from The Open University
Alternatively you can skip the navigation by pressing 'Enter'.
Is there a problem with OpenLearn? OpenLearn status updatesWe hope OpenLearn works normally, around the clock, through the year. If we know something's... Read more: Is there a problem with OpenLearn? OpenLearn status updates
The Met: Policing LondonThe Met: Policing London provides insight into one of the city’s most vital services and... Read more: The Met: Policing London
An introduction to Open Educational Resources (OER)In this free course, An introduction to Open Educational Resources (OER), you will look at some... Try: An introduction to Open Educational Resources (OER) now
Succeed with maths – Part 1If you feel that maths is a mystery that you want to unravel then this short 8-week course is for... Try: Succeed with maths – Part 1 now
Real functions and graphs
Sometimes the best way to understand a set of data is to sketch a simple graph. This...
Sometimes the best way to understand a set of data is to sketch a simple graph. This exercise can reveal hidden trends and meanings not clear from just looking at the numbers. In this unit you will review the various approaches to sketching graphs and learn some more advanced techniques.
By the end of this section you should be able to:
- understand the definition of a real function;
- use the notation for intervals of the real line;
- recognise and use the graphs of the basic functions described in the audio section;
- understand the effect on a graph of translations, scalings, rotations and reflections;
- understand how the shape of a graph of a function features properties of the function such as increasing, decreasing, even and odd.
- determine the x-intercepts and y-intercept of a given function f;
- determine the intervals on which a given function f is positive or negative;
- determine the intervals on which a given function f is increasing or decreasing, and any points at which f has a local maximum or local minimum;
- describe the asymptotic behaviour (if any) of a given function f;
- sketch the graph of a given function.
- sketch the graph of a combination of two functions, one of which is a trigonometric function;
- sketch the graph of a hybrid function, whose rule is defined by different formulas on different parts of its domain.
- define the hyperbolic functions cosh x, sinh x and tanh x, and be familiar with their properties;
- sketch the graphs of cosh x, sinh x and tanh x, and their reciprocals.
- plot a curve that is specified by a parametric representation;
- obtain the equation of a curve that is specified by a parametric representation, in simple cases;
- sketch a conic whose equation is given in standard form.
Study this free course
Enrol to access the full course, get recognition for the skills you learn, track your progress and on completion gain a statement of participation to demonstrate your learning to others. Make your learning visible!
Real functions and graphs
Many problems are best studied by working with real functions, and the properties of real functions are often revealed most clearly by their graphs. Learning to sketch such graphs is therefore a useful skill, even though computer packages can now perform the task. Computers can plot many more points than can be plotted by hand, but simply ‘joining up the dots’ can sometimes give a misleading picture, so an understanding of how such graphs may be obtained remains important. The object of this unit is to review the various techniques for sketching graphs that you may have met in your previous studies, and to extend these methods.
This unit is an adapted extract from the Open University course
Copyright & revisions
Originally published: Tuesday, 28th June 2011
- Creative-Commons: The Open University is proud to release this free course under a Creative Commons licence. However, any third-party materials featured within it are used with permission and are not ours to give away. These materials are not subject to the Creative Commons licence. See terms and conditions. Full details can be found in the Acknowledgements section.
- This site has Copy Reuse Tracking enabled - see our FAQs for more information.
If you enjoyed this, why not follow a feed to find out when we have new things like it? Choose an RSS feed from the list below. (Don't know what to do with RSS feeds?)
Remember, you can also make your own, personal feed by combining tags from around OpenLearn.