Algebra: Polynomials, transformations and partial fractions
AS level content
- Understand and use graphs of functions; sketch curves defined by simple equations including polynomials,
- y=a/x and y=a/x2 (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 graph
- Understand the effect of simple transformations on the graph of y=f(x) including sketching associated graphs: y=af(x), y=f(x)+a, y=f(x+a), y=f(ax)
A level content
- Understand and use composite functions; inverse functions and their graphs
- Understand the effect of combinations of simple transformations on the graph of y=f(x)
- Decompose rational functions into partial fractions (denominators not more complicated than squared linear terms and with no more than 3 terms, numerators constant or linear)
- Use of functions in modelling, including consideration of limitations and refinements of the models
Polynomials
This resource contains seven problems requiring students to explore the graphs of polynomials. Students are required to explain how they know whether a graph will cross the x axis, explain the differences and similarities of the graphs of two polynomials given just the equation, asked to devise questions that could be asked given the equation of a polynomial, find the equation of a cubic given a number of clues, use function notation and explain whether certain statements are true.
Functions and Graphs
This resource contains a number of activities which cover many aspects of functions and graphs. Topics appropriate to this collection are :
- Introduction to functions
- Composite functions
- Inverse functions
- Polynomial functions
Comprehensive notes, with clear descriptions, for each resource are provided, together with relevant diagrams and examples. Students wishing to review, and consolidate, their knowledge and understanding of functions and graphs will find them useful, as each topic includes a selection of questions to be completed, for which answers are provided.
Reciprocal Function
This interactive excel file shows the graph of a function and the graph of the corresponding reciprocal function. It begins with the graph of a linear function where the gradient and intercept can be changed. The reciprocal is plotted on the same diagram. The next two activities show graphs of quadratic and cubic functions whose coefficients can be altered. The graph is plotted and the graph of the reciprocal function shown. In the same way the graphs of trigonometric and exponential functions and their reciprocals are dealt with.
Exploring transformations of graphs
This activity requires students to use and interpret function notation, sketch graphs using key points, explore the common transformations of translate parallel to the y axis, translate parallel to the x axis, stretch parallel to the y axis from the x axis and stretch parallel to the x axis from the y axis.
Ideally students should have access to appropriate graph plotting technology to investigate the tasks. Students are required to explore the different transformations, record their results on the sheet and use their results to generalise the effect of each transformation.
It is recommended that students begin their investigation with linear functions, quadratic functions and a reciprocal function. Extension suggestions are to work with general equations, trigonometric equations and to explore reflections in the x axis and reflections in the y axis.
Transformation of Functions
This interactive excel resource explores the relationship between the transformation of functions their graphs.
Functions including linear, quadratic and cubic functions as well as sine and cosine curves are covered. Each type of function can be fully transformed in stages. The algebraic expression for each function can be hidden or revealed. By selecting "new" the interactive sheets present the student with the graph of a new function so understanding can be tested.
The resource also has twelve sheets of printable sets of questions which may be suitable for use within the classroom.
Quadratic Functions
This resource contains eleven problems requiring students to explore the graphs of quadratic equations, intersection points of quadratics and straight lines, explain what will be the same and what will be different about the graphs of two quadratic functions, transform quadratic graphs, find the odd one out, complete the square and deal with quadratic inequalities. In each problem students are encouraged to explain and justify their solutions.
Functions
The activity Periodic Functions, asks students to write down as many periodic functions as they can. The activity progresses to look at what happens to the period when graphs with different periods are combined. Topics covered are periodic functions, odd functions, even functions, composite functions and transformation of graphs.
The second activity, When does fg equal gf?, requires students to complete a number of tasks to investigate composite functions.
Functions
Composite Functions
This interactive excel workbook contains a series of sheets to help students explore different types of composite functions including linear, quadratic and reciprocal functions.
The first sheet allows students to input different numbers into a flowchart showing two different linear functions. Students work out and can check the expression for the composite function which is then simplified. The next four sheets involve linear-quadratic and linear-reciprocal composite functions.
In the final two sheets students combine two functions then solve an equation. Some of the solutions to the quadratic equations are given to 3 significant figures
Cubics (Factorising and Graphing)
The first six sheets allow students to investigate the relationships between cubic polynomials, the factorised form and the graph. Initially the factors are of the form (x-n) but later sheets deal with factors which could be (x±n) and with repeated factors. The sheet with tab "Evaluate f(a)" is intended to allow students to quickly find factors by inspection using the Remainder Theorem or the Factor Theorem. The graph of the function can be revealed. The final interactive sheet "Evaluate f(b)" is similar but the coefficient of x3 could be 1,2 or 4. There are also five sheets of questions which may be suitable for duplication for class use.
Using Partial Fractions to Evaluate the Definite Integral of an Algebraic Fraction Example One
This resource contains two tutorial videos produced by Casio.
The first explains how to express an algebraic fraction in partial fractions. The fraction used is a straight forward example solves by equating coefficients to obtain two simultaneous equations. The second part of the video uses the result of the first part of question in order to find the value of an integral between limits expressing the answer as a single logarithm.
The second video shows how a graphical calculator can be used to verify the solution. The steps show how to integrate a function between limits using the functions in the graphical calculator.
Using Partial Fractions to Evaluate the Definite Integral of an Algebraic Fraction Example Two
This resource contains two further Casio tutorial videos.
In the first video, how to express a function in partial fractions where the three denominators are linear is explained. The result is used to find the definite integral of the original function, the solution given in terms of natural logarithms in its simplest form before calculating the decimal value.
The second video explains how to use the graphic calculator to verify the solution found in the first video.