Download Exponential function : funzione esponenziale Base: base Exponent

VISUAL ORGANISERS
GLOSSARY
Exponential function : funzione esponenziale
Base: base
Exponent: esponente
Domain: dominio
To intersect: intersecare
Range: valori
Set: insieme
Table
y  ax
a>0 and
a≠1
increasing
y  ax
0 a 1
decreasing
Definition: An exponential
function is a function in
which…..
If a>0 the graph is…..
If the graph is….
The domain is……
The range is ……..
The intersection with y-axis
is………..
1
Exponential functions
Definition: exponential functions are functions in which the variable is the exponent
of a power.
If a is a positive number and different from 1 (a>0 and a≠1) the function is
defined by
y  a x and
the exponent x is any real number.
Sometimes they are called growth functions because they show how certain things
grow. The graph of an exponential function is usually hard to draw because of the
range of value of the y – variable.
.
Procedure: to define the most important properties of an exponential function, we
represent two graphs of two different exponential functions through their points, the
first one has
a1
and the second one has
0  a  1.
The curve trends that we get are typical of all exponentianl functions having a as
base .
2
First example
We represent on a Cartesian plane the function
y  2x
( a  1 ); the graph is obtained
by linking the points in the table below near the figure. Notice that the curve rises
slowly at first and then more and more steeply.
x
y  2x
3
2 3 
1
 0,125
8
2
2 2 
1
 0,25
4
1
2 1 
1
 0,5
2
0
2 0 1
1
21  2
2
22  4
3
23  8
If
a  1:
the graph of the exponential function is increasing
the domain is R;
the range is the set of strictly real numbers
the graph intersects the y_axis at (0,1)
3
Second example:
Now we represent on a Cartesian plane the function
1
y  
2
x
( 0  a  1 ); the graph is
obtained by linking the points in the table below near the figure:
x
x
1
y  
2
3
3
1
 
2
1
 
2
2
2
1
 
2
1
1
0
1
0
   2 1
2
1
1
1
    0,5
2
2
 
2
1
1
    0,25
2
4
 
3
1
1
    0,125
8
2
 23  8
 22  4
 21  2
0
1
2
3
If
0 a 1
the graph of the exponential function is decreasing
the domain is R;
the range is the set of strictly real numbers
the graph intersects the y_axis at (0,1)
Worksheet
Activity (work in pairs)
4
Task 1.Complete the table below and draw the graph of the exponential function:
x
1
y   1
 2
x
y
0
1
   1
2
0
2
1
....
1
  1
2
2
1
3
2
...
3
1
....   
2
y
1
1
 
2
1
1
 
2
1  2 1
x
....
 1  .....  1
....
 1  .....  1
Task 2
1. Define a table such as in the previous example and then draw the graph of the
following function
y  2 x 1
Worksheet
Task 3
First discuss the questions with a response partner, then answer them.
5
a) What is an exponential function?
b) Given the function, which value can a and x assume?
c) At which point does an exponential function intersect the y-axis?
d) Which are the domain and the range of aan exponential function?
Task 4
feedback
Two students summarize the aim of the different tasks.
Homework
After drawing the graphs of the following exponential functions, establish if they are
increasing or decreasing.
a)
y  2x
b)
e)
1
y 
 3
i)
y  2x 1
y  3 x 1
x 1
f)
1
y 
 3
c)
y  2 2 x
d)
1
y  
2
1
g)
h)
y  3 x
x
x
l)
1
y    1
2
y  3 x2
x
m)
1
1
y  
2
2
 
Next lesson
Exponential equations
6
x