teachers guide for general mathematics

7/25/2019 Teachers Guide for General Mathematics

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FIRST

QUARTER

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Weekb

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7

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Review/E

xam

Weekd

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11

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Review/Exam

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15

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Review/E

xam

Weekh

Lesson17

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19

Lesson19,20

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eeki

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21

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eekj

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Review/E

xam


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SECOND

QUARTER

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Lesson

27,

Review

Review/E

xam

W

eekc

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29

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eekd

Lesson29

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xam

W

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33

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eekf

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37

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eekh

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39

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eeki

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41

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College Readiness Standards

Foundational Skills

DepEd Functional Skills

Produce all forms of texts (e.g., written, oral,

visual, digital) based on:

1. Solid grounding on Philippine experience

and culture;

2.

An understanding of the self, community,and nation;

3. Application of critical and creative

thinking and doing processes;

4. Competency in formulating

ideas/arguments logically, scientifically,

and creatively; and

5. Clear appreciation of ones responsibility

as a citizen of a multicultural Philippines

and a diverse world

Visual and information literacies

Media literacy

Critical thinking and problem solving skills

Creativity

Initiative and self-direction


Foundational Skills


Systematically apply knowledge, understanding,

theory, and skills for the development of the self,

local, and global communities using prior

learning, inquiry, and experimentation

Global awareness

Scientific and economic literacy

Curiosity

Critical thinking and problem solving skills

Risk taking

Flexibility and adaptability

Initiative and self-direction


Foundational Skills


Work comfortably with relevant technologies and

develop adaptations and innovations for

significant use in local and global communities;

Global awareness

Media literacy

Technological literacy

Creativity

Flexibility and adaptability

Productivity and accountability


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Foundational Skills


Communicate with local and global communities

with proficiency, orally, in writing, and through

new technologies of communication;

Global awareness

Multicultural literacy

Collaboration and interpersonal skills

Social and cross-cultural skills

Leadership and responsibility


Foundational Skills


Interact meaningfully in a social setting and

contribute to the fulfillment of individual and

shared goals, respecting the fundamental humanity

of all persons and the diversity of groups andcommunities

Media literacy

Multicultural literacy

Global awareness

Collaboration and interpersonal skills

Social and cross-cultural skills

Leadership and responsibility

Ethical, moral, and spiritual values


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(x, y)

(x, y)

x

y


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f= {(1, 2), (2, 2), (3, 5), (4, 5)} g= {(1, 3), (1, 4), (2, 5), (2, 6), (3, 7)}

h= {(1, 3), (2, 6), (3, 9), . . . , (n, 3n), . . .}

f h x

y g (1, 3) (1, 4)

x y

f

g


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h

f g x X y Y h X

y x = 7 y = 11 13 x = 2

y= 17 19

x = a (a, b) (a, c)

y x= a


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(a) (b) (c) (d) (e)

x

y x

x y


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y= 2x + 1

y= x2 2x + 2 x2 + y2 = 1

y=

x + 1

y=2x + 1

x 1 y= x + 1 x

x y x = 0 y +1

1

x

R

R

[1, 1]

[1, +)

(, 1) (1, +) R

y f(x) y

x f

f(x) = 2x + 1

q(x) =x2 2x + 2 g(x) =

x + 1

r(x) =2x + 1

x 1 F(x) = x + 1 x


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C x

40

40 C(x) = 40x

A x

A = xy x

x+ 2y = 100 y = (100 x)/2 = 50 0.5x A(x) =x(50 0.5x) = 50x 0.5x2

300

1

m


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t(m)

t(m) =

300 0< m 100300 + m m >100

8.00

1.50

d

F(d)

F(d) =

8 0< d 48 + 1.5d d >4

d d d 4.1 = 4.9 = 4

25 0 0

100

100

100

T(x)

T(x)


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15

1, 000

400

f(x) =

1000 0 x 31000 + 400x 3 x >3

700


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f(x) = 700x4 x N

150

130 110

100

f(x) =

150x 0 x 20130x 21 x 50110x 51 x 100100x x >100

x N


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x

f a

a f f(a)

x= 1.5

f(x) = 2x + 1

q(x) =x2 2x + 2 g(x) = x + 1

r(x) =2x + 1

x 1 F(x) = x + 1 x

1.5 x

f(1.5) = 2(1.5) + 1 = 4

q(1.5) = (1.5)2 2(1.5) + 2 = 2.25 3 + 2 = 1.25 g(1.5) = 1.5 + 1 = 2.5

r(1.5) =2x + 1

x 1 =2(1.5) + 1

(1.5) 1 =3 + 1

0.5 = 8

F(1.5) = x + 1 = 1.5 + 1 = 1 + 1 = 2

g(4) r(1) g r

4 g(x) r(x)

f q

f(3x 1) q(2x + 3)


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f(3x 1) x f(x) = 2x + 1 (3x 1)

f(3x 1) = 2(3x 1) + 1 = 6x 2 + 1 = 6x 1 q(3x + 3) x q(x) =x2

2x + 2 (2x + 3)

q(2x + 3) = (2x + 3)2 2(2x + 3) + 2 = (4x2 + 12x + 9) 4x 6 + 2 = 4x2 + 8x + 5

f(x) =x 2

f(0) 2 f(3)

f(1)

f() 2 f(x + 1) x 1

f(3x) 3x 2

f(x) = 4

x

f(1) 4

f(2) 2

f(1) 4 f(

2) 2

2

f(1/x) 4x

f(2x) 2/x

f(x) =

x 3 f(3) 0

f(4) 1

f(12) 3

f(x 3) x 6 f

1

1 x

3x 21 x

f(x2 + 4x + 7)

x2 + 4x + 4 |x + 2|


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200

25

C(x) = 25x +200 x C(x)

