exponential function exercises - Eniscuolaclilinactionrepository.eniscuola.net/wp-content/uploads/2012/05/... · Flow Chart Think and Discuss ... scritti e per esercitazioni basate

  • View
    216

  • Download
    0

Embed Size (px)

Text of exponential function exercises -...

  • 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

  • 10

    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

  • 11

    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

  • 12

    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

  • 13

    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)

  • 15

    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

  • 16

    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

  • 17

    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

  • 18

    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

  • 19

    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

  • 21

    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