Grade 9 / Linear functions, affine functions and problems 58 exercises (100% corrected)

a
x-4-3-2-101234y-2-112CfCg xxyy-4-2024-6-4-2246 xyTemps (s)00,10,20,30,40,50,6Tension(V)123456 1. Linear functions and proportionality coefficient E.2569 In the reference frame below, the representative curves of two functions f and g have been rep-resented by straight lines passing through the origin of the reference frame. 1 By graphical reading, complete the table of values below x − 4 0 1 f ( x ) − 1 2 3 4 x − 1 2 1 g ( x ) 3 2 0 − 3 4 2 Establish that these two tables of values are also propor-tionality tables. Their proportionality coefficients will be given. E.2570 Consider the function f whose ta-ble of values is a proportionality table whose proportionality coefficient has the value 1.5 . 1 Complete the table of values for the function f below : x − 4 − 1 4 f ( x ) − 4.5 0 3 2 The plane is given a reference frame shown below : Attention to the orientation of graphique a Place the set of these points in the benchmark. What do you notice? b Draw the representative curve of the function f . c What feature does the representative curve of the func-tion f have? E.9282 A capacitor is an electronic component that stores electrical energy for later release. The following graph shows the evolution of the voltage mea-sured across a capacitor as a function of time when it is charg-ing. 1 Is this a proportionality situation? Justify. 2 What is the voltage measured after 0.2 s ? 3 After how long will the voltage across the capacitor have reached 60 % of the maximum voltage which is estimated to be 5 V ? E.5683 Consider a function f that admits the value table below : x − 1 0 3 4 6 f ( x ) − 3 0 9 12 18 1 Justify that the table of values is a proportionality table. 2 Which of the expressions below represents the expression for the function f ? a f ( x ) = x + 3 b f ( x ) = 3 x c f ( x ) = x 3 2. Linear function: expression https://chingmath.fr chapExoCorrec/2569 sacados/2569 x-4-3-2-101234y-2-112CfCg chapExoCorrec/2570 sacados/2570 xxyy-4-2024-6-4-2246 chapExoCorrec/9282 sacados/9282 xyTemps (s)00,10,20,30,40,50,6Tension(V)123456 chapExoCorrec/5683 sacados/5683
-4-3-2-12345I-2-123JO x-4-3-2-101234y-112 -4-3-2-1234I-2-12JOCfCgCh E.5685 Consider the linear f function whose image of the number 4 has value 2 . Give the expression for the function f . E.973 Consider a linear function f whose graphical representation passes through the point with coor-dinate (2 ; − 3) . Determine the algebraic expression of the function f . E.5684 Consider the linear f function with the slope 2 3 . 1 Determine the expression for the function f . 2 What are the images of the numbers 6 and 8 by the func-tion f ? 3 What is the antecedent of the number − 2 by the function f ? 3. Linear Function: Constructing the Graph E.11657 Proposition: dans le plan muni d’un repère, si une fonc-tion est une fonction linéaire alors sa courbe représentative est une droite passant par l’origine. On considère la fonction f définie par : f ( x ) = 0 ; 5 × x 1 Compléter le tableau de valeurs de la fonction f : x − 3 − 1 0 1 4 f ( x ) 2 Dans le repère, tracer la courbe C f représentative de la fonction f : E.11658 Proposition - réciproque: dans le plan muni d’un repère, si une fonction admet pour représentation une droite non-parallèle à l’axe des ordonnées alors cette fonction est une fonction affine. On considère la fonction f dont la courbe représentative C f est donnéee dans le repère: 1 Quelle est la nature de la fonction f ? 2 Compléter le tableau de valeurs ci-dessous : x − 2 − 1 0 1 2 3 f ( x ) 3 Donner l’expression de la fonction f . 4. Linear Function: Analysis of the Graph E.11659 Dans le plan muni d’un repère, on a représenté les courbes représentatives C f , C g , C h respective-ment des fonctions f , g , h : 1 Quelle fonction est une fonction linéaire? 2 Donner l’expression de cette fonction. https://chingmath.fr chapExoCorrec/5685 sacados/5685 chapExoCorrec/973 sacados/973 chapExoCorrec/5684 sacados/5684 chapExoCorrec/11657 sacados/11657 -4-3-2-12345I-2-123JO chapExoCorrec/11658 sacados/11658 x-4-3-2-101234y-112 chapExoCorrec/11659 sacados/11659 -4-3-2-1234I-2-12JOCfCgCh
