exponential function exercises -...
Transcript of exponential function exercises -...
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maths course exercises
Liceo Scientifico Isaac Newton - Roma
exponential function
in accordo con il
Ministero dellIstruzione, Universit, 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
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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
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1
Strategies Before Prerequisites
Exponential function
Rules of the powers
Injectivity of a function
Surjectivity of a function
Invertible function
Strictly growing function
Strictly decreasing function
Symmetries
Translations
Dilations
Compressions
Maths
the prerequisites are
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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?
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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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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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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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5
Strategies During Conceptual Map
Complete the conceptual map using the following words:
inverse
logarithmic function
0 < a < 1 a = e
decreasing
exponential function
injective and surjective a > 1
a R
+
growing exponential
function
straight line
natural
exponential function
a = 1
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6
Strategies After Multiple Choice
1. What transformations have you to apply to the function y = 2 to
obtain the following function y = 2 + ?
a. a translation by a vector having components (-3 , + )
b. a translation by a vector having components (+3 , + )
c. a dilation by the constants 3 and
d. a translation by a vector having components (+3 , - )
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 > - with a 0
b. - < a < 0
c. a > 0
d. it doesnt exist
3
4
3
4
3
4
3
4
3
4
x
x - 3
x
1
2
1
2
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3. What are the values of a, b, c, with b > 0 such that the graph of the
function of equation y = a b + c is the following ?
a. a = 4, b = + , c = + 1
b. a = 4, b = - , c = - 1
c. a = 4, b = + , c = - 1
d. a = 4, b = - , c = + 1
4. What is the equation of the function of the type y = a 2 + b,
the graph of which is symmetrical about the straight line y = -2 ?
a. y = - 2 + 4
b. y = - 2 - 4
c. y = +2 + 4
d. y = +2 - 4
x
0 X 1
1
2
1
2
1
2
1
2
x
x
x
x
x
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5. What is the equation of the exponential function of the type y = 2 the graph of which is symmetrical about the straight line x = 1 ?
a. y = 2
b. y = 2
c. y = 2
d. y = 2
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 doesnt exist
f(x)
+ 2 + x
- 2 + x
+ 2 - x
- 2 - x
y =
x 2a
a - 2
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7
Strategies After Matching
1) Match the equations of the exponential functions with the definitions:
3
8
x
y =
- x
9
8 y = -
8
9
- x
y = -
x
8
3 y = -
Strictly decreasing
Strictly growing
Strictly growing
a
b
c
da
Strictly growing
1 2 3 4
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Strategies After Matching
2) Match the graphs of the exponential functions with the equations:
Y
0 X c 1
1
3
2
4
c b d a
Y
1 0 X
Y
0 X
Y
1
Y
0 X 1
x
y =
1
3 5
2
x -1
y = 2
5
x
y = + 1 2
3
x
y =
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Strategies After Matching
3) Match the functions with the transformations:
y = 2 + 2
y =
1 x
2 2
1
x + 1
y =
- 2 - 1 3 x
c
a
b translation of the function by a vector having components (0 , 1) and symmetry about the x axis.
translation of the function by a vector having components (- 1 , 0)
translation of the function by a vector having components (0 , 2)
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Strategies After Matching
4) Match the graph of the exponential function with its inverse:
0
Y
X 1
Y
X 0
1 2
a b
Y
X 0 0
Y
X
1
1 1
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Strategies After Matching
5) Match the equation of the exponential function with its right graph:
c
0
0
b
0
a
Y
X
X
X
X
8
Y
- 8
8
8
d Y Y
y = - 1
y = - 1
y = + 1
y = - 1
2
2
y = - 1
x - 2
3
1
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8
Strategies After True or False
State if the sentences are true or false.
1) All exponential functions of the type y = a if a > 1 pass through the point
( 0 ; 1 ).
2) Every exponential function y = a lies above the x axis only if x is greater than
0.
3) If a is greater than 1, the exponential function y = a is strictly decreasing.
4) The exponential function is asymptotic to the y axis.
5) If a = 1 the exponential function y = a becomes a straight line that is parallel
to the x axis.
6) The functions y = a and y = a are symmetrical about the x axis.