2700 3950

t s(t) = 5t2 + 100 g= 10m/s2


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13

25

1

3+

2

5=

5

15+

6

15=

5 + 6

15

=11

15

1

x 3 2

x 5

(x 3)(x 5) (x2 8x + 15) 1

x 3+ 2

x 5= x 5

x2 8x + 15+ 2(x 3)

x2 8x + 15 =x 5 + 2x 6

x2 8x + 15=

3x 11x2 8x + 15

10

21

15

8

10

2115

8 =

2 53 7

3 52 2 2=

2 5 3 53 7 2 2 2=

25

28


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x2 4x 5x2 3x + 2

x2 5x + 6x2 3x 10

x2 4x 5x2 3x + 2

x2 5x + 6x2 3x 10=

(x + 1)(x 5)(x 2)(x 1)

(x 2)(x 3)(x 5)(x + 2)

=(x + 1)

(x 5)

(x 2)(x 3)

(x 2)(x 1)

(x 5)(x + 2)

=(x + 1)(x 3)(x 1)(x + 2) =

x2 2x 3x2 + x 2

f g

f+ g (f+ g)(x) =f(x) + g(x)

f g (f g)(x) =f(x) g(x)

f g (f g)(x) =f(x) g(x)

f /g (f /g)(x) =f(x)/g(x)

x g(x) = 0


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f(x) =x + 3

p(x) = 2x 7

v(x) =x2 + 5x + 4

g(x) =x2 + 2x 8 h(x) = x + 7

2 x t(x) = x 2

x + 3

(v+ g)(x)

(fp)(x) (f+ h)(x)

(p f)(x) (v/g)(x)

(v+ g)(x) =

x2 + 5x + 4

+

x2 + 2x 8= 2x2 + 7x 4 (fp)(x) = (x + 3) (2x 7) = 2x2 x 21

(f+ h)(x) = (x + 3) +x + 7

2 x = (x + 3) 2 x2 x +

x + 7

2 x =(x + 3)(2 x) + (x + 7)

2 x =

=6 x x2 + x + 7

2 x =13 x2

2 x =13 x2

2 x 11=

x2 13x 2

(p f)(x) = (2x 7) (x + 3) = 2x 7 x 3 =x 10

(v/g)(x) = (x2 + 5x + 4) (x2 + 2x 8) = x2 + 5x + 4

x2 + 2x 8

f(x) = 2x + 1 q(x) =x2 2x + 2 r(x) =2x + 1x 1

f1(x) =x2 + 3

q(x) f(x)

x2 + 3

q(x) + f(x) = (x2 2x + 2) + (2x + 1)=x2 + 3

=f1(x)

f1(x) =q(x) + f(x)

f2(x) =x24x + 1


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q(x) f(x) x2 4x + 1 q(x) f(x) = (x2 2x + 2) (2x + 1)

=x2 4x + 1

=f2(x) f2(x) =q(x) f(x)

f3(x) =2x2 + x

x 1

2x2 + x

x 1 x 1 r(x) = 2x + 1

x 1 f(x) r(x)

f(x) + r(x) = 2x + 1 +2x + 1

x 1

=(2x + 1)(x

1)

x 1 +2x + 1

x 1=

(2x + 1)(x 1) + (2x + 1)x 1

=(2x2 x 1) + (2x + 1)

x 1=

2x2 + x

x 1=f3(x)

(f+ g)(x) =f(x) + g(x)

f1(x) =q(x) + f(x) = (q+ f)(x)

f2(x) =q(x) f(x) = (q f)(x) f3(x) =f(x) + r(x) = (f+ r)(x)

g1(x) = 2x3 3x2 + 2x+ 2


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2x3 3x2 + 2x + 2 f(x) q(x)

f(x) q(x) = (2x + 1)(x2 2x + 2)= (2x)(x2 2x + 2) + (x2 2x + 2)

= (2x3

4x2

+ 4x) + (x2

2x + 2)= 2x3 3x2 + 2x + 2=g1(x)

g1(x) =f(x) q(x)

g2(x) = x 1

r(x) = 2x + 1

x 1 x 1 2x+ 1

f(x)

r(x)

f(x)

r(x) = (2x + 1) 2x + 1

x 1= (2x + 1) x 1

2x + 1

=2x + 1

2x + 1 (x 1)

=x 1=g2(x)

g2(x) =

f(x)

r(x)

g3(x) = 1

x 1

g3(x) = 1

x 1 r(x) = 2x + 1

x 1 2x + 1 r(x) f(x) = 2x + 1

r(x)

f(x)=

2x + 1

x 1 (2x + 1)

=

2x + 1

x 1 1

2x + 1

= 1

x 1=g3(x)

g3(x) = r(x)

f(x)


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f(x) = 2x + 1

q(x) =x2 2x + 2 (2x + 1)2 2(2x + 1 ) + 2

f g (f g)

(f g)(x) =f(g(x)).

f(x) = 2x + 1 q(x) =x2 2x + 2 r(x) =2x + 1

x 1

g(x) = x + 1 F(x) = x + 1

(g f)(x)

(g f)(x) =g(f(x))=

f(x) + 1

=

(2x + 1) + 1

=

2x + 2

(q f)(x) (f q)(x)


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(q f)(x) =q(f(x))= [f(x)]2 2 [f(x)] + 2

= (2x + 1)

2

2(2x + 1) + 2= (4x2 + 4x + 1) (4x + 2) + 2= 4x2 + 1

(f q)(x) =f(q(x))= 2(x2 2x + 2) + 1= 2x2 4x + 5

(q f)(x) (f q)(x)

(f r)(x)

(f r)(x) =f(r(x))= 2r(x) + 1

= 2

2x + 1

x 1

+ 1

=4x + 2

x 1 + 1

=(4x + 2) + (x 1)

x 1=

5x + 1

x 1

(F r)(5)