x-4-3-2-101234y-2-112CgCf xxyy-4-3-2-1O134-2-112CfMH xxyy-4-3-2-1O134-11CfMH x-4-3-2-101234y-2-112CfCgCh E.11660 On munit le plan d’un repère et on con-sidère les deux courbe C f , C g représentative respectivement des fonctions f et g : 1 Justifier que ces deux fonctions sont des fonctions linéaires. 2 Déterminer les expressions de ces deux fonctions 5. Graphical reading of the directrix of a linear function E.9273 Consider the linear function f of slope 0.75 . Its graphical representation is given in the refer-ence frame below : Consider the three points : O (0 ; 0) ; M (2 ; 1.5) ; H (2 ; 0) 1 Justify that the point M belong to the curve C f repre-sentative of the function f . 2 a Give the measures of segments [ OH ] and [ HM ] . b Which of the following quotients is equal to the slope of the function f : OH HM ; MH HO E.9274 Consider the linear function f of slope − 0.5 . Its graphical representation is given in the refer-ence frame below : Consider the three points : O (0 ; 0) ; M (2 ; − 1) ; H (2 ; 0) 1 Justify that the point M belong to the curve C f repre-sentative of the function f . 2 a Give the measures of segments [ OH ] and [ HM ] . b Which of the following quotients is equal to the slope of the function f : − OH HM ; − MH HO ; OH HM ; MH HO E.5125 In the reference frame below, the three curves C f are shown, C g and C g respectively represen-tative of the three functions f , g and h . 1 Graphically justify that the three functions f , g , and h are linear functions. 2 Determine graphically the slope of these three functions. 6. Introduction: Linear Functions and Proportionality https://chingmath.fr chapExoCorrec/11660 sacados/11660 x-4-3-2-101234y-2-112CgCf chapExoCorrec/9273 sacados/9273 xxyy-4-3-2-1O134-2-112CfMH chapExoCorrec/9274 sacados/9274 xxyy-4-3-2-1O134-11CfMH chapExoCorrec/5125 sacados/5125 x-4-3-2-101234y-2-112CfCgCh
xxyy-4-3-2-101234-2-112Cf xxyy-5-4-3-2-10123-1123 E.9275 We provide the plane with a refer-ence frame and consider the curve C f of a function f shown below : 1 Complete the table of values below : x − 3 0 2 3 f ( x ) − 1 0 2 Does the table of values reflect a proportionality situa-tion? E.9276 Consider the function f defined for any number x and whose image f ( x ) has the expression : f ( x )= − 0.5 x +0.5 1 a Complete, below, the table of values for the function f : ‘ .1 x − 4 − 1 2 ‘ .3 f ( x ) 2 0.5 0 b Justify that this table does not reflect a proportionality situation. 2 a Place the points associated with the array of values for the function f in the table below : b Complete the sentence below : ˇ the curve C f of the function f is a . . . . . . that does not pass through . . . . . . ı 3 The following questions use the point A ( − 4 ; 2.5) belong-ing to the curve C f and its coordinates. a Complete the table below : ‘ .1 x − 4 − 1 2 ‘ .2 x − ( − 4) 0 3 6 ‘ .3 f ( x ) 2 0.5 0 ‘ .4 f ( x ) − 2.5 − 0.5 − 2 − 2.5 b Justify that the lines ‘ .2 and ‘ .4 in the previous table represent a proportionality situation. The associated proportionality coefficient will be given. 7. Affine functions E.11640 Chacune des fonctions ci-dessous est une fonction affine. Compléter le tableau ci-dessous : Coefficient directeur Ordonnée à l’origine Expression de la fonction 2 3 f ( x ) = − 3 1 g ( x ) = 1 4 h ( x ) = − 2 − 1 j ( x ) = 0 5 k ( x ) = − 1 4 ‘ ( x ) = E.11641 Compléter le tableau ci-dessous : Expression de la fonction Nature Coefficient directeur Ordonnée à l’origine f ( x ) = 2 x + 1 f ( x ) = 3 x f ( x ) = x + 5 f ( x ) = 5 https://chingmath.fr chapExoCorrec/9275 sacados/9275 xxyy-4-3-2-101234-2-112Cf chapExoCorrec/9276 sacados/9276 xxyy-5-4-3-2-10123-1123 chapExoCorrec/11640 sacados/11640 chapExoCorrec/11641 sacados/11641
