# 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

#### Planes of Symmetry in 3D Shapes

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

Use isometric paper for hands-on learning and enhanced understanding.

#### GCSE Trigonometry Skills & SOH CAH TOA Techniques

Master GCSE Math: Get key SOH-CAH-TOA tips, solve triangles accurately, and tackle area tasks. Ideal for students targeting grades 4-5.

#### Regions in the Complex Plane

Explore Regions in the Complex Plane with A-Level Further Maths: inequalities, Argand diagrams, and geometric interpretations.