(F r)(5) =F(r(5))= r(5) + 1

=

2(5) + 1

5 1

+ 1

=

11

4

+ 1 = 2 + 1 = 3


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f g f+g f g f g f /g g/f

f(x) =x + 2 g(x) =x2 4 x2 + x 2 x2 + x + 6 x3 + 2x2 4x 8 1

x 2 x 2

f(x) =

x 1 g(x) =x2 + 4

x 1 + x2 + 4 x 1 x2 4 x 1(x2 + 4)

x 1

x2 + 4 x 2

f(x) =x 2x + 2

g(x) = 1

x

x 2x + 2

+1x

x 2x + 2

1x

x 2x(x + 2)

x(x 2)x + 2

x + 2x(x 2)

f(x) = 1

x + 2 g(x) =

x 2x

1

x + 2+

x 2x

1

x + 2x 2

x

x 2x(x + 2)

x

(x + 2)(x 2) (x + 2)(x 2)

x

f(x) = 1

x2 g(x) =

x

1

x2+

x

1

x2 x

x

x2

1

x2

x x2

x

f(x) =x2 + 3x g(x) =x 2 f g g f f f g g x2 x 2 x2 + 3x 2 x4 + 6x3 + 12x2 + 9x x 4

(f g) = x (g f) =x

f(x) = 3x 2 g(x) = 13(x + 2) f(x) = x

2 x g(x) = 2xx 1 f(x) = (x 1)3 + 2 g(x) = 3x 2 + 1


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p(x) =anxn + an1x

n1 + an2xn2 + + a1x + a0

a0, a1, . . . , an R an= 0 n a0, a1, a2, . . . , an

an anxn

a0

100, 000

y

x y=100, 000

x

x

y

750 g(x)

g(x) =100, 000

x + 750

x

y
https://www.geogebra.org/downloadhttps://www.desmos.com/calculator


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f(x) = p(x)

q(x) p(x) q(x)

q(x) q(x)0 f(x) x q(x)

= 0

v v(t)

t

v t

t

v

v(t) =10

t v t

c(t) = 5t

t2 + 1

t c(t)

t = 1, 2, 5, 10

t

c(t)


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a b c

x y
http://illuminations.nctm.org/Lesson.aspx?id=1968


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x

y

y = a

x b +c

x P(x)


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P(x) =2x2 + 800

x

b(t) = 50t

t + 1 0 t 20

t b(t)

t= 1, 2, 5, 10, 15, 20


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x2 + 3x + 2

x + 4

1

3x2

x2 + 4x 32

x + 1

x3 1

1

x + 2x 2

1

(x + 2)(x 2)

f(x) = p(x)

q(x)

p(x) q(x)

q(x)

2

x 3

2x=

1

5

5

x 3 2

x f(x) =

x2 + 2x + 3

x + 1

y=x2 + 2x + 3

x + 1

x

x y


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15x

1

5x4 6x7 + 1

5 x3x

y= 5x3 2x + 1

8

x 8 = x

2x 1

x 2 = 4

x 1x + 1

=x3

y=7x3 4x + 1

x2 + 3

6x 5x + 3

0


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x + 1

2x = 10

x + 1

2x 10

x 2

x 3

2x=

1

5

10x 10x


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10x

2

x

10x

3

2x

= 10x

1

5

20

15 = 2x

5 = 2x

x=5

2

x x

x + 2 1

x 2= 8

x2 4

x

x + 2 1

x

2

= 8

(x

2)(x + 2)

(x2)(x+2)

(x 2)(x + 2) xx + 2

(x 2)(x + 2) 1x 2= [(x 2)(x + 2)]

8

(x 2)(x + 2)

(x 2)x (x + 2) = 8x2 3x 10 = 0

x2

3x 10 = 0(x + 2)(x 5) = 0

x + 2 = 0 x 5 = 0x= 2 x= 5

x= 2 x= 5


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x

x

12 + x 25 + x

12 + x

25 + x= 0.6

25 + x

12 + x

25 + x = 0.6

12 + x= 0.6(25 + x)

12 + x= 0.6(25) + 0.6x

x 0.6x= 15 120.4x= 3

x= 7.5

x

60%

v= dt

v= dt t= dv

v v+ 10 5v

5v+10

43

5

v+

5

v+ 10=

4

3

3v(v+ 10)

5

v+

5

v+ 10=

4

3


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3v(v+ 10) 5v

+ 3v(v+ 10) 5v+ 10

= 3v(v+ 10) 43

15(v+ 10) + 15v = 4v(v+ 10)

30v+ 150 = 4v2 + 40v

4v2 + 10v 150 = 02v2 + 5v 75 = 0

(2v+ 15)(v 5) = 0v = 15

2 v = 5

v

(a, b) {x|a < x < b} a b[a, b] {x|a x b} a b[a, b) {x|a x < b} a b(a, b]

{x|a < x

b} a b

(a, ) {x|a < x} a

[a, ) {x|a x} a

(, b) {x|x < b} b

(, b] {x|x b} b

(, ) R


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x

2x

x + 1 1

2x

x + 1 1 0

2x (x + 1)x + 1

0x

1

x + 1 0

x= 1 x= 1 x = 1

x= 1

1


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1 1 x 1x + 1

x < 1 1< x 1 x= 2 x= 0 x= 2x 1 +x + 1 + +x 1x + 1

+ +

x < 1 x 1

1

{x R|x


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x < 1 1< x x


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8

x2 > x

8

x2 x >0

8 x3x2

>0

(2 x)(x2 + 2x + 4)x2

>0

x = 16 x= 28 x = 0 x= 4

4

x


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1120

x

1600

x + 4

1600

x + 41120

x 10

1600

x + 41120

x 10

160

x + 4112

x 1

160x + 4

112x

1 0160x 112(x + 4) (x2 + 4x)

x(x + 4) 0

160x 112x 448 x2 4xx(x + 4)

