IBDP Maths Applications and Interpretation: IB Style Question Bank HL Paper-2

Topic 2: Functions– SL content

• Topic: SL 2.1
  
  • Different forms of the equation of a straight line.
  • Gradient; intercepts.
  • Lines with gradients m₁ and m₂
  • Parallel lines m₁ = m₂.
  • Perpendicular lines m₁ × m₂ = − 1.
• Topic: 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).
• Topic: SL 2.3
  
  • The graph of a function; its equation \(y = f\left( x \right)\) .
  • 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.
• Topic: SL 2.4
  
  • Determine key features of graphs.
  • Finding the point of intersection of two curves or lines using technology
• Topic: SL 2.5
  
  • Modelling with the following functions:
  • Linear models.
    
    • f(x)=mx+c.
  • Quadratic models.
    
    • f(x)=ax²+bx+c ; a≠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^-x+c (for a>0)
    • f(x)=ke^rx+c
  • Equation of a horizontal asymptote.
  • Direct/inverse variation:
    
    • f(x)=axⁿ, n∈ℤ
    • The y-axis as a vertical asymptote when n<0.
  • Cubic models:
    
    • f(x)=ax³+bx²+cx+d.
  • Sinusoidal models:
    
    • f(x)=asin(bx)+d,
    • f(x)=acos(bx)+d.
• Topic: SL 2.5
  
  • Composite functions.
    
    • (f ∘ g)(x) = f(g(x))
  • Identity function.
  • Finding the inverse function f⁻¹(x)
    
    • (f ∘ f⁻¹)(x) = (f⁻¹∘ f)(x) = x
• Topic: 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
• Topic: SL 2.7
  
  • Solution of quadratic equations and inequalities. The quadratic formula.
  • The discriminant \(\Delta = {b^2} – 4ac\) and the nature of the roots, that is, two distinct real roots, two equal real roots, no real roots.
• Topic: SL 2.8
  
  • The reciprocal function f(x) = 1 x , x ≠ 0: its graph and self-inverse nature.
  • The rational function \(f(x)=\frac{{ax + b}}{{cx + d}}\)and its graph. Equations of vertical and horizontal asymptotes.

Topic 2: Functions– AHL content

• Topic: AHL 2.7
  
  • Composite functions in context.
  • The notation (f∘g)(x)=f(g(x)).
  • Inverse function f^-1, including domain restriction.
  • Finding an inverse function.
• Topic : AHL 2.8
  
  • 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= p f(x).
    • Horizontal stretch with scale factor \(\frac{1}{q}\): y = f(qx).
  • Composite transformations.
• Topic : 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+blnx
  • Sinusoidal models:
    
    • f(x)=asin(b(x-c))+d
  • Logistic models:
    
    • \(f(x)=\frac{L}{1+Ce^{-kx}};L,C,k>0\)
  • Piecewise models.
• Topic: 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.