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Liceo Scientifico Isaac Newton - Roma
exponential function
in accordo con il
Ministero dell’Istruzione, Università, Ricerca
e sulla base delle
Politiche Linguistiche della Commissione Europea
percorso formativo a carattere
tematico-linguistico-didattico-metodologico
scuola secondaria di secondo grado
teacher
Serenella Iacino
exponential function
Indice Modulo
Strategies – Before



Prerequisites
Linking to Previous Knowledge and Predicting con questionari basati su stimoli
relativi alle conoscenze pregresse e alle ipotesi riguardanti i contenuti da
affrontare
Italian/English Glossary
Strategies – During



Video con scheda grafica
Keywords riferite al video attraverso esercitazioni mirate
Conceptual Map
Strategies - After

Esercizi:
 Multiple Choice
 Matching
 True or False
 Completion
 Flow Chart
 Think and Discuss

Summary per abstract e/o esercizi orali o scritti basati su un questionario
e per esercizi quali traduzione e/o dettato

Web References di approfondimento come input interattivi per test orali e
scritti e per esercitazioni basate sul Problem Solving
Answer Sheets
2
exponential function
1
Strategies Before
Prerequisites
Maths
the prerequisites are
•Rules of the powers
•Injectivity of a function
•Surjectivity of a function
•Invertible function
•Strictly growing function
•Strictly decreasing function
•Symmetries
•Translations
•Dilations
•Compressions
•
•
•
Exponential
function
3
exponential function
2
Strategies Before
Linking to Previous Knowledge and Predicting
1. Do you know the rules of the powers?
2. Are you able to calculate the domain of a function?
3. Do you know the definition of asymptote of a function?
4. When is a function positive or negative?
5. When is a function strictly growing?
6. What is the definition of injectivity of a function?
7. What is the definition of surjectivity of a function?
8. When is a function invertible?
9. Do you know the equations of the symmetries, of the translations, of the dilations
and of the compressions?
4
exponential function
3
Strategies Before
Italian / English Glossary
Angolo
Angle
Ascissa
Abscissa
Asintotico
Asymptotic
Asse
Axis
Base
Base
Biiettiva
Bijective
Bisettrice
Bisecting - line
Codominio
Codomain
Coefficiente
Coefficient
Compressione
Compression
Crescente
Growing
Curva
Curve
Decrescente
Decreasing
Dilatazione
Dilation
Dominio
Domain
Equazione
Equation
Esponenziale
Exponential
Funzione
Function
Funzione esponenziale
Exponential function
Funzione inversa
Inverse function
Funzione logaritmica
Logarithmic function
Funzione polinomiale
Polynomial function
Grafico
Graph
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exponential function
Immagine
Image
Iniettiva
Injective
Insieme dei numeri reali
Set of real numbers
Invertibile
Invertible
Irrazionale
Irrational
Numero
Number
Ordinata
Ordinate
Parallela
Parallel
Pendenza
Slope
Piano
Plane
Piano cartesiano
Cartesian plane
Potenza
Power
Razionale
Rational
Retta
Straight-line
Simmetria
Symmetry
Simmetrico
Symmetrical
Strettamente
Strictly
Suriettiva
Surjective
Tangente
Tangent
Trascendente
Transcendental
Trasformazione
Transformation
Traslazione
Translation
Variabile
Variable
Vettore
Vector
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exponential function
4
Strategies During
Keywords
Circle the odd one out:
Real numbers – strictly growing – limit – decreasing – asymptotic curve – parabola - domain – straight
line – invertible – exponential function – bisecting
line –
intersection – axis – circle – symmetrical – injectivity – translation – positive
numbers – surjectivity – image – dilation – logarithmic function – slope - tangent –
power – equation – angle – bijective – polynomial – rational coefficient –
transformation – trigonometric function – abscissas – variable – ordinates – base –
irrational – codomain – set.
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exponential function
5
Strategies During
Conceptual Map
Complete the conceptual map using the following words:
injective and
surjective
a=1
natural
exponential
function
aєR
+
a>1
0<a<1
decreasing
exponential
function
a=e
straight
line
growing
exponential
function
inverse
logarithmic
function
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exponential function
6
Strategies After
Multiple Choice
1. What transformations have you to apply to the function y = 2
x-3
3
obtain the following function y = 2
+
?