7) The logarithmic function is the inverse of the exponential function.
x
T F
T F
T F
T F
T F
T F
T F
x
- x
x - x
x
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exponential function
8) The tangent to the natural exponential function in the point ( 0 ; 1 ) is parallel to
the bisecting line y = x.
9) The number e isnt solution of any polynomial equation with rational coefficients.
10) If 0 < a < 1, when x > 0 the exponential function y = a grows faster, while if
x < 0 the function decreases faster.
T F
T F
T F
x
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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, its
3) If the base 0 < a < 1 , the x axis is a
4) The functions y = a and y = a are
in fact if we apply the equations
5) The exponential function is invertible because
and its inverse
x - x
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6) Eulers number e is
and natural exponential function has
7) A function having equation of the type y = a + b with a > 0 and b < 0
represents
8) A function having equation of the type y = a with a > 0 and b < 0
represents
x
x + b
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10
Strategies After Flow Chart
Complete the flow chart using the terms listed below:
How many solutions does this equation have?
2 = - x + 2 x
I draw the symmetrical curve of the function y = 2 about the x
axis in its domain
The points of intersection between the parabola and the two exponential functions are the solutions of this equation
This equation is the solution of a system between the equation of
the exponential function and of the parabola
I draw the graph of the parabola having equation y = - x + 2
I draw the graph of the exponential function y = 2 in its domain
x
x
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end
start
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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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12
Strategies After Summary
We call general exponential function the function having equation y = a
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, its strictly growing and its 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, its strictly decreasing and its 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 its invertible; its inverse
function is logarithmic function whose equation is y = log x; its graph is
symmetrical about the bisecting line of the Cartesian plane y = x.
We call natural exponential function the exponential function having equation
y = e ; the base e is called Eulers 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.
x
a
x
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exponential function
1. Answer the following questions. The questions could be a answered in a
written or oral form, depending on the teachers 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 Eulers 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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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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13
Activities Based on Problem Solving
Solve the following problems:
1) Solve graphically the following equation:
2) Let f(x) be a function so defined:
f(x) = 2 + b;
3) Let f(x) be a function so defined: y = 2
Determine the values of x for which this function is worth eight.
4) Let f(x) be a function so defined: y = - 2 + 4;
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) on a Cartesian plane.
x - 1
x
= 2x + 1 2
1
x
x + a
-1
-x
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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5) Let f(x) be a function so defined y = 3 ;
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:
x
ln x = 1 - x
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Answer Sheets
Keywords:
Circle, intersection, limit, trigonometric function.
Conceptual Map:
inverse logarithmic
function
0 < a < 1 a = 1 a = e
growing exponential
function
straight line
decreasing
exponential function
injective and surjective
a > 1
a R +
natural exponential
function
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Multiple Choice:
1b, 2c, 3c, 4b, 5d, 6b
Matching: 1) 1a, 2d, 3b, 4c 2) 1d, 2a, 3b, 4c
3) 1c, 2a, 3b 4) 1b, 2a 5) 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 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, its
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 and y = a are symmetrical about the y axis, in fact if we apply the equations of this symmetry to the function y = a , we obtain the
curve y = a .
5) The exponential function is invertible because it is bijective and its inverse function
is logarithmic function.
6) Eulers number e is an irrational number and a transcendental number, and natural
exponential function has equation y = e .
x
x
- x
x
- x
x
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7) A function having equation of the type y = ax + b with a > 0 and b < 0
represents the curve y = a shifted down by b.
8) A function having equation of the type y = a with a > 0 and b < 0
represents the curve y = a shifted b points to the right.
Activities Based on Problem Solving: 1) x = 0
2) a = 2, b = - 1; y = log ( x + 1 ) - 2
3) x = -
4) D = R; C = { y R / y < 4}; asymptote y = 4; y = log ( x + 4 )
5) y = - + 2
6) x = ~ 0,5 e x = 1
1
2
x
x + b
x
2
2
3
1
x - 2
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exponential function
Flow Chart:
Materiale sviluppato da eniscuola nellambito del protocollo dintesa con il MIUR
end
start
The points of intersection between the parabola and the two exponential functions are the solutions of this equation
At first i draw the graph of the parabola having equation y = - x+2
This equation is the solution of a system between the equations of the exponential function and of the parabola
I draw the symmetric curve of the function y = 2 about the x axis
in its domain
x
Then i draw the graph of the exponential function y = 2 in its
domain
x