x-10123456y123 -4-3-2-1234I-2-12JOCfAB E.11642 Compléter le tableau ci-dessous : Expression de la fonction Nature Coefficient directeur Ordonnée à l’origine f ( x ) = 5 − 3 x f ( x ) = − 2 x f ( x ) = − 2 + x f ( x ) = 0 E.11643 Exemple commenté: On considère la fonction f affine dont le coefficient directeur vaut 3 et dont l’ordonnée à l’origine vaut 5 . L’expression de la fonction f est : f ( x ) = 3 x + 5 L’image du nombre 2 a pour valeur: f (2) = 3 × 2 + 5 = 6 + 5 = 11 Notons x l’antécédent du nombre 8 par la fonction f . Le nombre x est solution de l’équation: f ( x ) = 8 3 × x + 5 = 8 3 × x = 8 − 5 3 × x = 3 x = 3 3 = 1 On considère la fonction g affine dont le coefficient directeur a pour valeur 2 et dont l’ordonnée à l’origine à pour valeur − 3 . 1 Donner l’expression de la fonction g . 2 Déterminer l’image du nombre 3 par la fonction g . 3 Déterminer l’antécédent du nombre 5 par la fonction g . E.11644 On considère la fonction f affine dont le coefficient directeur a pour valeur − 4 et dont l’ordonnée à l’origine à pour valeur 1 . 1 Donner l’expression de la fonction f . 2 Déterminer l’image du nombre 5 par la fonction f . 3 Déterminer l’antécédent du nombre 17 par la fonction f . E.11645 On considère la fonction f affine dont le coefficient directeur a pour valeur − 3 et dont l’ordonnée à l’origine à pour valeur 7 . 1 Donner l’expression de la fonction f . 2 Déterminer l’image du nombre − 1 par la fonction f . 3 Déterminer l’antécédent du nombre 4 par la fonction f . E.11646 On considère la fontion f affine dont le coefficient directeur vaut 0 ; 5 et l’ordonnée à l’origine vaut 1 . 1 Donner l’expression de la fonction f . 2 a Déterminer les images des nombres 2 et 4 par la fonction f . b On note C f la droite représentative de la fonction f . Commpléter les pointillés suivants : Par la fonction f , l’image de 2 est . . . . . . . Donc la droite C f passe par le point de coordonnnées (2 ; ::: ) . Par la fonction f , l’image de 4 est . . . . . . . Donc la droite C f passe par le point de coordonnnées (4 ; ::: ) . 3 Dans le repère ci-dessous, tracer la droite C f représenta-tive de la fonction f : 4 Graphiquement, déterminer l’antécédent du nombre 0 ; 5 par la fonction f . 8. Directing coefficient and affine function E.9278 Proposition: Let f be an linear functionwith directrix a . For any number x 1 and x 2 such that x 1 = x 2 , we have : a = f ( x 2 ) − f ( x 1 ) x 2 − x 1 Consider the function f affine whose representative curve C f is shown belowbelow in the reference frame and are repre-sented the points A and B belonging to C f . 1 Give the imges of the numbers − 2.5 and 2 by the function f . 2 Determine the slope of the linear function f . https://chingmath.fr chapExoCorrec/11642 sacados/11642 chapExoCorrec/11643 sacados/11643 chapExoCorrec/11644 sacados/11644 chapExoCorrec/11645 sacados/11645 chapExoCorrec/11646 sacados/11646 x-10123456y123 chapExoCorrec/9278 sacados/9278 -4-3-2-1234I-2-12JOCfAB
-2-12345I-12JOCfCg xxyy-4-3-2-101234123ABC xxyy-3-2-10123-2-112(d1(d2(d3(d4 x0123456789y12345 E.9279 Definition: in the plane provided with a reference frame, consider a straight line ( d ) notparallel to the ordinate axis and two points A ( x A ; y A ) and B ( x B ; y B ) belonging to the line ( d ) . We call the slope of the line ( d ) the number a defined by: a = y B − y A x B − x A (this value does not depend on the points A and B ) Proposition: consider the plane provided with a reference frame Any line not parallel to the y-axis is the representative curve of an linear function The directrix of an linear functionis equal to the direc-trix of its representative line. a straight line Consider the plane provided with a reference frame and two linear functions f and g . The representative curves C f and C g of the functions f and g are shown below : Determine the slopes of the functions f and g . 