Mathematics - 9260 OxfordAQA

Functions, Graphs And Calculus

Übersicht

A graph is an algebraic relationship you can look at. Every straight line is a sentence about a constant rate of change; every curve is a sentence about a rate of change that is itself changing. This is the largest topic on the specification, ten references from A9 to A18, and it is where algebra and geometry finally become the same subject.

You will work with coordinates in all four quadrants, plot and interpret straight line graphs, use the form \( y = mx + c \), read gradients and intercepts both graphically and algebraically, recognise and sketch the standard curve shapes, find roots and turning points of quadratic functions, and read real context graphs including distance-time and speed-time graphs where the gradient is a rate of change. On the Extension Tier you will use function notation with domain and range, build composite and inverse functions, find equations of lines through given points, handle perpendicular gradients, meet exponential and trigonometric graphs, complete the square to locate a turning point, estimate gradients and areas under non-linear graphs, and differentiate to find tangents and stationary points.

Ziele

  1. [Core] interpret simple expressions as functions with inputs and outputs
  2. [Extension] definition of a function, use function notation of the form f(x) = …, understand and use the terms domain and range, understand and find the composite function fg and the inverse function f⁻¹
  3. [Core] work with coordinates in all four quadrants
  4. [Core] plot graphs of equations that correspond to straight line graphs in the coordinate plane
  5. [Core] use the form y = mx + c
  6. [Core] identify and interpret gradients and intercepts of linear functions graphically and algebraically
  7. [Core] understand the gradients of parallel lines
  8. [Extension] find the equation of the line through two given points, or through one point with a given gradient
  9. [Extension] understand and use the gradients of perpendicular lines
  10. [Core] recognise, sketch and interpret graphs of linear functions and quadratic functions including simple cubic functions and the reciprocal function y = 1/x with x ≠ 0
  11. [Extension] including exponential functions y = kˣ for positive values of k, and the trigonometric functions (with arguments in degrees) y = sin x, y = cos x and y = tan x for angles of any size
  12. [Core] understand and use the gradient function dy/dx
  13. [Extension] differentiation of kxⁿ where n is a positive integer or 0, and the sum of such functions (Notes: including expressions which need to be simplified first)
  14. [Extension] know that the gradient of a function is the gradient of the tangent at that point
  15. [Extension] work out the equation of a tangent at any point on a curve
  16. [Extension] use of differentiation to find stationary points on a curve: maxima, minima and points of inflection
  17. [Extension] sketch a curve with known stationary points
  18. [Core] identify and interpret roots, intercepts and turning points of quadratic functions graphically
  19. [Core] deduce roots algebraically
  20. [Extension] deduce turning points by completing the square (Notes: including the symmetrical property of a quadratic)
  21. [Core] plot and interpret graphs, and graphs of non-standard functions in real contexts, to find approximate solutions to problems such as simple kinematic problems involving distance, speed and acceleration
  22. [Core] interpret the gradient of a straight-line graph as a rate of change
  23. [Extension] calculate or estimate gradients of graphs and areas under graphs (including quadratic and other non-linear graphs), and interpret results in cases such as distance-time graphs and velocity-time graphs
  24. [Extension] express direct and inverse variation in algebraic terms and use this form of expression to find unknown quantities

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Lernkarten

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Lektionshinweis

A specimen Extension paper shows a straight line on axes running to 50 across and 400 up, tells you its equation is \( y = ax + b \), and asks for the values of \( a \) and \( b \) for two marks. No algebra is needed. The line crosses the vertical axis at 100, so \( b = 100 \), and it climbs 300 while moving 50 across, so \( a = 300 \div 50 = 6 \). The answer is \( y = 6x + 100 \), and the mark scheme awards both marks for that single line even if \( a \) and \( b \) were never separated out.

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Unterrichtsbewertung

Herzlichen Glückwunsch zum Abschluss der Lektion über Functions, Graphs And Calculus. Jetzt, da Sie die wichtigsten Konzepte und Ideen erkundet haben,

Sie werden auf eine Mischung verschiedener Fragetypen stoßen, darunter Multiple-Choice-Fragen, Kurzantwortfragen und Aufsatzfragen. Jede Frage ist sorgfältig ausgearbeitet, um verschiedene Aspekte Ihres Wissens und Ihrer kritischen Denkfähigkeiten zu bewerten.

Nutzen Sie diesen Bewertungsteil als Gelegenheit, Ihr Verständnis des Themas zu festigen und Bereiche zu identifizieren, in denen Sie möglicherweise zusätzlichen Lernbedarf haben.

  1. f(x) = 3x. Circle the expression for the inverse function f^-1(x). A. -3x B. 1/(3x) C. 3/x D. x/3 Answer: D
  2. A line has gradient 2/3. What is the gradient of a line perpendicular to it? A. 2/3 B. -2/3 C. 3/2 D. -3/2 Answer: D
  3. What is the gradient function of y = x^3 - 2x^2? A. 3x^2 - 4x B. 3x^2 - 2x C. x^2 - 4x D. 3x - 4 Answer: A
  4. On a speed-time graph, what does the area under the graph represent? A. the acceleration B. the average speed C. the distance travelled D. the time taken Answer: C
  5. The graph of y = 5 + 3x - 2x^2 crosses the horizontal axis twice. What are the solutions of 5 + 3x - 2x^2 = 0? A. -1 and 2.5 B. 1 and -2.5 C. -1 and 5 D. 2.5 and 5 Answer: A

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Empfohlene Bücher

OxfordAQA International GCSE Mathematics Core (9260)
OxfordAQA International GCSE Mathematics Core (9260)
Untertitel: Student Book
Verleger: Oxford University Press
Jahr: 2016
ISBN: 978-0-19-837586-9
Beschreibung: The endorsed student book for the Core Tier of the 9260 specification, written for first teaching in September 2016. 520 pages covering every Core content statement with graded practice and worked model answers.
OxfordAQA International GCSE Mathematics Extension (9260)
OxfordAQA International GCSE Mathematics Extension (9260)
Untertitel: Student Book
Verleger: Oxford University Press
Jahr: 2016
ISBN: 978-0-19-837587-6
Beschreibung: The endorsed student book for the Extension Tier. 560 pages: it carries the whole Core content plus every Extension statement, so it is the one book to own if you are entered for Papers 1E and 2E.
Alex's Adventures in Numberland
Alex's Adventures in Numberland
Untertitel: Dispatches from the Wonderful World of Mathematics
Verleger: Bloomsbury
Jahr: 2011
ISBN: 978-1-4088-0959-4
Beschreibung: Where the mathematics on this specification came from and what it is for: counting systems, the golden ratio, probability paradoxes and the people who argued about them. Read it for the reason behind the rule.

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