Functions, Graphs And Calculus
Overview
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.
Objectives
- [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
Mind map
This topic is mapped out so you can see how the ideas connect.
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Lesson Note
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Lesson Evaluation
Congratulations on completing the lesson on Functions, Graphs And Calculus. Now that youve explored the key concepts and ideas, its time to put your knowledge to the test. This section offers a variety of practice questions designed to reinforce your understanding and help you gauge your grasp of the material.
You will encounter a mix of question types, including multiple-choice questions, short answer questions, and essay questions. Each question is thoughtfully crafted to assess different aspects of your knowledge and critical thinking skills.
Use this evaluation section as an opportunity to reinforce your understanding of the topic and to identify any areas where you may need additional study. Don't be discouraged by any challenges you encounter; instead, view them as opportunities for growth and improvement.
- 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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