# Functions and Graphs

Scheme of work: Year 13 A-Level: Pure 2: Functions and Graphs

#### Prerequisite Knowledge

• Understand and use graphs of functions.
• Sketch curves defined by simple equations including polynomials and reciprocals including their vertical and horizontal asymptotes.
• Interpret algebraic solution of equations graphically.
• Use intersection points of graphs to solve equations.
• Understand and use proportional relationships and their graphs.
• Understand the effect of simple transformations on the graph of f(x) including sketching associated graphs.

#### Success Criteria

• Understand the different types of mapping.
• Understand and use the domain and range of a function.
• Understand and use graph of the modulus function .
• Solve modulus equations using sketched graphs.
• Understand and use composite functions, inverse functions, and their graphs.
• Understand the effect of combinations of the following transformations y=af(x), y = f(x) + a, y = f(x + a), y = f(ax) on the graph of f(x)
• Use of functions in modelling, including consideration of limitations and refinements of the models

#### Key Concepts

• Use a graphing program like Desmos to demonstrate the relationship between a function and its inverse. For example, understanding a function and its inverse are reflected in the line y = x is needed when solving equations of the form f-1(x) = f(x).
• When sketching graphs, students should label the coordinates where the function touches or passes through the coordinate axes. Then, students should use Desmos to check the features of their sketched graph.
• Encourage students to be as accurate and tidy as possible when sketching graphs so that important graphical properties are not lost.
• When sketching functions of form f (x) = a | x + b | ensure students state the coordinates of the maximum point, which helps to work out the range.
• When asked to work out the range of a function, encourage students to sketch a graph.
• When deciding whether a function has an inverse, students must consider whether the inverse would have a one-to-many or many-to-one mapping.

#### Common Misconceptions

• When sketching modulus functions, students can lose marks due to a lack of accuracy in the graph. Therefore, please encourage them to consider the reflective symmetry property.
• Sketching y = af ( x ) , instead of y = f (ax) , is a common error in exam papers.
• When solving equations of the form |ax + b| = x + c some students find only one solution when two exist. Encourage them to use a sketched graph to visualise the two intersections.
• When asked to work out f-1(x), some students confuse this with f'(x) and differentiate f(x).

## Functions and Graphs Resources

### Mr Mathematics Blog

#### Hypothesis Testing and Finding Critical Regions with the Normal Distribution

A-Level Mathematics: Statistics 2: How to hypothesis test and find critical regions with the normal distribution.

#### Convert to Standard Normal Distribution

Learn to convert any normal distribution into a standard normal distribution with step-by-step tutorials, lessons, and practice worksheets.

#### Grade 9 Coordinate Geometry Problems

How to solve geometrical problems involving the properties of shapes and straight line graphs.