4.+Functions+-+Teachers

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Syllabus Notes Textbook Analysis

Highlights from the Syllabus
Fundamental concepts: //The pictorial representation of a function is extremely useful and important, as is the idea that algebraic and geometrical descriptions of functions are both helpful in understanding and learning about their properties.//

//Examples of functions y = f(x) should be given which illustrate different types of domain, bounded and unbounded ranges, continuous and discontinuous curves, curves which display simple symmetries, curves with sharp corners and curves with asymptotes. Students are to be encouraged to develop the habit of drawing sketches which indicate the main features of the graphs of any functions presented to them. They should also develop at this stage the habit of checking simple properties of functions and identifying simple features such as: where is the function positive? negative? zero?; where is it increasing? decreasing?; does it have any symmetry properties?; is it bounded?; does it have gaps (jumps) or sharp corners?; is there an asymptote?// (p.36) and some technical notes: //It is also common usage to refer to ‘the function f(x)’ where f(x) is prescribed but no domain is given. In such cases, the understanding required to be developed is that the domain of f is the set of real numbers for which the expression f(x) defines a real number.//

//It is important to realise that use of the notation y = f(x) does not imply that the expression corresponding to f(x) is the same for all x.// (p.35)

Textbook Analysis
The major text books take very different approaches to the Functions topic.

Summary

 * Almost across the board, the fundamental function concepts are presented as a set of definitions and facts with little exposition, exploration, context or justification. Arguably this important topic should be offered to students in a way that provides for a richer and deeper learning experience. Mary Barnes points the way.
 * Function transformations and graphing techniques are only thoroughly explored in Pender. Grove makes an attempt at it through 'Investigations'.
 * Fitzpatrick 2U, while clear and well laid out, is extremely succinct (only 10 pages) and //does not cover the graphs and curves sketching ideas// (maybe more in the 3 Unit book?).
 * Extras: As part of this topic, Pender covers inverse functions and Grove attempts to cover limits and continuity.

Detailed Analysis

 * **The concept of a function**
 * Very rapid treatment of a critical concept. In some books (Grove especially), the keys ideas are reduced to a set of definitions and 'how to' rules.
 * Pender at least shows the function machine analogy (2F).
 * **Key Function Ideas**
 * Fitzpatrick has a good section on open and closed intervals (6.3) and the idea of restricting the domain of a function.
 * J&C explicitly address the idea of "Graphs with gaps or missing points" (7.4) and "Graphs with sharp corners" (7.5).
 * **The "Function Zoo" : a review of known functions**
 * Grove is fairly comprehensive, even adding exponential and logarithmic.
 * Pender does a nice short summary of each (2G) with key features and strategies for graphing. He doesn't give much time for cubics.
 * J&C reduce it to a bullet point "for-later" list.
 * Fitzpatrick ignores this aspect of the topic.
 * **Absolute Value**: gets a good treatment in most books, Fitzpatrick particularly strong, including a nice graph of absolute value of a //quadratic// function.
 * **Graphing Techniques**
 * Pender is strong here - developed in (3B) and (3C), and then consolidated in (3G) Asymptotes and a Curve Sketching Menu.
 * Only Pender suggests //why// these techniques work : The Intermediate Value Theorem.
 * Grove attempts a similar treatment in (5.11). Example using //oblique// asymptotes (which are not in the syllabus) without explanation of the obliqueness!
 * Fitzpatrick ignores this subtopic.
 * J&C have a brief but quality (7.6) "Ideas to help curve sketching".
 * **Transformations**
 * The syllabus doesn't really cover this subtopic explicitly which explains why Fitzpatrick & J&C ignore it completely at this stage. //(Do they return to it later in 3U books?)//
 * Pender is takes this subtopic very seriously - two entire sections (2I, 2J) - well worth taking the time to master this - especially for Extension 1 students.
 * Grove weaves the idea through the 'function zoo' sequence using investigation, however does not clearly develop the results.
 * **Locus**
 * Most books give this cursory treatment, saving it for later topics.
 * J&C have a short section looking at several examples of locus problems - a nice intro (7.7)
 * Main focus for this subtopic is the definition of a circle as a circle, transformed into geometric and algebraic representations.
 * **Regions**
 * Pender saves this for his Chapter 3: Graphs and Inequations. In effect, Pender has made inequations and graphing of inequations a distinct and separate topic.
 * **Extras**
 * **Grove (tries to) cover Limits and Continuity in the Functions topic** and quite frankly makes a mess of it. (5.10). The presentation is weak and pointless - devoid of meaning. The exercise becomes 'substitute the value' in most cases. Even worse, Grove starts evaluating functions at the 'value' infinity, writing f(infinity) and expressions like infinity squared. Note that the BOS syllabus detailed discussion on Topic 4 does //not// include limits or continuity. The syllabus only suggests students be prepared for the concepts by showing them functions with holes and asymptotes. See Limits and Discontinuity for more discussion.
 * **Pender has a light touch on Limits and Continuity** Pender shows the limits notation when discussing the hyperbola (2G). He saves further discussion of limits and continuity for Chapter 7B and 7I in the context of differentiation.
 * Pender covers **Inverse Relations**, and the concept of restricting to domain to get an inverse function. This section provides a good review of key function concepts, applied to new content. //This material would be good preparation for Extension 1 students.//

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