Working with Functions

Students learn how to use function notation to transform graphs by a translation and stretch.  As learning progresses they calculate composite and inverse functions.  Finally, students are challenged to solve equations through interation and calculate a rate of change along a non-linear graph using differentiation.

Before progressing on to working with functions students should have a good understanding of solving quadratic equations and trigonometrical graphs.  This is the final topic in the higher GCSE course as it takes place in Year 11 Term 3.

Working with Functions Lessons
Revising Functions

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) = 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) is a translation of ‘a’ units 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.
  • Students can confuse f-1(x) with f’(x).
  • The order of a composite function is often confused, fg(x) -> g acts on x first then f acts on the result.
  • Students are often able to differentiate functions with little understanding of how to apply the gradient function correctly.

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