1. function and algebra.ppt

7/27/2019 1. Function and algebra.ppt

http://slidepdf.com/reader/full/1-function-and-algebrappt 1/26

1

Axes of Graphs

• x axis: the horizontal line alongwhich values of x are measured

• x values increase from left to right

• y axis: the vertical line up which

values of y are measured

• y values increase from bottom to top

• Origin: the point at which the axes

intersect where x and y are both 0



2

Terms used in Plotting Points

• Coordinates: a pair of numbers ( x ,y ) that

represent the position of a point

the first number is the horizontal distance

of the point from the origin the second number is the vertical distance

• Positive quadrant: the area above the x

axis and to the right of the y axis whereboth x and y take positive values



3

Plotting Negative Values

• To the left of the origin x is negative

• As we move further left x becomes more

negative and smaller• Example:

– 6 is a smaller number than – 2

and occurs to the left of it on the x

axis

• On the y axis negative numbers occur

below the origin



4

Variables, Constants and

Functions

• Variable: a quantity represented by asymbol that can take differentpossible values

• Constant: a quantity whose value isfixed, even if we do not know itsnumerical amount

•

Function: a systematic relationshipbetween pairs of values of thevariables, written

y = f( x )



5

Working with Functions

• If y is a function of x , y = f( x )

• A function is a rule telling us how to

obtain y values from x values

• x is known as the independent

variable, y as the dependent variable

•

The independent variable is plottedon the horizontal axis, the dependent

variable on the vertical axis



6

Proportional Relationship

• Each y value isthe sameamount timesthecorresponding x value

• All points lie on

a straight linethrough theorigin

• Example:

y = 6 x

0

20

40

60

0 5 10x

y y = 6x



7

Linear Relationships

• Linear function: arelationship inwhich all the pairsof values formpoints on a straight

line• Shift: a vertical

movementupwards or

downwards of aline or curve

• Intercept: thevalue at which afunction cuts the y

0

20

40

60

80

0 5 10

x

y

y = 6x + 20

y = 6x



8

General Form of a Linear

Function

• A function with just a term in x and

(perhaps) a constant is a linear

function• It has the general form

y = a + bx

• b is the slope of the line• a is the intercept



9

Power Functions

• Power: an index indicating the number oftimes that the item to which it is applied ismultiplied by itself

• Quadratic function: a function in which the

highest power of x is 2 The only other terms may be a term in x and

a constant

• Cubic function: a function in which the

highest power of x is 3 The only other terms may be in x 2, x and a

constant



10

Writing Algebraic Statements

• The multiplication sign is often omitted, or

sometimes replaced by a dot

•

An expression in brackets immediatelypreceded or followed by a value implies

that the whole expression in the brackets

is to be multiplied by that value

• Example:y = 3(5 + 7 x ) = 15 + 21 x



11

The Order of Algebraic

Operations is

1. If there are brackets, do what is insidethe brackets first

2. Exponentiation, or raising to a power

3. Multiplication and division

4. Addition and subtraction

•You may like to remember the acronymBEDMAS, meaning brackets,exponentiation, division, multiplication,addition, subtraction



12

Positive and Negative Signs

• When two signs come together

– + (or + –) gives –

– – gives +

• Examples:

11 + ( – 7) = 11 – 7 = 4

12 – ( – 4) = 12 + 4 = 16

( – 9) ( – 5) = +45



13

Multiplying or Dividing by 1

• 1 x = x

•

( –

1)

x = –

x• x 1 = x



14

Multiplying or Dividing by 0

• Any value multiplied by 0 is 0

• 0 divided by any value except 0 is 0

• Division by 0 gives an infinitely large

number which may be positive or negative• 0 0 may have a finite value

• Example: When quantity produced Q = 0,

variable cost VC = 0but average variable cost = VC/Q

may have a finite value



15

Brackets

• An expression in brackets written

immediately next to another

expression implies that the

expressions are multiplied

together

• Example:

5 x (7 x – 4) = (5 x ) (7 x – 4)



16

Multiplying out Brackets 1

• One pair: multiply each of the terms in

brackets by the term outside

• Example:

5 x (7 x –

4) = 35 x 2

–

20 x



17

Multiplying out Brackets 2

• Two pairs: multiply each term in the second

bracket by each term in the first bracket

• Examples:

(3 x – 2)(11 + 5 x ) = 33 x + 15 x 2 – 22 – 10 x

= 15 x 2 + 23 x – 22

(a – b)( – c + d ) = – ac + ad + bc – bd



18

Results of Multiplying out

Brackets

(a + b)2 = (a + b) (a + b)

= a2 + 2 ab + b2

(a – b)2 = (a – b) (a – b)

= a2 – 2 ab + b2

(a + b)(a – b) = a2 – ab + ab – b2 = a2 – b2



19

Factorizing

•

Look for a common factor, or forexpressions that multiply together togive the original expression

• Example:

45 x 2 – 60 x = 15 x (3 x – 4)

• Factorizing a quadratic expressionmay involve some intelligent

guesswork• Example:

45 x 2 – 53 x – 14 = (9 x + 2) (5 x – 7)



20

Fractions

• Fraction: a part of a whole

• Amount of an item= fractional share of item total

amount• Ratio: one quantity divided by another

quantity

• Numerator: the value on the top of a

fraction• Denominator: the value on the bottom

of a fraction



21

Cancelling

• Cancelling is dividing both numerator

and denominator by the same amount

• Examples:

3

2

375

527

105

70

3322

22

52

23

7

2

74

24

28

8

z

x

z z x

z x x

z x

z x

..



22

Inequality Signs

• > sign: the greater than sign

indicates that the value on its left

is greater than the value on itsright

•

< sign: the less than signindicates that the value on its left

is less than the value on its right



23

Comparisons using Common

Denominator

• Example:

To find the bigger of 3/7 and 9/20multiply both numerator and denominator

of each fraction by the denominator of theother:

3/7 = (20 3)/(20 7) = 60/140

9/20 = (7 9)/(7 20) = 63/140Since 63/140 > 60/140

9/20 > 3/7



24

Adding and Subtracting

Fractions

• To add or subtract fractions first

write them with a common

denominator and then add or

subtract the numerators

• Lowest common denominator:

the lowest value that is exactly

divisible by all the denominatorsto which it refers



25

Multiplying and Dividing

Fractions

• Fractions are multiplied by

multiplying together the

numerators and also thedenominators

• To divide by a fraction turn it

upside down and multiply by it• Reciprocal of a value: is 1 divided

by that value



26

Functions of More Than 1 Variable

•

Multivariate function: the dependentvariable, y , is a function of more thanone independent variable

• If y = f( x ,z )

y is a function of the two variables x andz

• We substitute values for x and z to findthe value of the function

• If we hold one variable constant andinvestigate the effect on y of changingthe other, this is a form of comparativestatics analysis

1. function and algebra.ppt

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