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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

2

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

3

exponential function

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

4

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?

5

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

6

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

7

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.

8

exponential function

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

9

exponential function

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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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 + 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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exponential function

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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exponential function

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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exponential function

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 =

14

exponential function

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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exponential function

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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exponential function

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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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 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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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, 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

20

exponential function

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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exponential function

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

22

exponential function

end

start

23

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.

24

exponential function

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

25

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.

26

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

27

exponential function

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.

28

exponential function

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

29

exponential function

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

30

exponential function

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

31

exponential function

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

32

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