Working with Functions

Scheme of work: GCSE Higher: Year 11: Term 2: Working with Algebraic Fractions

Prerequisite Knowledge

  • Plot graphs of equations that correspond to straight-line graphs in the coordinate plane
  • Recognise, sketch and interpret graphs of linear and non-linear functions
  • Identify, describe and construct congruent and similar shapes, including on coordinate axes, by considering rotation, reflection, translation and enlargement.

Success Criteria

  • Where appropriate, interpret simple expressions as functions with inputs and outputs; interpret the reverse process as the inverse function; interpret the succession of two functions as a composite function.
  • Sketch translations and reflections of a given function
  • Calculate or estimate gradients of graphs (including quadratic and other non-linear graphs),
  • Find approximate solutions to equations numerically using iteration

Key Concepts

  • A function is any algebraic expression in which x is the only variable. It is denoted as f(x)
  • Understanding the notation for transformation of functions is critical to accessing this topic.
    • f(x) ± a = Vertical Translation
    • f(x ± a) = Horizontal Translation
    • af(x) = Horizontal stretch
    • f(ax) = Vertical stretch
  • Composite functions combine more than one function to an input.
  • Inverse functions perform the opposite operation to a function.
  • A gradient function calculates and approximate the instantaneous rate of change for given values of x.
  • Iterative solutions can diverge or converge.

Common Misconceptions

  • -f(x) is often incorrectly taken as a reflection in the y-axis rather than the x.
  • f(x + a) translates a unit to the left rather than to the right.
  • Students often struggle with writing the equation of the new function after a transformation.
  • Students need to be precise when drawing the transformed function.
  • The order of a composite function is often confused, fg(x) -> g acts on x first, then f acts on the result.
  • Students can often differentiate functions with little understanding of how to apply the gradient function correctly.

Working with Functions Resources

Mr Mathematics Blog

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Planes of Symmetry in 3D Shapes for Key Stage 3/GCSE students.

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