0

x2 44x + 448x(x + 4)

0(x 16)(x 28)

x(x + 4)

0

x= 16 x = 28 x= 0 x = 4

x 2 2+

x 2

x 3 2.5 2.1 2.01 2.001 2.0001 x 2f(x) f(x)

f(x) + x 2 f(x) x 2

x 2+

x 1 1.5 1.9 1.99 1.999 1.9999 x 2+f(x) 3 7 39 399 3999 39999 f(x)


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f(x) x 2+ f(x) x 2

x 2 x= 2

x= a f f

x a

a

x= a


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x

f(x) x x +

f(x) x +

x x +f(x) f(x) 1

f(x) x x f(x) x

x 5 10 100 1000 10000 x f(x) f(x) 1+

x f(x)

y= 1

y = b f f(x)

b x x + x

y= b b


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x 2 x + x

x

x =2 x= 2

x < 2 2< x 2 x= 3 x= 0 x= 3x 2 +x + 2 + +x 2x + 2

+ + x x x

x x= 2

y

f(x)< 1 x + f(x)> 1 x (2, 0)

f(x)

x 2

f(x)

x 2+

f(x)


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y=x 2x + 2

y = 1 f(x)


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f(x) (, 1) (1, +)

f(x) =4x2 + 4x + 1

x

2

+ 3x + 2

x x x +

x

x= 1000 4x2 + 4x + 1 4, 004, 001

4x2 4, 000, 000

x x2 + 3x+ 2 x2

x

f(x)

4x2

x2 = 4

f(x) x y= 4

f(x) = 2x2 53x2 + x 7

2x2 53x2 + x 7

2x2

3x2 =

2

3 x y= 23

f(x) = 3x + 4

2x2 + 3x + 1

3x + 4

2x2 + 3x + 1

3x

2x2 =

3

2x x x

3

2x

0 y= 0

f(x) = 4x3 1

3x2 + 2x 5

x

4x3

1

3x2 + 2x 5 4x3

3x2 =4x

3

x 4x

3 x

y


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n m

n < m y= 0

n = m y = ab a b

n > m

x

y

y

y

x= 0

x x

x

f(x) =3x2 8x 32x2 + 7x 4

x

f(x) (, 4) (4, 12) (12 , +)

f(x)

f(x) =3x2 8x 32x2 + 7x 4=

(3x + 1)(x 3)(2x 1)(x + 4)

y f(0) = 0 0 30 + 0 4=

3

4

x 3x + 1 = 0 x= 13 x 3 = 0 x= 3


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2x 1 = 0 x= 12 x + 4 = 0 x= 4

y= 32

f(x)

x

4, 13 , 12 3

x < 4 4< x < 13

13

< x < 12

1

2< x 3

x= 10 x= 2 x= 0 x= 1 x= 103x + 1 + + +x 3 +2x

1

+ +

x + 4 + + + +(3x + 1)(x 3)(2x 1)(x + 4) + + +

x x x x x


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y

x


82/342

R

x y

x


83/342

f(x) = 2

x + 1

f(x) = 2

x2 + 2x + 1

f(x) = 3x

x + 3

f(x) =2x + 3

4x 7

f(x) =(4x 3)(x 1)(2x + 1)(x + 1)

f(x) =(5x 2)(x 2)(3x 4)(x + 2)

f(x) = x2 x + 6x2

6x + 8

f(x) =x2 4x 5

x 4

f(x) = x 1x3 4x

f(x) =x2 9x2 + 4

N(t) t

N(t) = 75t

t + 5 t 0.

N

N(t) t t N(t)75

c

t

c(t) = 20t

t2 + 2 t 0.

c(t) c(t) t t c(t) 0

x = 3 x =3 y = 1 x 5 y 59 f(x) =

(x 5)2(x2 + 1)(x + 3)(x 3)(x2 + 5)


84/342

p = 5125000V2 449000V+ 19307

125V2(1000V 43) p V

p

V

V = 0 V = 43/1000

p= 0

p

V

V = 0.043


85/342


86/342

TC=5

9(TF 32)

TF =9

5TC+ 32

F

C


87/342

200F 93.33C

120C 248F

f x1 x2 f f(x1) =f(x2) y x

2


88/342

d

F(d) =

8.00 0< d 4(8.00 + 1.50 d) d >4

d d

F(3) = 8

F(3) =F(2) =F(3.5) = 8 F

y x
https://www.world-airport-codes.com/


89/342

y x 2

y= x24


90/342

y= 2x 1

x 4 3 2 1 y 9 7 5 3 1

y x

x y

x 9 7 5 3 1 y 4 3 2 1

x

y


91/342

x 4 3 2 1 y 1 1 1 1 0

y

x y= 1 x= 1, 2, 3, 4

x y

x 1 1 1 1 0 y 4 3 2 1

x= 1 y

x y

x y

f A B f

f1 B A f1(y) = x

f(x) =y y B

x y

y


92/342

x

y= f(x)

x y

y x

f(x) = 3x + 1

y= 3x + 1

x y x= 3y+ 1

y x

x= 3y+ 1

x 1 = 3yx 1

3 =y = y= x 1

3

f(x) = 3x + 1 f1(x) =x 1

3

f(f1(x)) f1(f(x))

f(x) f1(x)

f1(x) f(x)

