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PARTICIPANT HANDOUT – FET PHASE 1
Module 1: Functions
Sub-topic 1: Introduction to functions
CAPS extraction indicating progression from grades 10-
Grade 10 Grade 11 Grade 12 The concept of a function Relationships between variables using words, tables, graphs, and formulae Domain and range
Definition of a function
Introduction
When introducing functions to learners in grade 10, it is important to refer to the mathematics which they have come across in their earlier grades. Some examples of these are:
Working with the different operations Checking for relationships (between sets of numbers, in formulae, etc.) Substitution in equations/formulae Sketching, finding the equation of and interpreting the graph of (a straight line)
The approach used in grade 10 is developmental. In grade 12, learners are introduced to a more formal approach to a function. In this module, we used both approaches, starting with a developmental approach, which makes use of learners‟ prior knowledge.
Activity 1: Introduction to functions
Group organisation:
Time: Resources: Appendix:
Groups of 6 10 min Flip chart Permanent markers.
None
In your groups you will:
- Select a scribe and a spokesperson for this activity only.(Rotate from activity to activity)
- Use the flipchart and permanent markers and answer the questions as per the activity.
- Complete the table below: x^0 1 2 3 x^2
1.1 Write down some ordered pairs for the above table. 1.2 Write down the equation represented by the information in the table. 1.3 Draw the graph. 1.4 Is there any restriction on the domain and range? Why? 1.5 Write down the domain and range.
- The number of diagonals in a polygon is given by the following table: Number of sides (x)
3 4 5 6 7
Number of diagonals (y)
0 2 5
2.1 Complete the table. 2.2 Draw the graph. 2.3 Is there any restriction on the domain and range? 2.4 Write down the domain and range. 2.5 Determine the equation showing y in terms of x.
Consolidation & Terminology
The situations/relationships described in activity 1 represent functions. Use this information to explain to your learners in very simple terms what a function is. Even though this is handled in detail and formally assessed in grade 12, for grade 10 learners this is intuitive.
NB: On page 131 of the Report on the 2013 National Senior Certificate (Diagnostic report), it was noted that learners struggled to identify or define a function. They were not clear about terminology such as “one-to-many” and “many-to-many”.
Definition: A function is a rule by means of which each element of a first set, called the domain, is associated with only one element of a second set, called the range. Each element of the range is an image of corresponding elements of the range. Look at the following Venn diagrams.
If the vertical line cuts the graph at only one point, the graph represents a function; if it cuts at more than one point then the graph does not represent a function:
The vertical line cuts the graph at only one point so the graph will represent a function.
The vertical line cuts the graph at two points so the graph will not represent a function. (The circle and semi-circle is not prescribed in CAPS for this section. It is merely used to show the vertical line test.)
Vertical line
Vertical line
Worked Examples 1 & 2 : Functional Notation (10 mins)
The facilitator will now explain these suitable examples of functional notation.
- Remember to make notes as the facilitator is talking
- Ask as many questions as possible so as to clarify any misconceptions that may occur.
- Given that f x ( ) x^2 4 Determine the following: 1.1 f(1) 1.2 f(-3) 1.3 The value(s) of x if f x ( ) 0
Solution Once again, for 1.1 and 1.2 it is simple substitution:
(1) (1)^2
f
( 3) ( 3)^2
f
1.3 For this question please note that f x ( ) 0 is not the same as f (0). You may show learners this difference on a graph.
2 or 2
f x x x x x x
We note that in this question we used factorisation to obtain the values of x.
- Given the following function: f x ( ) x^2 9 Determine: 2.1 f (5) 2.2 The value(s) of x if (^) f x ( ) 2. 2.3 The domain of f.
- Use f x ( ) 2x 1 ;
2 (^) g x ( ) (^) x^ x 3 and h x ( ) 5 x to determine: 3.1 h(-4) – f(-1) 3.2 x if g(x) = 0 3.3 the domain of g
- Use g x ( ) (^) x^ x 32 to determine the domain of g.
Activity 5 : Types of Correspondence
Group organisation:
Time: Resources: Appendix:
Individual 5 min (^) Participants Handout None
- Study the graphs that follow and state whether they represent functions or not. Give your reason/s in terms of correspondence.
- Write your answers in the space provided.
(The circle and semi-circle is not prescribed in CAPS for this section. It is merely used to show the vertical line test.)
Misconceptions
Teachers should pay careful attention when working with functions which have restrictions as these affect the domain and range of the said function. Learners tend to ignore restrictions on certain functions.
Examples
- g x ( ) x 3 is a restricted function as x – 3 lies under the square root so x – 3 0, that is x 3
- h( x ) (^) x ^12 is also a restricted function as x + 2 cannot be 0 so x 2.
- The domain of f(^ x^ )^ ^2 x^^2 ^3 x ^1 is not restricted but its range is restricted.
Please make learners aware of the following: The difference between g(0) and g(x)= The difference between g(2) and g(x)=
Conclusion
When working with functions it is important to explain to learners how to get the correspondence. They should be able to look at any table, set of ordered pairs, equation or graph and state what the correspondence is. Functions are a major portion of the Mathematics paper 1 curriculum.
