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2 Functions

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  1. 2.1 Functions - SL content

    1. • Use gradient, intercepts and forms of a straight line. • Know parallel gradients m1=m2 and perpendicular gradients m1m2=-1.

    2. • Understand function, domain, range, graph and function notation. • Interpret functions as mathematical models. • Inverses undo functions, reflect in y=x and exist for one-to-one functions.

    3. • Draw/sketch graphs from equations, contexts or technology. • Label axes and key features, including sums and differences of functions.

    4. • Determine maxima, minima, intercepts, symmetry, vertex, roots/zeros and asymptotes. • Use technology to find intersections of curves or lines.

    5. • Use composite functions (f o g)(x)=f(g(x)). • Find inverse functions and use (f o f^-1)(x)=x for one-to-one functions.

    6. • Use standard, factorized and vertex forms of quadratic functions. • Interpret y-intercept, x-intercepts, axis of symmetry and vertex. • Convert between quadratic forms.

    7. • Solve quadratic equations and inequalities by factorization, completing the square and formula. • Use discriminant Delta=b^2-4ac to determine nature of roots.

    8. • Graph f(x)=1/x and rational functions of the form (ax+b)/(cx+d). • Include vertical/horizontal asymptotes and axis intercepts.

    9. • Graph exponential functions a^x and e^x. • Graph logarithmic functions log_a(x) and ln x. • Understand exponential and logarithmic functions as inverses.

    10. • Solve equations graphically and analytically. • Use technology when no suitable analytic method exists; apply to real contexts.

    11. • Apply translations, reflections and vertical/horizontal stretches. • Understand order of transformations and composite transformations. • At SL, transformations of f(ax+b) are not required.

  2. 2.2 Functions - AHL content

    1. AHL 2.12 (HL)—Polynomial functions

      • Work with polynomial graphs, equations, zeros, roots and factors. • Use factor and remainder theorems. • Use sum/product of roots of polynomial equations.

    2. AHL 2.13 (HL)—Further rational functions

      • Graph rational functions with quadratic numerator or denominator forms. • Include horizontal, vertical and oblique asymptotes plus intercepts.

    3. AHL 2.14 (HL)—Odd/even and inverse functions

      • Use even f(-x)=f(x) and odd f(-x)=-f(x) definitions, including periodic functions. • Find inverse functions, using domain restrictions where needed. • Recognize self-inverse functions.

    4. AHL 2.15 (HL)—Inequalities between functions

      • Solve g(x) >= f(x) graphically and analytically. • Use simple polynomial methods up to degree 3 and technology for wider cases.

    5. AHL 2.16 (HL)—Further graph transformations

      • Graph y=|f(x)|, y=f(|x|), y=1/f(x), y=f(ax+b) and y=[f(x)]^2. • Solve modulus equations and inequalities.