4
x
to
a. a translation by a vector having components (-3 , + 3 )
4
b. a translation by a vector having components (+3 , +
c. a dilation by the constants 3 and
3
4
)
3
4
d. a translation by a vector having components (+3 , - 3 )
4
x
2. Let f(x) be the function having equation y = ( 2 a + 1 ) ; what is the
value of a for which f(x) is a strictly growing exponential function ?
a. a > b. - 1
2
1
2
with a
≠
0
<a<0
c. a > 0
d. it doesn’t exist
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exponential function
3. What are the values of a, b, c, with b > 0 such that the graph of the
function of equation y = a ∙ b
0
a. a = 4, b = +
b. a = 4, b = c. a = 4, b = +
1
2
1
2
1
x
+ c is the following ?
1
X
,c=+1
,c=-1
,c=-1
2
d. a = 4, b = -
1
2
,c=+1
4. What is the equation of the function of the type
y=a∙2
x
+ b,
the graph of which is symmetrical about the straight line y = -2 ?
a. y = - 2
b. y = - 2
x
x
+ 4
- 4
c. y = +2 x + 4
d. y = +2 x - 4
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exponential function
5. What is the equation of the exponential function of the type
y = 2 f(x) the graph of which is symmetrical about the straight
line x = 1 ?
a. y = 2
b. y = 2
c. y = 2
d. y = 2
+2+x
-2+x
-2-x
+2-x
2a
6. What are the values of a for which the equation
represents a strictly growing exponential function ?
a. – 2 < a < 0
b.
a<-2 v a>2
c.
0<a<+2
d. a doesn’t exist
11
y=
a-2
x
exponential function
7
Strategies After
Matching
1) Match the equations of the exponential functions with the definitions:
x
8
y=
3
x
- x
3
8
y=-
1
y=-
9
2
a
Strictly growing
b
Strictly growing
c
Strictly growing
d
a
Strictly decreasing
- x
8
3
12
9
y=-
8
4
exponential function
Strategies After
Matching
2) Match the graphs of the exponential functions with the equations:
Y
0
1
1
x
1
+1
2
c
X
5
0
X
4
1
X
x
x
x -1
2
3
y=
y=
3
a
1
Y
3
0
y=
0
X
Y
2
Y
1
b
2
c
13
y=
5
d
exponential function
Strategies After
Matching
3) Match the functions with the transformations:
y=2
x
+2
1
x+1
1
y=
2
2
y= -2
x
-1
3
a
translation of the function by a vector having components (- 1 , 0)
b
translation of the function by a vector having components (0 , 1) and symmetry
about the x axis.
c
translation of the function by a vector having components (0 , 2)
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exponential function
Strategies After
Matching
4) Match the graph of the exponential function with its inverse:
Y
Y
0
1
0
X
1
X
2
1
Y
Y
0
1
0
X
1
X
b
a
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exponential function
Strategies After
Matching
5) Match the equation of the exponential function with its right graph:
Y
Y
8
8
a
b
X
2
y=+1
0
X
y=-1
Y
Y
c
8
d
X
0
X
2
0
y=-1
-8
x-2
1
y=
-1
3
16
y=-1
exponential function
8
Strategies After
True or False
State if the sentences are true or false.
1) All exponential functions of the type
y=a
x
if a > 1 pass through the point
( 0 ; 1 ).
T
2) Every exponential function y = a
x
F
lies above the x axis only if x is greater than
0.
T
F
3) If a is greater than 1, the exponential function y = a x is strictly decreasing.
T
F
T
F
4) The exponential function is asymptotic to the y axis.
5) If a = 1 the exponential function y = a - x becomes a straight line that is parallel
to the x axis.
6) The functions y = a
T
x
and y = a
-x
F
are symmetrical about the x axis.
7) The logarithmic function is the inverse of the exponential function.
17
T
F
T
F
exponential function
8) The tangent to the natural exponential function in the point ( 0 ; 1 ) is parallel to
the bisecting line y = x.
T
F
9) The number e isn’t solution of any polynomial equation with rational coefficients.
T
10) If 0 < a < 1, when x > 0 the exponential function y = a
x
F
grows faster, while if
x < 0 the function decreases faster.
T
18
F
exponential function
9
Strategies After
Completion
Complete the following definitions.
