algebraic simplification

Use of electronic calculator

 

PROPERTIES OF QUADRILATRALS;

LESSON PLAN                          

MATHEMATICS                                                                                 

Term                                                                                                                Grade

Week                                                                                                                          Periods

From :

LESSON TITLE:  PROPERTIES OF QUADRILATRALS;

                               ANGLESUM PROPERTY OF TRIANGLE AND QUADRILATERALS

PRE-REQUISITE KNOWLEDGE:, angle at a point, angle in a straight line and  angles in parallel lines                                              

TEACHING AND LEARNING PROCESS                                               

PERIOD   1 Learning Objectives: Students will be able to                                         Appreciate the properties of quadrilaterals and use them to solve problems
Key Vocabulary:  parallel; diagonal; bisect; quadrilateral; parallelogram; square; rhombus; kite; trapezium
Key Questions: 1.      What are the properties of various quadrilaterals? 2.      How will you use them to solve problems?
TEACHER ACTIVITY	STUDENT ACTIVITY
Explains & Describes properties of quadrilaterals	Understands the properties of quadrilaterals
Explains the questions and helps the students in finding out the solution.	Tries to solve the question and asks if they have any doubt.
Gives more questions for practice and explains the questions so that the student can easily find out the solution using all the concepts	Understands and tries to find out the solution from the method taught by the teacher
Gives some more questions as home-work and some in the weekend assignment	Complete their work and ask the teacher if they have any doubt

                                                                                                                                                                                   

PERIOD   1 Learning Objectives: Students will be able to                                         Solve problems involving angles at a point, angles on a straight line
Key Vocabulary:  point, straight line, sum
Key Questions: 1.      What is the sum angles at a point? 2.      What is the sum of angles on a straight line?
TEACHER ACTIVITY	STUDENT ACTIVITY
Explains & Describes the sum angles at a point and sum of angles on a straight line	Understands the properties to solve problems.
Explains the questions and helps the students in finding out the solution.	Tries to solve the question and asks if they have any doubt.
Gives more questions for practice and explains the questions so that the student can easily find out the solution using all the concepts	Understands and tries to find out the solution from the method taught by the teacher
Gives some more questions as home-work and some in the weekend assignment	Complete their work and ask the teacher if they have any doubt

                                   

PERIOD   1 Learning Objectives: Students will be able to                                         Solve problems involving angles in parallel lines
Key Vocabulary:  alternate, allied, corresponding
Key Questions: 1.      How many angles can be created on parallel lines? 2.      How will you use parallel line properties to solve problems?
TEACHER ACTIVITY	STUDENT ACTIVITY
Explains & Describes alternate, allied, corresponding angles	Understands the properties to solve problems.
Explains the questions and helps the students in finding out the solution.	Tries to solve the question and asks if they have any doubt.
Gives more questions for practice and explains the questions so that the student can easily find out the solution using all the concepts	Understands and tries to find out the solution from the method taught by the teacher
Gives some more questions as home-work and some in the weekend assignment	Complete their work and ask the teacher if they have any doubt

                                                                                                           

PERIOD  5 Learning Objectives: Students should be able to know ·         Sum of interior angles of a triangle is 180˚ ·         Sum of interior angles of a Quadrilateral  is 360˚
Key Vocabulary:   sum of interior angles,
Key Questions: How many triangles can be formed from a quadrilateral simultaneously? What is the sum of exterior angles of a triangle/quadrilateral?
TEACHER ACTIVITY	STUDENT ACTIVITY
Asks the students about triangle and quadrilateral.	Tries to recall a triangle and quadrilateral
Asks about angle sum property of triangle and quadrilateral.	Recall  the angle sum property of triangle and a quadrilateral.
Shows that any quadrilateral can be drawn as combination of two triangles.	Recognize that one diagonal separates a quadrilateral into two triangles.

………………………………………….

                  (Subject Teacher)

FUNCTIONS

FUNCTIONS


1. The functions f,g and h are defined by
f : x--->4x-5
g : x---> 2x2
h : x--->x6+4x

a) find the value of (i) f(4) (ii)g(-3) , (iii) h(1 /2)

b) solve the equations (i) f(x) = 3, (ii) g(x) = 32, (iii) h(x) = 7

c) express the composite functions (i) gf (ii)fh in the form of gf  : x---> ……., fh : x--->
you need not simplify your answers
d) express the inverse f -1and h -1 in the form of f -1: x---> , h -1 : x--->

e) show that the equation g(x) = 5[f(x)] + 7 can be written in the form ax2 + bx + c = 0 and state the values of a,b and c.