f(f1(x)) =x x f1

f1(f(x)) =x x f


93/342

g(x) =x3 2

y= x3 2 x y x= y3 2 y x

x= y3 2x + 2 =y3

3

x + 2 =y = y= 3x + 2 g(x) =x3 2 g1(x) = 3x + 2

f(x) =2x + 1

3x 4

y=2x + 1

3x 4

x y x=2y+ 1

3y

4 y x

x=2y+ 1

3y 4x(3y 4) = 2y+ 13xy 4x= 2y+ 13xy 2y= 4x + 1 y

y

y(3x

2) = 4x + 1

y=4x + 1

3x 2

f(x) f1(x) =4x + 1

3x 2


94/342

f(x) =x2 + 4x 2

y= x2 + 4x 2

x y x= y2 + 4y 2 y x

x= y2 + 4y 2

x + 2 =y2 + 4y

x + 2 + 4 = y2 + 4y+ 4

x + 6 = (y+ 2)2

x + 6 =y + 2x + 6 2 =y = y= x + 6 2

y =x + 6 2 x y x = 3 y 1 5 f(x) =x2 + 4x 2

f(x) = |3x|

y= |3x|

f1 f

f(1) =f(1) = 3 x 1 y f

y= |4x| x y x= |4y|


95/342

y x

x= |4y|x= (4y)

2 |x| =

x2

x2 = 4y2

x2

4 =y2

x2

4 =y = y=

x2

4

x = 2 y = 1 y =1 y =

x2

4

f(x) = |3x|

k(t) = 59(t32)+273.15 t

k= 59(t 32) + 273.15 k t

t k

k=5

9

(t

32) + 273.15

k 273.15 =59

(t 32)9

5(k 273.15) =t 32

9

5(k 273.15) + 32 =t = t= 9

5(k 273.15)

t(k) = 95(k 273.15) k


96/342

f(x) =1

2x + 4 f1(x) = 2x 8

f(x) = (x + 3)3 f1(x) = 3x 3 f(x) =

3

x 4 f1(x) =

4x + 3

x

f(x) =x + 3

x 3 f1(x) =

3x + 3

x 1 f(x) =

2x + 1

4x 1 f1(x) =

x + 1

4x 2

f(x) = |x 1|


97/342

y= x

y = x

y= x

y =x

y= x
http://artforkidshub.com/5-free-symmetry-art-activity/


98/342

y= x

x y

f

f1 f1(f(x)) =x f1(x) y f(x) x

x f(x) y f

1(y) x

y= x

y = f1(x) y = f(x) = 2x+ 1

{x |2 x 1.5} f(x)


99/342

y= 2x + 1 y= x

{y R | 3 y 4}

f1(x) =x 1

2

f1(x) = [3, 4] f1(x) = [2, 1.5]


100/342

f(x) f1(x)

[2, 1.5] [3, 4] [3, 4] [2, 1.5]

f(x) = 1

x

f(x) = 1

x y = x

y= x f(x) f1(x) =f(x)


101/342

f1(x) =f(x) = 1

x

f(x) = 3

x + 1

f(x) y= x


102/342

f(x) =

3

x + 1

y = x

3

1

f1(x) =x3 1

f(x) = 5x 1x + 2


103/342

f(x) = (, 2) (2, ) f(x) = (, 5) (5, )

x= 2

y= 5 y= x

x

y

y= x

x= 5 y= 2

f(x) f1(x)

(, 2) (2, ) (, 5) (5, ) (, 5) (5, ) (, 2) (2, )


104/342

y= x

y= x

f1

1(x) =f(x)

f(x) = (x + 2)2 3 2 =3(x + 2)2

2

x 0


105/342

x 0 x y x=

3(y+ 2)2

2 , y 0

y x

x=3(y+ 2)2

22x

3 = (y+ 2)2

2x

3 =y + 2 y 2

2x

32x

3 2 =y = y=

2x

3 2

x= 54

f1(54) =

2(54)

3 2 =

108

3 2 =

36 2 = 6 2 = 4

t

d

t(d) = 12.5d3

t=

12.5

d

3

d t

d t

t= 12.5d3

3

t=12.5

d

d=12.5

3

t

d(t) =12.5

3

t


106/342

t= 6.5 d(6.5) = 22.53

6.5= 12.06

12.06

f(x) = x2 + 1

{0, 0.5, 1, 1.5, 2, 2.5, 3}

x 0 0.5 1 1.5 2 2.5 3

f(x) 1 1.25 2 3.25 5 7.25 10 f

x 1 1.25 2 3.25 5 7.25 10

f1(x) 0 0.5 1 1.5 2 2.5 3 A ={(4, 4), (3, 2), (2, 1), (0, 1), (1, 3), (2, 5)}

A1 = {(4, 4), (2, 3), (1, 2), (1, 0), (3, 1), (5, 2)}

f(x) = 2

x 2 + 3 [2, )

[3, ) [2, )


107/342

f(x) =3x + 2

x 4 (, 3) (3, )

(, 4) (4, )

x= a

f1

(a) = 0 1/2

w l

w= (3.24 103)l2

l 0

l=

w/(3.4 103)


108/342


109/342

n

s n= 2s

b f(x) = bx y = bx

b >0 b

= 1

x = 3, 2, 1, 0, 1, 2 3

y=13

x y= 10x y= (0.8)x


110/342

x 3 2 1

y=13

x

1

3

1

9

1

27

y= 10x 11000

1100

110

y= (0.8)x

f(x) = 3x f(2) f(2) f12 f(0.4) f()

f(2) = 32 = 9

f(2) = 32 = 132

=1

9

f

1

2

= 31/2 =

3

f(0.4) = 30.4 = 32/5 = 5

32 = 5

9

3.14159 3 f() = 3 33.14 33.14159

3

3

b

b

g(x) =a bxc

+ d

a c d


111/342

t t= 0

t

t= 0

t= 100 20(2)

t= 200 = 20(2)2

t= 300 = 20(2)3

t= 400 = 20(2)4

y= 20(2)t/100

y T y0

y t y= y0(2)t/T

t

t= 0

t= 10

t= 20

t= 30


112/342

y= 1012

t/10

100, 000 6%

t

t= 0 = 100, 000

t= 1 = 100, 000(1.06) = 106, 000

t= 2 = 106, 000(1.06) = 112, 360

t= 3 = 112, 360(1.06) 119, 101.60 t= 4 = 119, 101.60(1.06)