1) We call general exponential function ……………………………………………………………………
……………………………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………………………
2) The domain of the exponential function ………………………………………………………………
and it passes through …………………………………………………; its graph lies…………………
and if the base a > 1, it’s ………………………………………………………………………………………
…………………………………………………………………………………………………………………………………
3) If the base 0 < a < 1 , the x axis is a ………………………………………………………………
4) The functions y = a
x
and y = a
-x
are …………………………………………………………………
in fact if we apply the equations ……………………………………………………………………………
……………………………………………………………………………………………………………………………………
5) The exponential function is invertible because ………………………………………………………
and its inverse ……………………………………………………………………………………………………………
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exponential function
6) Euler’s number e is …………………………………………………………………………………………………
and natural exponential function has ………………………………………………………………………
7) A function having equation of the type
y=a
x
+ b with a > 0
and
b<0
represents …………………………………………………………………………………………………………………
8) A function
having
equation of
the
type y = a
x+b
with a > 0
and
b<0
represents …………………………………………………………………………………………………………………
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exponential function
10
Strategies After
Flow Chart
How many solutions does this equation have?
2
x
= - x² + 2
Complete the flow chart using the terms listed below:
I draw the graph of the parabola having equation y = - x² + 2
The points of intersection between the parabola and the two
exponential functions are the solutions of this equation
I draw the graph of the exponential function y = 2 x
in its domain
This equation is the solution of a system between the equation of
the exponential function and of the parabola
I draw the symmetrical curve of the function y = 2 x about the x
axis in its domain
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exponential function
start
end
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exponential function
11
Strategies After
Think and Discuss
The following activity can be performed in a written or oral form. The teacher will
choose the modality, depending on the ability (writing or speaking) that needs to be
developed.
The contexts in which the task will be presented to the students are:
A)The student is writing an article about the chain letter and the exponential
function.
B)The student is preparing for an interview on a local TV about the compound
interest.
The student should:
1) Write an article or prepare an interview.
2) Prepare the article or the debate, outlining the main points of the argument, on
the basis of what has been studied.
3) If the written activity is the modality chosen by the teacher, the student should
provide a written article, indicating the target of readers to whom the article is
addressed and the type of magazine / newspaper / school magazine where the
article would be published.
4) If the oral activity is the modality chosen by the teacher, the student should
present his point of view on the topics to the whole class and a debate could start
at the end of his presentation.
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exponential function
12
Strategies After
Summary
We call general exponential function the function having equation y = a
x
where its domain is the set of real numbers, while its codomain is the set of real
positive numbers; a is a number greater than 0 and we can have three types of
exponential functions according to the following values of a:
a>1
0<a<1
a=1
The exponential function having base a > 1 passes through the point (0 ; 1), it
always lies above the x axis, it’s strictly growing and it’s asymptotic to the negative x
axis.
Instead if the base 0 < a < 1 it passes through the same point (0 ; 1), it always lies
above the x axis, it’s strictly decreasing and it’s asymptotic to the positive x axis.
If a = 1
for every positive and negative value of x, the function becomes y = 1
which represents a straight-line parallel to the x axis and passing through the same
point (0 ; 1).
If a > 1, when we increase its value, if x > 0 the function grows faster, while if
x < 0 the function decreases faster.
If 0 < a < 1, when we decrease its value, if x > 0 the function decreases faster,
while if x < 0 the function grows faster.
The exponential function is injective and surjective, so it’s invertible; its inverse
function is logarithmic function whose equation is y = log x; its graph is
a
symmetrical about the bisecting line of the Cartesian plane y = x.
We call natural exponential function the exponential function having equation
x
y=e
; the base e is called Euler’s number in honor of this mathematician who
discovered it.
It is an irrational number and a transcendental number because it isn’ t solution of
any polynomial equation with rational coefficients.
Its value is approximately 2.7 .
Furthermore we can easily draw the graphs of other non - elementary exponential
functions using some transformations of the plane as for example symmetries,
translations, dilations or compressions.
24
exponential function
1. Answer the following questions. The questions could be a answered in a
written or oral form, depending on the teacher’s objectives.
a) What is the equation of general exponential function?
b) How many types of general exponential functions do you know?
c) What are the properties of general exponential function?
d) Is the exponential function invertible?
e) What is its inverse function ?
f) What are the properties of logarithmic function?
g) How do you define the natural exponential function?
h) What type of number is Euler’s number?
i) Can you easily draw the graphs of other non – elementary exponential
function?
