### Standard 6: Functions

In grades 9-12, the mathematics curriculum should include the continued study of functions so that students can:
• Model real-world phenomena with a variety of functions
• Benchmarks 4F (The Physical Setting: Motion)
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)
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.

• Represent and analyze relationships using tables, verbal rules, equations, and graphs
• Benchmarks 11D (Common Themes: Scale)
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.

• Translate among tabular, symbolic, and graphical representations of functions
• Benchmarks 9C (The Mathematical World: Shapes)
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)
Graphs and equations are useful (and often equivalent) ways for depicting and analyzing patterns of change.

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

• Recognize that a variety of problem situations can be modeled by the same type of function

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• Analyze the effects of parameter changes on the graphs of functions
• Benchmarks 4F (The Physical Setting: Motion)
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)