9. Directing coefficient and direction of variation E.9280 Consider the plane provided with a reference frame and two linear functions f and g whose rep-resentative curves are noted C f and C g . Furthermore, we know that : function f has directional coefficient 0.5 . function g has slope − 0.5 . point A ( − 2 ; 1.5) belongs to both curves C f and C g . each of the points B (0.5 ; 2.75) and C (1 ; 0) belongs to one of these two curves. 1 a Determine the slope associated with the line ( AB ) . b Determine the slope associated with the line ( AC ) . 2 Associate with each curve, C f and C g , its representative line. E.9281 In the plane provided with a refer-ence frame, consider the four lines ( d 1 ) , ( d 2 ) , ( d 3 ) and ( d 4 ) shown below : Associate each of these lines with the curve of representative of one of the four linear functions below : Consider the four linear functions : f : x ↦−→ 0.5 x + 0.75 g : x ↦−→ − 0.75 x + 0.75 h : x ↦−→ x − 0.25 j : x ↦−→ − 0.25 x − 0.25 10. Drawing representative curves E.4052 We consider the plane with the following reference frame : https://chingmath.fr chapExoCorrec/9279 sacados/9279 -2-12345I-12JOCfCg chapExoCorrec/9280 sacados/9280 xxyy-4-3-2-101234123ABC chapExoCorrec/9281 sacados/9281 xxyy-3-2-10123-2-112(d1(d2(d3(d4 chapExoCorrec/4052 sacados/4052 Aix-Marseille Juin 2006 x0123456789y12345
x-4-3-2-101234y-2-1123Cf xxyy-2-10123456-112AB Consider the two functions f and g defined by: f ( x ) = − 1 4 · x + 4 ; g ( x ) = 1 3 · x 1 a Give the nature of the two functions f and g . b Complete the following two tables of values : x 0 4 6 8 f ( x ) x 0 3 6 9 g ( x ) 2 Perform the plot of the representative curves of the func-tions f and g in the frame below. 3 The representative curves C f and C g will provide approx-imate values of the images and antecedents of numbers by these two functions. Leave the construction lines to answer the following questions : a Determine the approximate value of the image of 2 by the function f . b Determine the approximate value of the image of 4 by the function g . c Determine le (s) valeur (s) approchée (s) of the antécé-dent (s) of the number 2 by the function f . d Determine le (s) valeur (s) approchée (s) of the antécé-dent (s) of the number 2.5 by the function g E.5139 Consider the plane provided with a reference frame and the curve C f of a function f . 1 a What is the nature of the function f ? b Determine the slope of the function f . 2 Consider the two functions g and h defined by: g ( x ) = x − 1 ; h ( x ) = − 1 2 x + 2 We denote C g and C h their representative curves : a Give the associated director coefficients of the func-tions g and h . b Which of the four points below belong to the curve C g ? Which ones belong to the curve C h ? A (0 ; 2) ; B (0 ; − 1) ; C (3 ; 2) ; D (2 ; 1) c Perform curve plots C g and C h . E.9277 Consider the linear function f whose slope is 1.5 and whose image of 2 is 1 . We provide the plane with a reference frame and consider the two points A (2 ; 1) and B (1 ; 2) . We will note C f the representative curve of the function f . 1 Which of the two points A and B belongs to the curve C f . 2 Draw the curve C f justifying your approach. 11. Calculation of images and priors E.5682 Consider the linear function f of slope 2 and y-intercept 1 1 Determine the algebraic expression for the function f . 2 Determine the image of the number 3 by the function f . 3 Determine the antecedent of the number 5 by the func-tion f . https://chingmath.fr chapExoCorrec/5139 sacados/5139 x-4-3-2-101234y-2-1123Cf chapExoCorrec/9277 sacados/9277 xxyy-2-10123456-112AB chapExoCorrec/5682 sacados/5682
