Working with Functions

August 23, 2016

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
Algebra: Interactive Lessons
Estimating the Area Under a Curve
4 Part Lesson
Estimating the Area Under a Curve
How to estimate the area under a curve using the trapezium rule.
Algebra: Interactive Lessons
Instantaneous Rates of Change
4 Part Lesson
Instantaneous Rates of Change
How to calculate and interpret instantaneous rates of change in different contexts.
Algebra: Interactive Lessons
Working with Functions
4 Part Lesson
Working with Functions
How to use the correct notation to calculate outputs and inputs for complex functions.
Algebra: Interactive Lessons
Translating Functions
4 Part Lesson
Translating Functions
How to translate graphs in the form f(x+a) and f(x)+a.
Algebra: Interactive Lessons
Stretching Functions
4 Part Lesson
Stretching Functions
How to stretch a graph in the form f(ax) and a(fx).
Algebra: Interactive Lessons
Solving Equations Through Iteration
4 Part Lesson
Solving Equations Through Iteration
How to use numerical methods to solve equations through iteration.
Algebra: Interactive Lessons
Inverse Functions
4 Part Lesson
Inverse Functions
How to find the inverse of a function using function machines.
Algebra: Interactive Lessons
Composite Functions
4 Part Lesson
Composite Functions
How to find the output of composite functions using the correct notation.
Additional Resources
Algebra: Interactive Lessons
Extended Learning - Function Notation
Extended Learning
Function Notation
Additional practice using the correct notation to calculate outputs and inputs for complex functions.
Algebra: Interactive Lessons
Extended Learning - Rates of Change
Extended Learning
Rates of Change
Additional practice finding the instantaneous rate of change of a point along a curve.
Algebra: Interactive Lessons
Revising Transforming Graphs
Revision
Transforming Graphs
Students revise how to perform vertical and horizontal translations of graphical functions.
Algebra: Interactive Lessons
Revising Gradients of Curves
Revision
Gradients of Curves
Students revise how to estimate the gradient at a point along a curve.
Algebra: Interactive Lessons
Revising Inverse and Composite Functions
Revision
Inverse and Composite Functions
Students revise how to find the inverse of a function using the correct notation.
Algebra: Interactive Lessons
Revising Area Under a Curve
Revision
Area Under a Curve
Students revise how to estimate the distance travelled in a speed-time graph by finding the area under a curve.
Algebra: Interactive Lessons
Students revise how to substitute known values and expressions into basic and composite functions
Revision
Function Notation and Composite Functions
Revision lesson on function notation and composite functions for exam students.
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.

Mr Mathematics Blog

Pull Up Nets

March 9, 2022

Home learning project teaching how to create pull up nets for 3D shapes.

Finding the Area of a Sector

January 26, 2022

How to find the area of a sector as a fraction of a full circle.

How to Write 3 Part Ratios

January 19, 2022

How to teach calculating 3-part ratios.