2. Write a short abstract of the summary (max 150 words) highlighting the
main points of the video.
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exponential function
Web References
This site is intended to help students on maths.
http://www.videomathtutor./
This site offers students the opportunity to expand their knowledge on the study of a
function.
http://mathworld.wolfram.com/ExponentialFunction.html
This site offers students the opportunity to expand their knowledge on the study of
the exponential function.
http://www.themathpage.com/acalc/exponential-function.htm
This site offers students the opportunity to expand their knowledge on the kinds of
discontinuity of a function.
http://www.purplemath.com/modules/exponential-function.htm
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exponential function
13
Activities Based on Problem Solving
Solve the following problems:
1) Solve graphically the following equation:
x
1
2
=
2x + 1
2) Let f(x) be a function so defined:
x+a
f(x) = 2
+ b;
determine the values of a and b knowing that it passes through the point ( 3; 31 )
and it has a point of intersection of abscissas x = 1 with the straight - line of equation
y = 2x + 5;
draw f(x) on a Cartesian plane;
determine its inverse function;
draw f(x) -1 on a Cartesian plane.
x-1
3) Let f(x) be a function so defined:
y=2 x
Determine the values of x for which this function is worth eight.
4) Let f(x) be a function so defined:
y=-2
-x
+ 4;
determine its domain, its codomain and its asymptote; write the equation of the
symmetrical curve about the bisecting line of the Cartesian plane y = - x.
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exponential function
5) Let f(x) be a function so defined y = 3
x
;
apply the equations of the symmetry about the y axis and then about the straight
line of equation y = - 1 ;
finally apply the equations of the translation by a vector having components
(2 , 4);
write the equation of the function so obtained and draw it.
6) Solve graphically the following equation:
ln x
= 1 - x²
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exponential function
Answer Sheets
Keywords:
Circle, intersection, limit, trigonometric function.
Conceptual Map:
+
aєR
0<a<1
decreasing
exponential
function
a>1
a=1
straight
line
growing
exponential
function
injective and
surjective
inverse
logarithmic
function
29
a=e
natural
exponential
function
exponential function
Multiple Choice:
1b, 2c, 3c, 4b, 5d, 6b
Matching:
1)
2)
3)
4)
5)
1a, 2d, 3b, 4c
1d, 2a, 3b, 4c
1c, 2a, 3b
1b, 2a
c
True or False:
1T, 2F, 3F, 4F, 5T, 6F, 7T, 8T, 9T, 10F
Completion:
1) We call general exponential function the function having equation y = a
x
where a
is a fixed number greater than 0 and the power x is the variable that could be a
negative or positive number.
2) The domain of the exponential function is the set of real numbers R and it passes
through the point (0;1); its graph lies above the x axis and if the base a > 1, it’s
strictly growing and the x axis is a horizontal left asymptote for the curve.
3) If the base 0 < a < 1, the x axis is a horizontal right asymptote for the curve.
4) The functions y = a
x
and y = a
-x
are symmetrical about the y axis, in fact if
we apply the equations of this symmetry to the function y = a
curve y = a
-x
x
, we obtain the
.
5) The exponential function is invertible because it is bijective and its inverse function
is logarithmic function.
6) Euler’s number e is an irrational number and a transcendental number, and natural
exponential function has equation y = e
x
30
.
exponential function
7) A function having equation of the type y = ax + b
represents the curve y = a
x
with
a > 0 and
shifted down by b.
8) A function having equation of the type y = a x + b
with
x
represents the curve y = a
shifted b points to the right.
a > 0 and
Activities Based on Problem Solving:
1) x = 0
2) a = 2, b = - 1;
3) x = -
y = log ( x + 1 ) - 2
2
1
2
4) D = R; C = { y є R / y < 4}; asymptote y = 4; y = log ( x + 4 )
2
x-2
5) y = -
1
b<0
+ 2
3
6) x = ~ 0,5 e x = 1
31
b<0
exponential function
Flow Chart:
start
This equation is the solution of a system between the equations of
the exponential function and of the parabola
At first i draw the graph of the parabola having equation y = - x²+2
Then i draw the graph of the exponential function y = 2
domain
I draw the symmetric curve of the function y = 2
in its domain
x
x
in its
about the x axis
The points of intersection between the parabola and the two
exponential functions are the solutions of this equation
end
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