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maths course exercises 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 5 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 6 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. 7 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 8 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 9 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 10 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) 14 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 15 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 …………………………………………………………………………………………………………… 19 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 ………………………………………………………………………………………………………………… 20 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 21 exponential function start end 22 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. 23 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. 25 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 26 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. 27 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 28 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 Materiale sviluppato da eniscuola nell’ambito del protocollo d’intesa con il MIUR 32