocket launch can be modeled by a polynomial function like $h(t) = -4.9t^2 + 100t + 10$ , where $t$ is time and $h(t)$ is height.

Zeros (of a polynomial function)

Criticality: 3

The x-values where a polynomial function crosses the x-axis, meaning the function's output is zero ($f(x) = 0$).

Example:

If a company's profit is modeled by $P(x)$ , the zeros of $P(x)$ would indicate the break-even points where profit is zero.