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3 Sequences, functions and graphs

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  1. 3.1 Sequences

    1. generate terms of a sequence using term-to-term and position-to-term definitions of the sequence Including odd, even, squares, multiples and powers

    2. find subsequent terms of an integer sequence and the rule for generating it 5, 9, 13, 17, … (add 4) 1, 2, 4, 8, … ( multiply by 2)

    3. use linear expressions to describe the nth term of arithmetic sequences 1, 3, 5, 7, 9, … nth term is 2n – 1 nth term is 4n + 3, write down the first 3 terms of the sequence

    4. understand and use common difference (d) and first term (a) in an arithmetic sequence e.g. given 2nd term is 7 and 5th term is 19, find a and d

    5. know and use the nth-term formula a + (n − 1)d for an arithmetic sequence

    6. find the sum of the first n terms of an arithmetic series (Sn) e.g. given 4 + 7 + 10 + 13 + … find sum of first 50 terms

  2. 3.2 Function notation

    1. Understand a function as a mapping between elements of two sets.

    2. use function notation in the forms f(x) = … and f : x ↦ …

    3. understand domain and range, including values excluded from a domain, such as x = 2 for f(x) = 1/(x − 2)

    4. understand and find the composite function fg and the inverse function f⁻¹; fg means apply g first, then f

  3. 3.3 Graphs

    1. 3.3.AInterpreting linear and non-linear graphs

      interpret information presented in a range of linear and non-linear graphs To include speed/time and distance/time graphs

    2. 3.3.BCartesian coordinate conventions

      understand and use conventions for rectangular Cartesian coordinates

    3. 3.3.CPoints (x, y) in any of the four quadrants or locate

      plot points (x, y) in any of the four quadrants or locate points with given coordinates

    4. 3.3.DThe coordinates of points identified by geometrical

      determine the coordinates of points identified by geometrical information

    5. 3.3.EMidpoint coordinates

      Determine a line segment’s midpoint from the coordinates of its endpoints.

    6. 3.3.FStraight-line conversion graphs

      Draw and interpret straight-line conversion graphs, including currency-conversion graphs.

    7. 3.3.GGradient of a straight line

      find the gradient of a straight line gradient = (increase in y) ÷ (increase in x)

    8. 3.3.HStraight-line graphs in the form y = mx + c

      recognise that equations of the form y = mx + c are straight line graphs with gradient m and intercept on the y-axis at the point (0, c) Write down the gradient and coordinates of the y intercept of y = 3x + 5; Write down the equation of the straight line with gradient 6 that passes through the point (0, 2)

    9. 3.3.ILinear and quadratic function graphs

      Recognise, generate points and plot linear and quadratic function graphs, including tables and equations of the form ax + by = c.

    10. 3.3.HAPolynomial, reciprocal and trigonometric graphs

      recognise, plot and draw graphs of y = Ax³ + Bx² + Cx + D; y = Ax³ + Bx² + Cx + D + E/x + F/x² where at least three constants are zero; and y = sin x, y = cos x and y = tan x for angles of any size in degrees; x and y may be replaced by other variables

    11. 3.3.HBTransformations of graphs

      apply to the graph of y = f(x) the transformations y = f(x) + a, y = f(ax), y = f(x + a), y = af(x) for linear, quadratic, sine and cosine functions

    12. 3.3.HCAnalysing transformations of functions

      interpret and analyse transformations of functions and write the functions algebraically

    13. 3.3.HDThe gradients of non-linear graphs By drawing a tangent

      find the gradients of non-linear graphs By drawing a tangent

    14. 3.3.HEIntersections of linear and non-linear graphs

      find intersections of a linear graph y₁ and a non-linear graph y₂, and recognise that their x-coordinates solve y₂ − y₁ = 0

    15. 3.3.HFGradient from two points

      Calculate a straight line’s gradient from the coordinates of two points.

    16. 3.3.HGParallel and perpendicular line equations

      Find equations of straight lines parallel or perpendicular to a given line.

  4. 3.4 Calculus

    1. Understand the concept of a variable rate of change.

    2. differentiate integer powers of x

    3. determine gradients, rates of change, stationary points, turning points (maxima and minima) by differentiation and relate these to graphs Find the coordinates of the maximum and minimum points

    4. distinguish between maxima and minima by considering the general shape of the graph only

    5. apply calculus to linear kinematics and other simple practical problems