2. f and g are defined by
f : x--->4x –1 and g : x---> 2x2 + 3

a) find the value of f(2)

b) express the inverse of the function f in the form f -1: x--->
c) express fg in the form fg : x--->
d) solve the equation fg(x) = f(x) + 24.

3. The functions f and g are defined by
f : x---> 2x2 + 1, g : x---> 7x – 5
a) find the value of x for which f(2) = g(x)
b) copy and complete the following
i) fg : x---> ii) inverse of g : x--->
c) find the values x for which f(x) = g (x)


4. Given that f : x---> x2 +3
a) find f(4), b) complete and simplify the statement ff : x--->

5. f : x ---->3x2 + 1 , g ; x 2x –1
copy , complete and simplify the following
a) gg : x---> b) gf : x---> c) g -1 : x--->

6. f : x---> 5x + 4, g : x---> 4x + 3
a) find g(2)
b) solve, for x, the equation f(x) = g(2)
c) copy and complete the following, simplifying as appropriate
i) f -1: x---> ii) ff : x--->
d) show that f (g(x)) = g (f(x))
e) solve the equation g(x) = 2 giving your answers to 2 decimal places.

7. The functions f, g and h are defined by
f : x---> 3x + 2, g : x---> x2 – 4, h : x---> x + 1
a) find the value of i) f(2) ii) g(-3) iii) h(1/4)
b) solve the equation i) f(x) = 12 ii) f(x) = h(x)
c) express the composite function gf in the form gf : x ….,simplify your answer.
8. f : x x2 – 5x≠
a) write down the value of f (-2)
b) write down the value of ff(2)
c) find the range of f when the domain is {-1, 0, 1}

9. The functions f and g are defined by
f : x---> 2x – 1 , g : x---> x2 + 8,
a) find the values of i) f(3) ii) g(-2)
b) solve the equations i) g(x) = 12 ii) g(x) = 4 f(x)
c) Express the composite functions i) fg, ii) gf in the form fg : x--->, gf : x--->
d) Show that the equation fg(x) = gf(x) can be expressed in the form x2 + bx + c = 0. Where b and c are integers, and state the values of b and c.
e) Express the inverse function, f –1, in the form, f –1 : x--->


10. The functions f,g and h are defined by f : x 2x + 1, g : x 3x2 , h : x , x ≠ 1
a) find the value of i) g (-4) ii) h(1 /2)
b) solve the equations i) g(x) = 75 ii) h(x) = 4, iii) f(x) = g(x)
c) express hf(x) in terms of x, simplifying your answer.


11. f : x---> , x ≠ -2

Given that f(x) = 3, a) find x b) find f (f(4))
Given that the domain of f is the set s = {-1,0,1}
c)find the range of s under f.

12. f : x---> x2 + 3x – 4 , g : x---> x +1
a) find i) f(-2) ii) g -1(1/2 )
b) solve the equation fg(x) = 0
c) solve the equation fg(x) = gf(x)


13. f : x---> 1 – 3x , g : x x ≠ 0

find the values of a) f(3) b) gf(1)


14. f : x---> 2x2 – 3 , g : x---> 3x + 5, copy , complete and simplify
a) gg : x---> b) fg : : x---> c) g -1 : x--->

15. f : x---> x2 – 3x – 5, g : x---> 2x + 1
a) find the value of I) f(2) ii) fg(-1 /2 )
b) express the inverse function g -1 in the form g -1 : x--->
c) express the composite function fg in the form fg : x--->
simplifying your answer
d) find the values of x for which f(x) = g(x)

16. The functions f and g are defined by
f : x---> 2x – 3, g : x , x ≠ 0

a) find the value of I) f(3), ii) gf(2)
b) find and simplify I) f -1 : x---> ii) fg : x--->
c) solve the equation fg(x) = gf(x)


FUNCTIONS ( continued)

17. The functions f and g are defined by f: x---> 3x – 2 , g : x--->2x+8 , x ≠ 2
a) find the value of I) f(2) ii) gf(10)
b) express
i) the inverse function g -1, in the form g -1 : x--->
ii) the composite function fg, in the form fg : x--->
c) solve the equation f(x) = g(x)