x-4-3-2-101234y-4-3-2-11234(d1(d2(d3(d4(d5ABCDABCDABCDABCDABCD AnnéePourcentage197319801990200220142026203810%20%30%40%50%60%70%80% xxyy-4-3-2-1234I-2-12JOCf E.963 In the coordinate system shown be-low, five lines ( d 1 ) , ( d 2 ) , ( d 3 ) , ( d 4 ) , and ( d 5 ) are plotted : We also consider the following five linear functions : f ( x ) = − 0 ; 2 · x − 2 ; 4 ; g ( x ) = − 1 ; 5 · x + 1 ; 5 h ( x ) = 0 ; 8 · x ; j ( x ) = − 0 ; 5 · x ; k ( x ) = 2 · x − 3 1 a Find the coordinates of the point A . b Determine the two linear functions whose representa-tive lines pass through the point A . c Find the coordinates of the point B . d From this, determine the linear function whose graph is the line ( d 1 ) . 2 a Find the coordinates of point C . b Determine the unique linear function whose graph passes through point C . 3 a Using the coordinates of point D , determine the two linear functions whose representative lines pass through point D . b Which function has the line ( d 4 ) as its representative curve? E.9283 We study the energy consump-tion in France and we can observe the evolution of the share of oil over the years from a graphical representation like the one proposed below. The dotted lines indicate that it is assumed that the decline in the share of oil will continue at the rate observed since 2002 . Following this assumption, we can model the share of oil (ex-pressed as a percentage) as a function of year a by the function P , defined as : P ( a )= − 17 48 a +743.5 1 Write the calculation to verify that : P (1990) ≈ 38.7 2 According to this model, starting in what year will oil’s share be zero? 12. Affine functions: calculating images E.11152 Consider the f affine function whose representative curve C f is shown below in the reference frame. below in the reference frame and are represented the points A and B belonging to C f . 1 Graphically, give the image of the number 2 by the func- tion f . 2 Which of the following algebraic expressions defines the function f : a g ( x ) = 0.5 x + 0.25 b h ( x ) = 0.5 x − 0.25 c j ( x ) = − 0.5 x + 0.25 d k ( x ) = − 0.5 x − 0.25 https://chingmath.fr chapExoCorrec/963 sacados/963 x-4-3-2-101234y-4-3-2-11234(d1(d2(d3(d4(d5ABCDABCDABCDABCDABCD chapExoCorrec/9283 sacados/9283 AnnéePourcentage197319801990200220142026203810%20%30%40%50%60%70%80% chapExoCorrec/11152 sacados/11152 xxyy-4-3-2-1234I-2-12JOCf
xxyy-2-123456I-2-1JOCf fx1234ABCDEFGHx-3-2-10123f(x)22171272-3-8g(x)138545813C2−5×C17 I012345678910P100020003000400050006000700080009000Partie B: représentation de la fonctiong E.11216 We consider the affine function f whose representative curve C f is shown below below in the coordinate system, and the points A and B belonging to C f are shown. 1 Graphically, give the image of the number 2 using the function f . 2 Which of the algebraic expressions defines the function f : a g ( x ) = 0.5 x + 1.25 b h ( x ) = 0.5 x − 1.25 c j ( x ) = − 0.5 x + 1.25 d k ( x ) = − 0.5 x − 1.25 13. Problems E.5921 A spreadsheet was used to calculate the images of different values of x by an linear function f and by another function g . A copy of the resulting screen is given below. 1 What is the image of − 3 by f ? 2 Calculate f (7) . 