Functions, Graphs And Calculus
Muhtasari
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.
Malengo
- [Core] interpret simple expressions as functions with inputs and outputs
- [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
- [Core] work with coordinates in all four quadrants
- [Core] plot graphs of equations that correspond to straight line graphs in the coordinate plane
- [Core] use the form y = mx + c
- [Core] identify and interpret gradients and intercepts of linear functions graphically and algebraically
- [Core] understand the gradients of parallel lines
- [Extension] find the equation of the line through two given points, or through one point with a given gradient
- [Extension] understand and use the gradients of perpendicular lines
- [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
- [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
- [Core] understand and use the gradient function dy/dx
- [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)
- [Extension] know that the gradient of a function is the gradient of the tangent at that point
- [Extension] work out the equation of a tangent at any point on a curve
- [Extension] use of differentiation to find stationary points on a curve: maxima, minima and points of inflection
- [Extension] sketch a curve with known stationary points
- [Core] identify and interpret roots, intercepts and turning points of quadratic functions graphically
- [Core] deduce roots algebraically
- [Extension] deduce turning points by completing the square (Notes: including the symmetrical property of a quadratic)
- [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
- [Core] interpret the gradient of a straight-line graph as a rate of change
- [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
- [Extension] express direct and inverse variation in algebraic terms and use this form of expression to find unknown quantities
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Tathmini ya Somo
Hongera kwa kukamilisha somo la Functions, Graphs And Calculus. Sasa kwa kuwa umechunguza dhana na mawazo muhimu, ni wakati wa kuweka ujuzi wako kwa mtihani. Sehemu hii inatoa mazoezi mbalimbali maswali yaliyoundwa ili kuimarisha uelewaji wako na kukusaidia kupima ufahamu wako wa nyenzo.
Utakutana na mchanganyiko wa aina mbalimbali za maswali, ikiwemo maswali ya kuchagua jibu sahihi, maswali ya majibu mafupi, na maswali ya insha. Kila swali limebuniwa kwa umakini ili kupima vipengele tofauti vya maarifa yako na ujuzi wa kufikiri kwa makini.
Tumia sehemu hii ya tathmini kama fursa ya kuimarisha uelewa wako wa mada na kubaini maeneo yoyote ambapo unaweza kuhitaji kusoma zaidi. Usikatishwe tamaa na changamoto zozote utakazokutana nazo; badala yake, zitazame kama fursa za kukua na kuboresha.
- 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
- 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
- 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
- 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
- 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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