- Model real-world phenomena with a variety of functions
- Represent and analyze relationships using tables, verbal rules, equations, and graphs
- Translate among tabular, symbolic, and graphical representations of functions
- Recognize that a variety of problem situations can be modeled by the same type of function
- Analyze the effects of parameter changes on the graphs of functions

*Benchmarks* 4F (The Physical Setting: Motion)

Grades 9-12, page 91

The change in motion of an object is proportional to the applied force
and inversely proportional to the mass.

*Benchmarks* 4G (The Physical Setting: Forces of Nature)

Grades 9-12, page 96

Gravitational force is an attraction between masses. The strength of
the force is proportional to the masses and weakens rapidly with increasing
distance between them.

*Benchmarks* 11D (Common Themes: Scale)

Grades 9-12, page 279

As the number of parts of a system grows in size, the number of possible
pair-wise interactions increases much more rapidly, roughly with the square
of the number of parts.

*Benchmarks* 9C (The Mathematical World: Shapes)

Grades 6-8, page 219

Graphs can show a variety of possible relationships between two variables.
As one variable increases uniformly, the other may do one of the following:
always keeps the same proportion to the first, increase or decrease steadily,
increase or decrease faster and faster, get closer and closer to some limiting
value, reach some intermediate maximum or minimum, alternately increase
and decrease indefinitely, increase or decrease in steps, or do something
different from any of these.

*Benchmarks* 11C (Common Themes: Constancy and Change)

Grades 9-12, page 275

Graphs and equations are useful (and often equivalent) ways for depicting
and analyzing patterns of change.

*Benchmarks* 9B (The Mathematical World: Symbolic Relationships)

Grades 9-12, page 221

Tables, graphs, and symbols are alternative ways of representing data
and relationships that can be translated from one to another.

*Benchmarks* 9B (The Mathematical World: Symbolic Relationships)

Grades 9-12, page 220

In some cases, the more of something there is, the more rapidly it
may change (as the number of births is proportional to the size of the
population). In other cases, the rate of change of something depends on
how much there is of something else (as the rate of change of speed is
proportional to the amount of force acting).

*Benchmarks* 4F (The Physical Setting: Motion)

Grades 9-12, page 91

All motion is relative to whatever frame of reference is chosen, for
there is no motionless frame from which to judge all motion.

*Benchmarks* 11D (Common Themes: Scale)

Grades 9-12, page 279

Representing large numbers in terms of powers of ten makes it easier
to think about them and to compare things that are greatly different.

*Benchmarks* 11D (Common Themes: Scale)

Grades 9-12, page 279

Because different properties are not affected to the same degree by
changes in scale, large changes in scale typically change the way that
things work in physical, biological, or social systems.

*Benchmarks* 11C (Common Themes: Constancy and Change)

Grades 9-12, page 275

In many physical, biological, and social systems, changes in one direction
tend to produce opposing (but somewhat delayed) influences, leading to
repetitive cycles of behavior.