Transcript FUNCTIONS 3 ESO BIL IES MANUEL CAÑADAS MATHS NFC (Source: Fina Cano and Boardworks)

FUNCTIONS
3 ESO BIL
IES MANUEL CAÑADAS
MATHS NFC
(Source: Fina Cano and Boardworks)
T.6 FUNCTIONS AND GRAPHS
1.
2.
3.
4.
5.
6.
Introduction
Functions and their graphs
Variations in a function
Maxima and minima
Trends and periodicity
Continuity and discontinuity
Introduction
• A little of history:“La gripe española”
Between 1918 and 1919 the Spanish flu (H1N1) killed 50 million people all over the
world, 300.000 in Spain, 650.000 in USA. It was much worse than the black death of
the XIV century.
• A little of history:“La gripe española”
• A little of history:“La gripe española”
 Modeling the real world
 Modeling the real world
• In every day life, many quantities depend on one or more changing
variables. For example:
• Plant growth depends on sunlight and rainfall.
• Distance travelled depends on speed and time taken.
• Test marks depend on attittude, attention, study (among
many others variables!).
 Modeling the real world
• In this unit we are going to study functions.
• Functions are very useful to model real situations where one
quantity depends on another quantity.
• Graphs are a good way of presenting a function, they give us a
visual picture of the function.
 Modeling the real world
Dan´s journey on his bike.
Dan start his journey covering 10 miles in 1
hour. Then he stops for 1 hour to rest.
Forthe following half an hour he goes faster
because he goes downhill. After that he
stops for one hour and a half to visit a
friend. Finally, he returns home covering 25
miles in 3 hours.
 Modeling the real world
This graph shows Dan´s journey on his bike.
Filling flasks 1
Filling flasks 2
 Modeling the real world
This graph models the depth of the water flowing in or out of a container at a constant
rate.
Matching graphs to statements
Functions and their graphs
 What´s a function?
is a function but
is not.
 What´s a function?
is a function but
is not.
Function is a relation between two variables such that for every
value of the first (usually x), there is only one corresponding
value of the second (usually y).
 What´s a function?
is a function but
is not.
Function is a relation between two variables such that for every
value of the first (usually x), there is only one corresponding
value of the second (usually y).
 Function or not?
is a function but
is not.
Not all the graphs are functions!
 Function or not?
is a function but
is not.
Not all the graphs are functions!
 Function or not?
is a function but
is not.
Not all the graphs are functions!
 What´s a function?
is a function but
is not.
We say that the second variable is a function of the first variable.
Lots of functions can be given by formulas, called analytical
expressions of the functions.
 What´s a function?
is a function but
is not.
We say that the second variable is a function of the first variable.
Lots of functions can be given by formulas, called analytical
expressions of the functions.
We normally write the function as f(x), and read
this as “function of x” (y and f(x) are the same).
y  x
3
f (x)  x
3
Using an equation
A function is a rule which maps one number, sometimes called the input or x,
onto another number, sometimes called the output or y.
A function can be illustrated using a function diagram to show the operations
performed on the input.
For example:
x
×3
+2
y
A function can be written as an equation.
For example,
y = 3x + 2.
A function can can also be be written with a mapping arrow.
For example,
x  3x + 2.
Using an equation: The function machine
Is there any difference between
x
×2
+1
y
×2
y
and
x
+1
?
The first function can be written as y = 2x + 1.
The second function can be written as y = 2(x + 1) or 2x + 2.
Using a table
We can use a table to record the inputs and outputs of a function.
For example,
We can show the function y = 2x + 5 as
×2
x 4613
3, 1, 6,
3, 4,
1,
3,1.5
6,
1,
3,
y77,
11
11,
11,
11,
11,
7,
7,17
17,
17, 13
13, 8
+5
and the corresponding table as
x
3
1
6
4
1.5
y
11
7
17
13
8
Using a table with ordered values
It is often useful to enter inputs into a table in numerical order.
For example,
We can show the function y = 3(x + 1) as
x54321
1, 2,
1, 3,
2,
1,4,
3,
2,
1,
×3
+1
66,
6,
6,
6,
9y9,
9,
9,12
12,
12,15
15, 18
When the inputs
are ordered
and the corresponding table as
x
1
2
3
4
5
y
6
9
12
15
18
the outputs form
a sequence.
Drawing a function: Coordinate pairs
When we write a coordinate, for example,
(3, 5)
x-coordinate
y-coordinate
the first number is called the x-coordinate and the second number is called the ycoordinate.
coordinate.
Together, the x-coordinate and the y-coordinate are called a coordinate pair.
Drawing graphs of functions
The x-coordinate and the y-coordinate in a coordinate pair can be linked
by a function.
What do these coordinate pairs have in common?
(1, 3), (4, 6), (–2, 0), (0, 2), (–1, 1) and (3.5, 5.5)?
In each pair, the y-coordinate is 2 more than the x-coordinate.
These coordinates are linked by the function:
y=x+2
We can draw a graph of the function y = x + 2 by plotting points that obey this
function.
Drawing graphs of functions
Given a function, we can find coordinate points that obey the function by
constructing a table of values.
