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 m1m_1 and m2m_2

    • Parallel lines m1=m2m_1=m_2 . Perpendicular lines m1×m2=1m_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)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=xy=x , and the notation f1(x)f^{-1}(x)

  • SL 2.3

    • The graph of a function; its equation y=f(x)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+cf(x)=mx+c

    • Quadratic models. f(x)=ax2+bx+c;a0f(x)=ax^2+bx+c; a \neq 0 . Axis of symmetry, vertex, zeros and roots, intercepts on the xx-axis and yy-axis.

    • Exponential growth and decay models. f(x)=kax+cf(x)=ka^x+c , f(x)=ka1+cf(x)=ka^{-1}+c (for a>0)a>0) , f(x)=kerx+cf(x)=ke^{rx}+c

    • Equation of a horizontal asymptote.

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

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

    • Cubic models: f(x)=ax3+bx2+cx+df(x)=ax^3+bx^2+cx+d

    • Sinusoidal models: f(x)=asin(bx)+d,f(x)=acos(bx)+df(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 (fg)(x)=f(g(x))(f \circ g)(x)=f(g(x))

    • Inverse function f1f^{-1} , including domain restriction.

    • Finding an inverse function.

  • AHL 2.8

    • Transformations of graphs.

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

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

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

    • Horizontal stretch with scale factor 1q:y=f(qx)\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+blnxf(x)=a+b \ln x

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

    • Logistic models: f(x)=L1+Cekx;L,C,k>0f(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.

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