126, 247.70

t= 5 = 26, 247.70(1.06) 133, 822.56

y= 100, 000(1.06)t

P r

t A= P(1 + r)t

y= 100000(1.06)t t t

t = 8 y = 100, 000(1.06)8 159, 384.81 t = 10 y = 100, 000(1.06)10 179, 084.77 200, 000


113/342

e2.71828 e e

e

f(x) =ex

T t

T= 170165e0.006t t

t

T

50, 000

A= 50, 000(1.044)t t

t= 18 A= 50, 000(1.044)18 108, 537.29 100, 000

A = 20, 000(1.05)t t

t = 10, A = 20, 000(1.05)10 32, 577.89 32, 577.89


114/342

y = 10012

x/250 x = 0

x= 500 y= 10012

500/250= 100

12

2= 25

y = 1, 000(3)x/80 x = 0

x= 100 y= 1, 000(3)100/80 = 3, 948.22 3, 948

10, 000

A = 10, 000(1.02)t t

A= 10, 000(1.02)12 = 12, 682.42

12, 682.42


115/342

4x1

= 16x

y= 2x

2x

26

a bx

c

+d

b > 0

b = 1

f(x) =bx b >0

b = 1

72xx2

= 1

343

52x

5x+1

0

f(x) = ( 1.8)x y =

(1.8)x

x

x y

f(x) = 2x3

f(x) = 2x

y= ex

22(5x+1) = 500

625 5x+8


116/342

a = 0

a0 = 1

an = 1

an

r s

aras =ar+s

ar

as =ars

(ar)s =ars

(ab)

r

=a

r

b

r

ab

r=

ar

br


117/342

49 = 7x+1

7 = 2x + 3

3x = 32x1

5x1 = 125

8x= x2 9 x2 = 3x3 + 2x 1 2x + 3> x 1 2x2 >8

x1=x2 bx1 =bx2 bx1 =bx2 x1= x2

4x1 = 16

4x1 = 16

4x1 = 42

x 1 = 2x= 2 + 1

x= 3

4x1 = 16

(22)x1 = 24

22(x

1)

= 24

2(x 1) = 42x 2 = 4

2x= 6

x= 3


118/342

x= 3 431 = 42 =

16

125x1 = 25x+3

125x1 = 25x+3

(53)x1 = (52)x+3

53(x1) = 52(x+3)

3(x 1) = 2(x + 3)3x 3 = 2x + 6

x= 9

9x2

= 3x+3

(32)x2

= 3x+3

32x2

= 3x+3

2x2 =x + 3

2x2 x 3 = 0(2x 3)(x + 1) = 0

2x

3 = 0 x + 1 = 0

x=3

2 x= 1


119/342

b > 1 y = bx x

bx < by x < y

0 < b < 1 y = bx x

bx > by x < y

bm < bn

m < n m > n b

3x


120/342

110 (0.36)x1


121/342

(0.6)x3 >(0.36)x1

(0.6)x3 >(0.62)x1

(0.6)x3

>(0.6)2(x1)

(0.6)x3 >(0.6)2x2

x 3> 2x 23x >1

x >1

3

(0.6)x3 >(0.36)x1

(0.6)x3 >(0.62)x1

(0.6)x3

>(0.6)2(x1)

(0.6)x3 >(0.6)2x2

x 3< 2x 23x (0.6)2x2 x

3 >

2x

2

x > 13

x

162x3 = 4x+2 x= 83

1

2

2x= 23x

1

2 21 x= 3

42x+7 322x3

29

6 , +

255x1 254 254 25

2

(, 15 ]

x

7x+4 = 492x1 x= 2

4x+2 = 82x x= 1

2

35x+2

= 3

22x

2

7, +


122/342

f(x) =bx b >1

f(x) =bx 0< b


123/342

f(x) = 2x f(x) = 2x

x

y x

y= 0

g(x) =

1

2

x

g(x)

x 3 2 1

f(x) 1

2

1

4

1

8

1

16


124/342

g(x) =1

2x

g(x) =1

2x

x

x y= 0

b > 1 0 < b < 1 f(x) = bx

b > 1 0< b < 1


125/342

f(x) =bx b > 0 b = 1 R

(0, +)

y x

y= 0 x

f(x) = 2x g(x) = 3x

x 4 3 2 1 f(x)

g(x)

x 4 y 1 f(x) g(x)

y

f(x) =

1

2

x g(x) =

1

3

x

x 4 3 2 1 f(x)

g(x)

x 4 y 1 f(x) g(x)


126/342

y

f(x) = 5x y

5x = 1

5x = 1

5x


127/342

x 3

x

4 y

1 y= 2x

y= 2x

y= 2x y= 3 2x y= 25 2x

y= 2x + 1

y= 2x 1 y= 2x+1

y= 2x1


128/342

x 3 2 1 y= 2x

y= 2

x

y= 2x 0.125 .25 .5 1 2 4 8 y 1

y= 3 2x y

y= 25 2x y

2/5

y= 2x + 1

y= 2x 1 0.875 0.75 0.5

y= 2x+1

y= 2x1

y= 2x y= 2x y= 2x

y

x 3 2 1 y= 2x

y= 2x 0.125 0.25 0.5 1 2 4 8y= 2x

y y =2x y y= 2x y= 2x y= 2x x

y = 2x x y = 2x x y = 2x y= 2x y


129/342

y= f(x) x y= f(x)

y= f(x) y y= f(x)