18. The function f is defined by f : x , x ≠3
Given that f(x) = 2
a) find the value of x
b) find the value of ff(0).

19. f : x 3x – 4 , g : x x2 + 1
copy, complete and simplify
a) fg : x ……. b) gf : x …….
c) find the two values of x for which fg(x) = gf(x)

20. Given that f : x 3x –1 ,
a) evaluate f(-1/3), b) find f -1 (x)

21. Write down the range of each of the following functions
a) f : x x2 , -3< x < 3,
b) g : x sinxo 0 < x < 90


22. f : x a 5x + 3, g : x a + 15

a ) calculate i) f (3) , ii) fg(4)
b) solve i) f(x) = 1, ii) f(x) = g(x)

23. (i) Given that f : x 6 – x2
a) state the maximum value of f(x).
b) write down the range of f.
(ii) Given that g : x , state the value of x which must be excluded from the domain of g

24. The functions f and g are defined by f : x 1 – x, g : x 2x2 + 3,
a) find the values of I) f(-2) ii) g( 2)
b) Express the inverse function f -1 in the form , f -1 : x ……
c) Express the composite function gf in the form gf : x ………simplifying your answer
d) solve the equation
i) g(x) = 53 ii) ff(x) = f(x)

25. f : x x 2 – 1, g : x 3x – 4 ,
a ) complete the following statements, simplifying your answers where appropriate
i) g-1 : x …….. ii) fg : x ……..

b) find the values of x which satisfy the equation fg(x) = 9 – 3x

FUNCTIONS

FUNCTIONS


The idea of functions is used in almost every branch of Mathematics.
The two common notations are
a) f(x) = x2 + 4
b) f : x2 + 4

Some more notations used in functions are as follows

Simple Functions : f(x)
Inverse Functions : f-1(x)
Composite Functions : fg(x) or f(g(x))

Grade 9
Example:-
1) Given f(x) = 3x - 1 and g(x) = 1 – x2
Find the following:-
a) i) f(2) = 5 b) i) g(2) = - 3
ii) f(0) = - 1 ii) g(-2) = -3
iii) f(-3) = -10 iii) g(1/2) = ¾
iv) f(-x) = -3x – 1 iv) g(m) = 1 – m2
v) f(k) = 3k + 1 v) g(1/t) = 1 – 1/t2

Exercise

1) Given the functions h : x x2 + 1 and g: x 10x + 1
Find the following:-

a) h(2) b) h(-1) c) h(0)
d) h(+5) e) h(1/2) f) g(k)
g) g(-m) h) g(k) i) g(1/10)
j) g(-5)



2) Given the functions f(x) = 2x – 4 , g(x) = , h(x) = (7 – 3x)2
Use the above functions to find the following

a) f -1(x)
b) f -1(8)
c) g -1(x)
d) g -1(16)
e) h-1(x)


3) Given the functions f(x) = 2x2/3 , g(x) =10 – x2 , h(x) = 2x2 + 1
Use the above functions to solve the following
a) f(x) = 5
b) f(x) = x
c) h(x) = 0
d) g(x) = 6
e) g(x) = h(x)
f) h(k) = k + 1

Grade 10

Take all the above questions also for Gr 10
These questions are also included

4) Given the functions f: x x/4 , h : x x2 + 1 and g: x 10x + 1
Find the following

a) fg(2) b) gh(3) c) f-1g(4)

d) f(x) = g(x) e) h-1f(-2) f) hg(5)


Past Paper Questions

1. (P2May/June- 2001)
Given f(x) = for x>0 and g(x) = 3 – 3x for any value of x.
a) find f( ) , giving your answer as a fraction
b) if f(x) = g(x), find the value of x
c) find f-1(x) and g-1(x)
d) g-1(18)



2.(P2, May/June 2001)


f(x) = x1/3 and g(x) = 2x2 – 5 for all values of x.
a) Find
i) g(4)
ii) f(27)
b) Find an expression for g-1(x) in terms of x.
c) Find f1(x)


3.



a) calculate i) ii)



b) find and simplify as a single fraction
of the composite function
find the inverse h-1, of the function.
c) solve for x if
Give your answer , correct to, 2 decimal places.


4. The functions f(x) and g(x) are defined as follows.

a) find the value of g (- 4)
b) find and simplify as a single fraction f(x) – g (x).
c) find an expression for f -1.
d) find an expression for g -1.