Suppose we want to plot points that obey the function
y=x+3
We can use a table as follows:
x
–3
–2
–1
0
1
2
3
y=x+3
0
1
2
3
4
5
6
(–3, 0)
(–2, 1)
(–1, 2)
(0, 3)
(1, 4)
(2, 5)
(3, 6)
Drawing graphs of functions
For example,
y
to draw a graph of y = x – 2:
y=x–2
1) Complete a table of values:
x
–3
–2
–1
0
1
2
3
y=x–2
–5
–4
–3
–2
–1
0
1
2) Plot the points on a coordinate grid.
3) Draw a line through the points.
4) Label the line.
5) Check that other points on the line fit the rule.
x
Drawing graphs of functions
 Function or not?
is a function but
is not.
Not all the analytical expressions are functions!
 Function or not?
is a function but
is not.
Not all the analytical expressions are functions!
y  x
2
y   2x
• What´s a function?
As a conclusion:
A Function is a relation between two variables, x and y, which
associates each value of x to a single value of y.
This relation can be expressed using:
A. Equation (mathematical relationship)
B. Table (pair of values for x and y)
C. Graph (drawing of the function)
D. Description (describing the relation between x and y)
 Domain of a function
The set of input values for x
 Domain of a function
The set of input values for x
http://mathdemos.org/mathdemos/domainrange/domainrange.html
 Range of a function
The set of output values for y
 Range of a function
The set of output values for y
http://mathdemos.org/mathdemos/domainrange/domainrange.html
•Graph of a function
x is the independent variable and is represented on the horizontal axis
(x axis). The x-value is called abscissa.
y or f(x) is the dependent variable and is represented on the vertical
axis (y axis). The y-value is called ordinate.
The values of x and y together, written as (x , y) are called the
coordinates.
•Plotting a function
Each axis must be graded with its own scale!
•Plotting a function
Each axis must be graded with its own scale!
Number of days, d
Cost (£)
Cost in £, c
1
2
3
4
5
55
80
105
130
155
Cost of car hire
150
140
130
120
110
100
90
80
70
60
50
40
30
0
It is most accurate to use a small cross for each
point.
If appropriate, join the points together using a ruler.
Lastly, don’t forget to give the graph a title.
0
1
2
3
4
Number of days
5
Variations in a function
 Increase
A function is “increasing” if the y-value increases as the x-value
increases.
 Increase
A function is “increasing” if the y-value increases as the x-value
increases.
 Increase
A function is “increasing” if the y-value increases as the x-value
increases.
Ex.: The height of a kid versus his age.
 Decrease
A function is “decreasing” if the y-value decreases as the
x-value increases.
 Decrease
A function is “decreasing” if the y-value decreases as the
x-value increases.
 Decrease
A function is “decreasing” if the y-value decreases as the
x-value increases.
Ex.: The height of an elderly person versus his age.
 Increasing and Decreasing intervals
Many functions have increasing intervals and decreasing intervals.
 Increasing and Decreasing intervals
Many functions have increasing intervals and decreasing intervals.
 Increasing and Decreasing intervals
Many functions have increasing intervals and decreasing intervals.
Ex.: The height of a person versus his age.
 Constancy
A function is constant if the graph is horizontal, parallel to the X-axis.
y
All of the points lie on a straight
line parallel to the x-axis.
y=1
x
Example: Graphs parallel to the x-axis
What do these coordinate pairs have in common?
(0, 1), (3, 1), (–2, 1), (2, 1), (1, 1) and (–3, 1)?
The y-coordinate in each pair is equal to 1.
Look what happens when these points are plotted on a graph.
All of the points lie on a straight line parallel
to the x-axis.
y
y=1
Name five other points that will lie on this line.
x
This line is called y = 1.
Graphs parallel to the x-axis
All graphs of the form
y = c,
where c is any number, will be parallel to the x-axis and will cut the yaxis at the point (0, c).
y
y=5
y=3
x
y = –2
y = –5
Maxima and minima
 Maxima and Minima (local)
A function has a relative (or local) maximum at a point if its ordinate is
higher than the ordinate of the points around it.
A function has a relative (or local) minimum at a point if its ordinate is less
than the ordinate of the points around it.
 Maxima and Minima (local)
A function has a relative (or local) maximum at a point if its ordinate is
higher than the ordinate of the points around it.
A function has a relative (or local) minimum at a point if its ordinate is less
than the ordinate of the points around it.
 Maxima and Minima (local and global)
A function has an absolute (or global) maximum at a point if its ordinate is
the largest value that the function takes.
A function has an absolute (or global) minimum at a point if its ordinate is
the smallest value that the function takes.
 Maxima and Minima (local and global)
A function has an absolute (or global) maximum at a point if its ordinate is
the largest value that the function takes.
A function has an absolute (or global) minimum at a point if its ordinate is
the smallest value that the function takes.
Trends and periodicity
•Trends
We can predict the shape of the graph for some functions for large values of x
because they describe events with a very clear trend.
Ex.: Water temperature versus time passed during the
heating and turning off the heat.
•Periodicity
Periodic functions are those whose behavior is repeated each time the
independent variable covers a certain interval. The length of this
interval is called a period.
Continuity and discontinuity
 Continuity and discontinuity
A function is be continuous when you can plot its graph without
lifting your pencil off the paper.
A function is be discontinuous when it has discontinuities or
“jumps” on its graph.
.
 Continuity and discontinuity
A function is be continuous when you can plot its graph without
lifting your pencil off the paper.
A function is be discontinuous when it has discontinuities or
“jumps” on its graph.
.