y= 2x y= 3(2x) y= 0.4(2x)

y

x 3 2 1 y= 2x

y= 3(2x

)

y= 0.4(2x)

y y= 3(2x) y

y= 2x y y= 0.4(2x)

y y= 2x


130/342

R

y y y= 3(2x)

y y= 0.4(2x)

y= 0

(0, +

)

a >0 y= af(x) y

y= f(x) a

a >1 0< a


131/342

R

y= 2x + 1 (1, +) y= 2x 3 (3, +)

y

y

y= 2x

y= 2x y= 0

y = 2x + 1 y = 1 y = 2x 3 y= 3

d y = f(x) + d d

d >0 d d


132/342

x 3 2 1 y= 2x

y= 2x2

y= 2x+4

R

(0, +) y x= 0 y

y= 2x+4 24 = 16 y y= 2x2 22 = 0.25

y= 0

c y = f(x c) c c > 0 c c < 0

y= f(x)


133/342

f(x) =a bxc + d

b b >1 0< b 0 d d 0 c c


134/342

x

24 =x 43 =x 51 =x 16

12 =x

1

5

1

4


135/342

5x = 625

3x =1

9

7x = 0

10x = 100, 000

x

x b logb x b

x log381 = 4 34 = 81

log232 = 5 25 = 32

log51 = 0 50 = 1

log61

6

= (1) 61

=

1

6

log232

log9729

log55

log 12

16

log71

log51

5

4

15

1/2


136/342

a b b = 1 a b logb a b

logba =a logb a

b a

log232 = 5 25 = 32

log9729 = 3 93 = 729

log55 = 1 51 = 5

log1/216 = 412

4= 16

log71 = 0 70 = 1

log515

= 12 51/2 = 15

logb a= c

bc =a

b c c = logb a

logb a a log2(8)

logb x log51

125= 3 53 = 1

125

log x log10 x

e

e

ln ln x loge x


137/342

53 = 125

72 = 1

49 102 = 100

2

32

=4

9

(0.1)4 = 10, 000

40 = 1

7b = 21

e2 =x

(

2)2 = 4

log5125 = 3

log71

49= 2

log 100 = 2

log 23

4

9

= 2

log0.110, 000 = 4 log41 = 0

log721 =b

b= 3 73 = 21 b 71.5645

ln x= 2

log m= n

log381 = 4

log55 = 2

log 34

64

27= 3

log42 =1

2 log100.001 = 3

ln 8 =a


138/342

10n =m

34 = 81

(5)2 = 5

3

4

3=

64

27

41/2 = 2

103 = 0.001 103 = 1

1, 000 ea = 8

1031

log 1031 = 31

R=2

3log

E

104.40

E 104.40

1012


139/342

E= 1012 R=2

3log

1012

104.40 =

2

3log 107.6

log107.6 = 7.6

107.6 log 107.6 = 7.6

R=2

3 (7.6) 5.1

1012/104.40 = 107.6 39810717
http://www.phivolcs.dost.gov.ph/index.php?option=com_content&task=view&id=45&Itemid=100


140/342


141/342

D= 10log I

1012

I 2 1012 2

106


142/342

60

85

90 100

106 m2

D = log 106

1012 = 10log106 log 106

106 log106 = 6

D= 10(6) = 60

106

1012 = 106 = 100, 000


143/342

[H+]

pH= log[H+]

pH= log 1

[H+]

105

log 105 log105

105 log 105 = 5 log 105 = (5) = 5


144/342

log3243 5

log6

1

216

3

log0.2516

2

49x = 7 log497 =x

63 = 1

216 log6216 = 3

102 = 100 log 100 = 2

log 112

4

121= 2 11

2 2

= 4

121

ln 3 =y ey = 3

log0.001 = 3 103 = 0.001
http://www.sciencebuddies.org/science-fair-%20projects/project_ideas/Chem_AcidsBasespHScale.shtml


145/342

log3(x 2) = 5 ln x 9 y= log 12

x

logx2 = 4 ln x2 >(ln x)2 g(x) = log3 x

x


146/342

x y

logx2 = 4

log2 x= 4 log24 =x

logx2 = 4 x

log2 x= 4 4

log24 =x

x

log2 x= 4


147/342

f(x) = 3x

x f(x)

4

1

g(x) = log3 x

x

x g(x)181

13

log2 x= 4 logx16 = 2 log 1000 = x

log2 x= 4

24 =x

x= 16

logx16 = 2

x2

= 16x2 16 = 0(x + 4)(x 4) = 0x= 4, +4 x= 4

log 1000 = x


148/342

log 1000

3 = xx= 3

log381

34 = 81

log41 log33 log442

log43 = 0 40 = 1

log33 = 1 31 = 3

log442 = 2 42 = 42


149/342

b x b > 0 b = 1

logb1 = 0

logb bx =x

x >0 blogbx =x

logb1 b b? = 1

logb bx

b bx

x

logb x b x b

x

log214

23 = 8 24 = 16

23.8074 14.000

2log214 = 14

logb1 = 0 logb bx = x blogbx = x

log10

ln e3

log464

log5

1

125

5log52

log1

log 10 = log10101 = 1

ln e3 = loge e3 = 3

log464 = log443 = 3

log5

1

125

= log55

3 = 3


150/342

5log52=2

log1 = 0

102 m2

D= 10log

I

I0

D= 10log

102

1012

D= 10log1010

D= 10 10 D= 100

pH= log[H+]3.0 = log[H+]3.0 = log[H+]103.0 = 10logH

+

103.0 = [H+]

