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Achievement objective M7-9

In a range of meaningful contexts, students will be engaged in thinking mathematically and statistically. They will solve problems and model situations that require them to:

  • sketch the graphs of functions and their gradient functions and describe the relationship between these graphs.


  • Sketches graphs of gradient functions of a variety of functions represented in various forms such as an equation or a graph (differentiable over the given domain).
  • Sketches graphs of functions from the graph of the gradient function.
  • Relates the gradient at a point on a curve to the instantaneous rate of change.
  • Relates significant features of the functions such as:
    • turning points on the original function and the zeros of the gradient function, or vice versa
    • the increasing or decreasing of the original function and the sign of the gradient function, or vice versa.
  • Make links with graphs M7-2 and differentiation of polynomials M7-10.
  • See key mathematical ideas on NZmaths.


M7-9 links from NA6-8 and to M8-10, M8-11.

Possible context elaborations

Given y, sketch y'

If you cannot view or read this diagram, open this text version.

Possible context elaboration 1.

Given y', sketch y

A graph of the positive linear function y = 3x + 2.


  • Variety of graphs includes piecewise, log, non-polynomial.
  • Students will develop an understanding of the relationship between the degree of a polynomial and the degree of its gradient function.
  • Oil spills: Exploring rates of change
  • Rates of change: Sketching graphs of situations and describing the rates of change.
  • Use a video camera to record things like a student running, walking, cycling. You will need to have marked out on the ground the distance in metres. You can then record quite accurate measurement of distance and time by using the frame advance on the video.

Assessment for qualifications

NCEA achievement standards at level 1, 2 and 3 have been aligned to the New Zealand Curriculum. Please ensure that you are using the correct version of the standards by going to the NZQA website.

The following achievement standard(s) could assess learning outcomes from this AO:

  • AS91262 Mathematics and statistics 2.7 Apply calculus methods in solving problems

Refer to the mathematics and statistics matrix.

Last updated September 26, 2013