Topic 2: Functions

Concepts

• Essential understandings:

  • Models are depictions of real-life events using expressions, equations or graphs while a function is defined as a relation or expression involving one or more variable. Creating different representations of functions to model the relationships between variables, visually and symbolically as graphs, equations and/or tables represents different ways to communicate mathematical ideas.

• Suggested concepts embedded in this topic:

  • Representation, relationships, space, modelling, change.

  • AHL: Generalization, validity.

• Content-specific conceptual understandings:

  • Different representations of functions, symbolically and visually as graphs, equations and tables provide different ways to communicate mathematical relationships.

  • The parameters in a function or equation may correspond to notable geometrical features of a graph and can represent physical quantities in spatial dimensions.

  • Moving between different forms to represent functions allows for deeper understanding and provides different approaches to problem solving.

  • Our spatial frame of reference affects the visible part of a function and by changing this “window” can show more or less of the function to best suit our needs.

  • Changing the parameters of a trigonometric function changes the position, orientation and shape of the corresponding graph.

  • Different representations facilitate modelling and interpretation of physical, social, economic and mathematical phenomena, which support solving real-life problems.

  • Technology plays a key role in allowing humans to represent the real world as a model and to quantify the appropriateness of the model.

  • AHL: Extending results from a specific case to a general form and making connections between related functions allows us to better understand physical phenomena.

  • AHL: Generalization provides an insight into variation and allows us to access ideas such as half-life and scaling logarithmically to adapt theoretical models and solve complex real-life problems.

  • AHL: Considering the reasonableness and validity of results helps us to make informed, unbiased decisions.

SL Content

• SL 2.1

  • Different forms of the equation of a straight line. Gradient; intercepts.
• SL 2.2

  • Concept of a function, domain, range and graph.

  • Informal concept that an inverse function reverses or undoes the effect of a function.
• SL 2.3

  • Creating a sketch from information given or a context, including transferring a graph from screen to paper.

  • Using technology to graph functions including their sums and differences.

• SL 2.4

  • Determine key features of graphs.

  • Finding the point of intersection of two curves or lines using technology.

• SL 2.5

  • Modelling with the following functions:

  • Equation of a horizontal asymptote.
• SL 2.6

  • Modelling skills:

  • Use the modelling process described in the “mathematical modelling” section to create, fit and use the theoretical models in section SL2.5 and their graphs.

  • Develop and fit the model:

  • Given a context recognize and choose an appropriate model and possible parameters.

  • Determine a reasonable domain for a model.

  • Find the parameters of a model.

  • Test and reflect upon the model:

  • Comment on the appropriateness and reasonableness of a model.

  • Justify the choice of a particular model, based on the shape of the data, properties of the curve and/or on the context of the situation.

  • Use the model:

  • Reading, interpreting and making predictions based on the model.

AHL Content

• AHL 2.7

  • Composite functions in context.

  • Finding an inverse function.

• AHL 2.8

• • Transformations of graphs.

  • Composite transformations.

• AHL 2.9

  • In addition to the models covered in the SL content the AHL content extends this to include modelling with the following functions:

  • Exponential models to calculate half-life.

  • Piecewise models.

• AHL 2.10

  • Scaling very large or small numbers using logarithms.

  • Linearizing data using logarithms to determine if the data has an exponential or a power relationship using best-fit straight lines to determine parameters.

  • Interpretation of log-log and semi-log graphs.