4.11 FUNCTIONS AND GRAPHS
FUNCTIONS AND GRAPHS LESSON IDEA 1
TITLE: THE FUNCTION MACHINE
TOPIC: FUNCTIONS AND GRAPHS
AIM:
- To give students an intuitive idea of functions using the analogy of a function machine.
- To give students the opportunity to complete a graph table by reading the ordered pairs from the graph.
- To show how the calculator can be used as a learning tool.
RESOURCES:
A copy of the "function machine" for display using an overhead projector.
METHOD:
- Ask students to consider the function f (x) = 2x2 - 3x - 5 in the domain -3 = x = 3, x R. The image of 2 is -3. A similar pattern exists for the other elements of the domain. Each input element is coupled with its output image in the form (x, y), for example (2,-3).
- In passing, students could be shown how to use the PLAYBACK and the DEL functions on a suitable calculator to substitute the 2 by another number and get a new result or output.
- While constructing the traditional long form of the table is foolproof, in many respects the logic behind how functions operate can get somewhat lost. What is required ultimately is a set of couples that can be graphed. This is where the "function machine" can help.
- Explain to students that the elements which are fed into the top of the function machine are plotted along the x-axis, and the elements which correspond to the output are plotted along the y-axis.
- Now, questions such as "Find the value of x such that f (x) = -3" take on the meaning "For what values of x inserted at the top of the function machine will an output of -3 be produced?" Students readily relate to this type of treatment and see the concept of a function as an operation (turning the handle) on a set of elements.
- It is instructive to drop - into the top of the machine and to observe that the image is also -3, giving the couple (- ,-3). Students soon observe that the "processing" or number crunching takes place inside the machine and corresponds to "adding up" the values in the graph table normally used.
- The reverse approach to graph drawing is also of educational benefit. Working from a graph such as that illustrated overleaf, the student is asked to complete the table to the best of his/her ability by reading the ordered pairs from the graph. These pairs can be recorded in a table similar to that used at step 3.
CLASSROOM MANAGEMENT IMPLICATIONS:
None.
NOTE:
To create a graph such as the one shown, enter the x and y values into a spreadsheet (e.g. Excel) in the following format:
Select the table and click on the chart wizard. Choose XY scatter and select the scatter with data points connected by smoothed lines. More detailed changes to the graph can be made by using the option tags presented as the graph is produced.
FUNCTIONS AND GRAPHS LESSON IDEA 2
TITLE: MAXIMUM AND MINIMUM VALUES FOR QUADRATIC FUNCTIONS
TOPIC: FUNCTIONS AND GRAPHS
AIM: To show students the method of "completing the square" for finding the maximum and minimum values of quadratic functions.
RESOURCES:
None.
METHOD:
- Consider the following question.
Find the minimum value of the function:
f(x) = x2 - 6 x + 13, x R
f(x) = x2 - 6 x + 13
=> f(x) = x2 - 6 x + 9 - 9 + 13
= x2 - 6 x + 9 + 4
= (x - 3)2 + 4
Therefore the minimum value is 4 and this will occur when x = 3.
In passing, it could be pointed out to students that x = 3 is the axis of symmetry of the graph 2. Similarly:
Find the maximum value of the function:
f(x) = 5 - 2x - x2, x R
f(x) = 5 - 2x - x2
=> f(x) = 5 + 1 - 1 - 2x - x2
= 6 - (1 + x)2
giving a maximum value of 6, which occurs when x = -1
CLASSROOM MANAGEMENT IMPLICATIONS:
None
NOTE:
The syllabus (p.16) mentions the "maximum and minimum values of quadratic functions estimated from graphs" for Higher Level students. It would be desirable if the algebraic method of "completing the square" for evaluating the maximum or minimum was treated at least once for students taking the Higher level.