# 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.

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