A-Level Maths: Graphs, Transformations and Partial Fractions

Much of A level pure mathematics depends on reading and changing graphs quickly and on handling functions precisely. This material trains the functions-and-graphs part of the course, based on the algebra and functions content of the DfE A level Mathematics subject content: transformations, the modulus function, domain and range, composite and inverse functions, and partial fractions.

The transformation questions ask where a key point or turning point moves under $y=af(x)$, $y=f(ax)$, $y=f(x)+a$, $y=f(x+a)$, $y=-f(x)$ and $y=f(-x)$, and how to describe a combination of transformations in the correct order, for example mapping $y=x^2$ onto $y=2(x+3{)}^2-5$. The modulus questions use the graphs of $y=\mathrm{\mid }f(x)\mathrm{\mid }$ and $y=f(\mathrm{\mid }x\mathrm{\mid })$ to solve equations such as $|2x - 3| = |x + 3|$ and inequalities such as $|x - 4| < 3x$, and to count solutions by seeing how many times a horizontal line meets a reflected curve.

Function questions ask for the range of a function by completing the square, the largest domain of a composite function (the output of the inner function must lie in the domain of the outer one) and an inverse function together with its domain, which is the range of the original function. The last group practises partial fractions, first with distinct linear factors and then with a repeated linear factor, where a term is needed for each power of the repeated factor. These are the forms you will later integrate and expand as series.

Every question is self-contained and all answers are exact.

The material offers three formats. The quiz has 12 multiple-choice questions whose explanations show each step and the typical mistake behind each wrong option, such as moving a graph the wrong way or forgetting that the domain of an inverse is the range of the function. The flashcards summarise the transformation rules, the modulus graphs, the domain rules for composite and inverse functions and the partial-fraction forms. The written work gives 8 longer questions to answer by hand, from describing the images of several points to finding the values of $k$ for which a modulus equation has exactly four solutions.

• Describe and apply transformations of graphs, including combinations
• Sketch and use the graphs of y = |f(x)| and y = f(|x|)
• Solve equations and inequalities involving the modulus function
• Find the domain and range of functions and composite functions
• Find an inverse function and state its domain
• Express rational functions in partial fractions with distinct or repeated linear factors

Practice material written by Zestly, based on the DfE A level Mathematics subject content (algebra and functions: modulus, graphs of functions and transformations, composite and inverse functions, partial fractions).