View
214
Download
0
Category
Preview:
Citation preview
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
7/27/2019 1. Function and algebra.ppt
http://slidepdf.com/reader/full/1-function-and-algebrappt 2/26
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
7/27/2019 1. Function and algebra.ppt
http://slidepdf.com/reader/full/1-function-and-algebrappt 3/26
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
7/27/2019 1. Function and algebra.ppt
http://slidepdf.com/reader/full/1-function-and-algebrappt 4/26
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 )
7/27/2019 1. Function and algebra.ppt
http://slidepdf.com/reader/full/1-function-and-algebrappt 5/26
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
7/27/2019 1. Function and algebra.ppt
http://slidepdf.com/reader/full/1-function-and-algebrappt 6/26
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
7/27/2019 1. Function and algebra.ppt
http://slidepdf.com/reader/full/1-function-and-algebrappt 7/26
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
7/27/2019 1. Function and algebra.ppt
http://slidepdf.com/reader/full/1-function-and-algebrappt 8/26
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
7/27/2019 1. Function and algebra.ppt
http://slidepdf.com/reader/full/1-function-and-algebrappt 9/26
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
7/27/2019 1. Function and algebra.ppt
http://slidepdf.com/reader/full/1-function-and-algebrappt 10/26
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
7/27/2019 1. Function and algebra.ppt
http://slidepdf.com/reader/full/1-function-and-algebrappt 11/26
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
7/27/2019 1. Function and algebra.ppt
http://slidepdf.com/reader/full/1-function-and-algebrappt 12/26
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
7/27/2019 1. Function and algebra.ppt
http://slidepdf.com/reader/full/1-function-and-algebrappt 13/26
13
Multiplying or Dividing by 1
• 1 x = x
•
( –
1)
x = –
x• x 1 = x
7/27/2019 1. Function and algebra.ppt
http://slidepdf.com/reader/full/1-function-and-algebrappt 14/26
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
7/27/2019 1. Function and algebra.ppt
http://slidepdf.com/reader/full/1-function-and-algebrappt 15/26
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)
7/27/2019 1. Function and algebra.ppt
http://slidepdf.com/reader/full/1-function-and-algebrappt 16/26
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
7/27/2019 1. Function and algebra.ppt
http://slidepdf.com/reader/full/1-function-and-algebrappt 17/26
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
7/27/2019 1. Function and algebra.ppt
http://slidepdf.com/reader/full/1-function-and-algebrappt 18/26
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
7/27/2019 1. Function and algebra.ppt
http://slidepdf.com/reader/full/1-function-and-algebrappt 19/26
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)
7/27/2019 1. Function and algebra.ppt
http://slidepdf.com/reader/full/1-function-and-algebrappt 20/26
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
7/27/2019 1. Function and algebra.ppt
http://slidepdf.com/reader/full/1-function-and-algebrappt 21/26
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
..
7/27/2019 1. Function and algebra.ppt
http://slidepdf.com/reader/full/1-function-and-algebrappt 22/26
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
7/27/2019 1. Function and algebra.ppt
http://slidepdf.com/reader/full/1-function-and-algebrappt 23/26
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
7/27/2019 1. Function and algebra.ppt
http://slidepdf.com/reader/full/1-function-and-algebrappt 24/26
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
7/27/2019 1. Function and algebra.ppt
http://slidepdf.com/reader/full/1-function-and-algebrappt 25/26
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
7/27/2019 1. Function and algebra.ppt
http://slidepdf.com/reader/full/1-function-and-algebrappt 26/26
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
Recommended