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 variables. Creating different representations of functions to model the relationships between variables, visually and symbolically as graphs, equations and tables represents different ways to communicate mathematical ideas.

• Suggested concepts embedded in this topic:

  • Representation, relationships, space, quantity, equivalence.

  • AHL: Systems, patterns.

• 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 correspond to 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.

  • Equivalent representations of quadratic functions can reveal different characteristics of the same relationship.

  • Functions represent mappings that assign to each value of the independent variable (input) one and only one dependent variable (output).

  • AHL: Extending results from a specific case to a general form can allow us to apply them to a larger system.

  • AHL: Patterns can be identified in behaviours which can give us insight into appropriate strategies to model or solve them.

  • AHL: The intersection of a system of equations may be represented graphically and algebraically and represents the solution that satisfies the equations.

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

  • Composite functions.

  • Identity function. Finding the inverse function $f^{-1}(x)$

• SL 2.6

  • The quadratic function $f(x)=ax^2+bx+c$ its graph, $y$-intercept $(0, c)$ . Axis of symmetry.

  • The form $f(x)=a(x-p)(x-q)$ , $x$-intercepts $(p, 0)$ and $(q, 0)$ .

  • The form $f(x)=a(x-h)^2+k$ , vertex $(h, k)$ .

• SL 2.7

• • Solution of quadratic equations and inequalities. The quadratic formula.

  • The discriminant $\bigtriangleup=b^2-4ac$ and the nature of the roots, that is, two distinct real roots, two equal real roots, no real roots.

• SL 2.8

  • The reciprocal function $f(x)=\frac{1}{x}, x \neq 0$ : its graph and self-inverse nature.

  • Rational functions of the form $f(x)=\frac{ax+b}{cx+d}$ and their graphs.

  • Equations of vertical and horizontal asymptotes.

• SL 2.9

  • Exponential functions and their graphs: $f(x)=a^x, a>0, f(x)=e^x$

  • Logarithmic functions and their graphs: $f(x)=log_a x, x>0, f(x)=\ln x, x>0$

• SL 2.10

  • Solving equations, both graphically and analytically.

  • Use of technology to solve a variety of equations, including those where there is no appropriate analytic approach.

  • Applications of graphing skills and solving equations that relate to real-life situations.

• Transformations of graphs. Translations: $y=f(x)+b; y=f(x-a)$

• Reflections (in both axes): $y: -f(x); y=f(-x)$

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

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

• Composite transformations.

AHL Content

• AHL 2.12

  • Polynomial functions, their graphs and equations; zeros, roots and factors.

  • The factor and remainder theorems.

  • Sum and product of the roots of polynomial equations.

• AHL 2.13

  • Rational functions of the form: $f(x)=\frac{ax+b}{cx^2+dx+e}$ and $f(x)=\frac{ax^2+bx+c}{dx+e}$

• AHL 2.14

  • Odd and even functions.

  • Finding inverse function, $f^{-1}(x)$ , including domain restriction.

  • Self-inverse functions.

• AHL 2.15

  • Solutions of $g(x) \geq f(x)$ , both graphically and analytically.

• AHL 2.16

  • The graphs of the functions, $y=|f(x)|$ and $y=f(|x|), y=\frac{1}{f(x)}, y=f(ax+b), y=[f(x)]^2$

  • Solution of modulus equations and inequalities.