103.0

log749

log273 1

3

ln e


151/342

log7

73 78 log773 + log778 log7

49

7

log749 log77

log775

5 log77

log2

24210

log22

4 log2210 6

log3(27 81) log327 + log381

b > 0, b = 1 n R u >0, v >0

logb(uv) = logb u + logb v

logb

uv

= logb u logb v

logb(un) =n logb u


152/342

logb(uv) = logb u + logb v log7

73 78= log773 + log778logb

uv

= logb u logb v

log7

49

7

= log749 log77

log2

24

210

= log22

4 log2210

logb(un) =n logb u

log775 = 5 log77

log3(27 81) = log327 + log381

r= logb u s= logb v u= br

v= bs

logb(uv) = logb(brbs)

logb(uv) = logb br+s

logb(uv) =r + s logb(uv) = logb u= logb v

r= logb u u= br

un =brn

un =brn

logb(un) = logb(brn) logb(un) =rn

logb(u

n) =n logb u

logb un

(logb u)n

n

u n logb u


153/342

log2(5 + 2) = log25 + log22 log2(5 + 2) = (log25)(log22) log2(5

2)

= log25

log22

log2(5 2) =log25log22 log2(5

2 2) = 2log2(5 2)

log(ab2)

log3

3

x

3 ln[x(x 5)]

log(ab2) = log a + log b2 = log a + 2 log b

log3

3

x

3= 3log3

3

x

= 3(log33 log3 x) = 3(1 log3 x) = 3 3log3 x

ln[x(x 5)] = ln x + ln(x 5)

log 2 + log 3 2 ln x ln y

log5(x2

) 3log5 x 2 log5

log 2 + log 3 = log(2 3) = log 6 2 ln x ln y= ln(x2) ln y= ln

x2

y

log5(x2) 3log5 x= log5(x2) log5(x3) = log5

x2

x3

= log5

1

x

= log5(x

1) = log5 x 2 log5

2 = 2(1)

= 2(log 10) 1 = logb b

= log 102 n logb u= logb un

= log 100

2 log 5 = log 100 = log

100

5

= log 20


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log3729 6

log9729 3

log27729 2

log1/27729 2 log729729 1

log81729 3/2

log81729

811/4 = 3 36 = 729 (811/4)6 = 729

816/4

= 813/2

= 729

log81729 = 32

log3n729 = 6n

a b x a = 1, b = 1logb x=

loga x

loga b

log3n729 = 6

n

log3n729 =log3729

log33n

= 6

n.

log832

log2431

27

log251

5


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log832 =log232

log28 =

5

3

log2431

27=

log31

27log3243

=3

5

log251

5=

log5 15

log525 =

1/22

= 14

log64

log 12

2 e

log64 = log24

log26=

2

log26

log 12

2 = ln 2

ln

1

2

= ln 2ln 1 ln 2=

ln 2

0 ln 2= ln 2

ln 2= 1

log

x3

2

3log x log2

ln(2e)2 (2e)2 = 4e3 2 ln 2 + 2

log4(16a) 2 + log4 a

log(x + 2) + log(x 2) log(x2 4) 2log35 + 1 log375

2 ln

3

2

ln 4 ln

9

16

(log32)(log34) = log38

(log32)(log34) = log36


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log22 = (log 2)2

log4(x 4) =log4 x

log44

3log9 x2 = 6log9 x

3(log9 x)2

= 6log9 x

log32x2 = log32 + 2 log3 x

log32x2 = 2log32x

log52 0.431 log58, log5

1

16

, log5

2, log252 log258

1.2920,

1.7227, 0.2153, 0.2153, 0.6460

log6 0.778 log4 0.602 log64, log 24, log46, log

2

3

log

3

2

0.7737, 1.3802, 1.2925, 0.1761, 0.1761

b x logb x


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b x logb x

f(x) = logb x logb u= logb v u= v

ab= 0 a= 0 b= 0

x

log4(2x) = log410

log3(2x 1) = 2 logx16 = 2

log2(x + 1) + log2(x 1) = 3 log x2 = 2

(log x)2 + 2 log x 3 = 0

log4(2x) = log410

2x= 10

x= 5


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log4(2 5) = log4(10)

log3(2x 1) = 22x 1 = 32 2x 1 = 92x= 10

x= 5

log3(2 (5)1) = log3(9) = 2

logx16 = 2

x2 = 16

x2

16 = 0(x + 4)(x 4) = 0 a2 b2 = (a + b)(a b)x= 4, 4

log4(16) = 2 4 log4(16)

log2(x + 1) + log2(x 1) = 3log2[(x + 1)(x 1)] = 3 logb u + logb v = logb(uv)(x + 1)(x

1) = 23

x2 1 = 8 x2 9 = 0(x + 3)(x 3) = 0 a2 b2 = (a + b)(a b)x= 3, 3

log2(3 + 1) log2(31

3 log2(3 + 1) = log2(2)

x= 3

log x2 = 2

log x2 = 2

x2 = 102

x2 = 100


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x2 100 = 0(x + 10)(x 10) = 0x= 10, 10 log(10)2 = 2 log(10)2 = 2

log x2 = 2

log x2 = log 102 2 = 2(1) = 2(log 10) = log 102

x2 = 102

x2 100 = 0(x + 10)(x 10) = 0x= 10, 10 log(10)2 = 2 log(10)2 = 2

logb un =n logb u

log x2 = 2

2log x= 2

log x= 1

x= 10

log x2 = 2log x x >0

(log x)2 + 2 log x 3 = 0

log x= A

A2 + 2A 3 = 0(A + 3)(A 1) = 0A= 3 A= 1

log x= 3 log x= 1x= 103 =

1

1000 x= 10

log

1

1000

log 10

x 2x = 3

2x = 3


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log2x = log 3

x log 2 = log 3 logb un =n logb u

x= log3

log2 1.58496

x log 12

x

18

14

12

x log2 x

18

14

12

x log 12

x

18

14

12

1 2 3

x log2

teachers guide for general mathematics

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