NB: In each lesson please ensure that the lesson incorporates the following: Kinaesthetic learning – learners are given hands-on activities/exercises to work through. Auditory learning – using one‟s voice effectively without confusing learners. Visual learning – learners must be able to see. In the cases of graphs, there should be some colour and the various features shown.
Sub-topic 2: Introduction to quadratic
functions
{( ; x y ) : y a x ( p )^2 q }
CAPS extraction indicating progression from grades 10-
Grade 10 Grade 11 Grade 12 Point-by point plotting of y x^2 Shape of functions, domain & range; axis of symmetry; turning points, intercepts on the axes The effect of a and q on functions defined by y a f x. ( ) q where f x ( ) x^2 Sketch graphs and find the equation of graphs and interpret graphs
Revise the effect of the parameters a and q and investigate the effect of p and the function defined by y a x ( p )^2 q Sketch graphs and find the equation of graphs and interpret graphs
Introduction
It is important to note that the quadratic function is also known as the parabola. Study the picture below:
This is an example of a parabola in real life. It is the St Louis Arch, which is located in the state of Missouri in the United States of America. Accessed from: http://www.tattoodonkey.com/parabolas-in-real-life-tattoo/2/
We can use simple substitution to obtain points on this function:
x^0 1 -^1 -^2 2 -^3
y x^20 1 1 4 4 9
We can plot these points on the Cartesian plane:
NB: The above sketch was drawn using GeoGebra
For the graph of y x^2 the y values are negative.
x^0 1 -^1 -^2 2 -^3
y x^20 -^1 -^1 -^4 -^4 -^9 -^9
Worked Examples 1- 3 : Simple sketches of quadratic
Functions(10 Mins)
The facilitator will now take you through a number of sketches of quadratic functions.
Example 1 1.1 Sketch the graphs of: 2 2 2 2
2
y x y x y x y x
y x
1.2 What do you observe?
Solution We can bring down the table and extend the number of rows:
x^0 1 -^1 -^2 2 -^3
y x^20 1 1 4 4 9
y 2 x^20 2 2 8 8 18
y 3 x^20 3 3 12 12 27
y 12 x^20 12122 2 4,5^ 4,
1 2 4
y x^01 4
Solution 2.1 We can bring down the table and extend the number of rows:
x^0 1 -^1 -^2 2 -^3 y x^20 -^1 -^1 -^4 -^4 -^9 -^9
y 2 x^20 -^2 -^2 -^8 -^8 -^18 -^18
y 3 x^20 -^3 -^3 -^12 -^12 -^27 -^27
1 2 2
y x^0 1 2
y 41 x^20 14 14 -^1 -^1 - 2,25^ - 2,
2.2 As a decreases the graph becomes narrower or stretches. As a increases the graph becomes wider or flatter.
Some properties of the function f {( ; x y ) : y ax^2 } When a 0 , it has a minimum value. When a 0 , it has a maximum value The y-axis (x = 0) is the axis of symmetry The turning point is (0;0) Domain = : ∈ 𝑅 When a 0 , the range = * : ≥ 0; ∈ 𝑅+ When a 0 , the range = : ≤ 0; ∈ 𝑅
Example 3 3.1 Sketch the graphs of 2 2 2
4 4
y x y x y x
Activity 1 - 3 : Basic exercise
Group organisation:
Time: Resources: Appendix:
Pairs 30 min (^) Graph Paper None
- Read each question carefully.
- Discuss the question with your partner and proceed to answer the questions. You may draw the graphs on the graph paper supplied and answer the questions alongside the graphs.
- Given the function: f {( ; x y ) : y 2 x^2 } 1.1 Sketch the graph of f. 1.2 If f is moved 3 units downwards, to form the graph of g, determine the equation defining graph g and its maximum value. 1.3 Describe how the graph of y x^2 may be transformed firstly to f and then to g.
We can move y = -2 down 2 units to get y = -4.
y 4 is a tangent to y x^2 4.
If we move y = -4 further down to say y= -5 then we have:
There are no points of intersection.
Question:
The graphs of y x^2 4 and y d are drawn. For what value(s) of d will there be: (a) Two points of intersection? (two roots) (b) One point of intersection? (one root) (c) No points of intersection? (no roots)
Solution We get the value(s) of d from the graphs (a) d 4 (b) d 4 (c) d 4
This question can be phrased in another way. See below: Use your graph to determine the value(s) of d for which the roots of x^2 4 d (a) Real and unequal (two points of intersection). (b) Real and equal (one point of intersection). (c) Non-real or imaginary (no points of intersection).
Misconceptions
Learners must know the following: If the y-intercept of a graph is 6, then the coordinate of the point is (0;6). If the x-intercepts are - 2 and 4, then the coordinates are (-2;0) and (4;0).
Conclusion
The work done in this module only included examples of parabolas which had the y-axis as the axis of symmetry. The graphs drawn were translated up and down the y-axis. These examples provide more than suitable preparation for the ones that follow in the next module.
