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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.

- 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.

- 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

- 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.

- -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.

July 6, 2019

Earlier this week, my school took part in a trial OFSTED inspection as part of getting ready for the new inspection framework in September 2019. This involved three Lead Inspectors visiting our school over the course of two days. The first day involved a ‘deep dive’ by each of the Lead Inspectors into Mathematics, English […]

June 30, 2019

The method of how to solve quadratics by factorising is now part of the foundational knowledge students aiming for higher exam grades are expected to have. Here is an example of such a question. Solve x2 + 7x – 18 = 0 In my experience of teaching and marking exam papers students often struggle with […]

June 24, 2019

When learning how to write 3-part ratios students need to understand how ratios can be made equivalent. The start of the lesson reminds students by asking which of six ratios is the odd one out. This is presented to the class as they come into the lesson. Writing Equivalent Ratios A few students immediately go […]