# 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.

##### 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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