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.

  • Lines with gradients $m_1$ and $m_2$

  • Parallel lines $m_1=m_2$ . Perpendicular lines $m_1 \times m_2=-1$

• SL 2.2

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

  • Function notation, for example $f(x), v(t), C(n)$ . The concept of a function as a mathematical model.

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

  • Inverse function as a reflection in the line $y=x$ , and the notation $f^{-1}(x)$

• SL 2.3

  • The graph of a function; its equation $y=f(x)$

  • 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:

  • Linear models. $f(x)=mx+c$

  • Quadratic models. $f(x)=ax^2+bx+c; a \neq 0$ . Axis of symmetry, vertex, zeros and roots, intercepts on the $x$-axis and $y$-axis.

  • Exponential growth and decay models. $f(x)=ka^x+c$ , $f(x)=ka^{-1}+c$ (for $a>0)$ , $f(x)=ke^{rx}+c$

  • Equation of a horizontal asymptote.

  • Direct/inverse variation: $f(x)=ax^n, n \in \mathbf{Z}$

  • The $y$-axis as a vertical asymptote when $n < 0$ .

  • Cubic models: $f(x)=ax^3+bx^2+cx+d$

  • Sinusoidal models: $f(x)=a \sin (bx) + d, f(x)=a \cos (bx) + d$

• 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.

  • The notation $(f \circ g)(x)=f(g(x))$

  • Inverse function $f^{-1}$ , including domain restriction.

  • Finding an inverse function.

• AHL 2.8

• • Transformations of graphs.

  • Translations: $y=f(x)+b; y=f(x-a)$

  • Reflections: in the $x$ -axis $y=-f(x)$ , and in the $y$ -axis $y=f(-x)$ .

  • Vertical stretch with scale factor $p: y=pf(x)$

  • Horizontal stretch with scale factor $\frac{1}{q}: y=f(qx)$

  • 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.

  • Natural logarithmic models: $f(x)=a+b \ln x$

  • Sinusoidal models: $f(x)=a \sin (b(x-c))+d$

  • Logistic models: $f(x)=\frac{L}{1+Ce^{-kx}}; L, C, k>0$

  • 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.