3 Give the expression for f ( x ) 4 We know that g ( x )= x 2 +4 . A formula was entered in cell B3 and then copied to the right to complete the cell range C3:H3 . What is this formula? E.9284 In physics, the voltage U across a ˇresistorı is proportional to the intensity I of the current flowing through it, i.e.: U = R × I where R (value of the resistor) is the coefficient of propor-tionality. Remember that the unit of current is the ampere and the unit of voltage is the volt. The current I (in amperes) 0 ; 02 0 ; 03 0 ; 04 0 ; 08 Voltage U (in volts) 3 4 ; 5 6 12 1 We call f the function that gives the voltage U as a func-tion of the current I . a Justify that the function f is a linear function. Specify the slope of the function f . b Determine the exact value of the current when U =9 volts. In physics, the power P of the ˇresistanceı is the product of the voltage U across its terminals and the current I flowing through it, i.e.: P = U × I Remember that the unit of power is the watt. 2 Using the expression obtained in question 3 of part A , justify that : P = 150 × I 2 We call g the function that gives the power P as a function of the intensity I and we represent, in the plane equipped with a coordinate system, the curve C g representative of the function g https://chingmath.fr chapExoCorrec/11216 sacados/11216 xxyy-2-123456I-2-1JOCf chapExoCorrec/5921 sacados/5921 fx1234ABCDEFGHx-3-2-10123f(x)22171272-3-8g(x)138545813C2−5×C17 chapExoCorrec/9284 sacados/9284 I012345678910P100020003000400050006000700080009000Partie B: représentation de la fonctiong
Temps (en s)020406080100Vitessederotation(ennombredetoursparseconde)510152025Inspiréde:https://www.sciencesetavenir.frlfondamental/combien-de-temps-peut-tourner-votre-hand-spinner-112808 3 Graphically read the power P when I =5 amperes (the construction lines used for reading will appear on the graph) . 4 Graphically read a antecedent of 2 500 by the function g (the construction lines used to read the graph will be shown) . 5 Is the power P proportional to the intensity I ? Justify your answer. E.6301 There are different units of temperature measurement : in France the degree Celsius ( o C ) is used, in the United States the degree Fahrenheit ( o F ) . To change from degrees Celsius to degrees Fahrenheit, multi-ply the starting number by 1.8 and add 32 to the result. 1 What would a thermometer read in degrees Farhenheit if it were dipped into a pan of freezing water? Recall that water freezes at 0 o C . 2 What would a Celsius thermometer indicate if it were immersed in a pan of water heated to 212 o F ? What happens? 3 a If we denote x the temperature in degrees Celsius and f ( x ) the temperature in degrees Farhenheit, ex-press f ( x ) as a function of x . b What is the name of this type of function? c What is the image of 5 by the function f ? d What is the antecedent of 5 by the function f ? e Translate the relationship f (10)=50 into temperature conversion terms. E.9286 The ˇ hand-spinner ı is a kind of flat top that spins on itself. The ˇ hand-spinner ı has been given an initial rotational speed at time t =0 , then, over time, its rotational speed de-creases until the ˇ hand-spinner ı comes to a complete stop. Its rotational speed is then equal to 0 . Using a measuring device, the rotational speed was recorded, expressed in revolutions per second. This speed is plotted against time in seconds on the graph below : Launching the ˇ hand-spinner ı with an initial speed of 20 revo-lutions per second, the rotational speed of the ˇ hand-spinner ı as a function of time t , denoted V ( t ) , is expressed as : V ( t )= − 0.214 × t +20 1 Calculate its rotational speed after 30 s . 2 After how long will the ˇ hand-spinner ı stop? Justify with a calculation. 3 Is it true that, in general, if you spin the ˇ hand-spinner ı twice as fast at the start, it will spin twice as long? Jus-tify. https://chingmath.fr chapExoCorrec/6301 sacados/6301 chapExoCorrec/9286 sacados/9286 Temps (en s)020406080100Vitessederotation(ennombredetoursparseconde)510152025Inspiréde:https://www.sciencesetavenir.frlfondamental/combien-de-temps-peut-tourner-votre-hand-spinner-112808
Nombredejours012345678910111213Coût(eneuros)102030405060708090100110120130140150160170180190200210220230240250260CfCg 0123456789101112131415162000400060008000100001200014000Tarif en FTarif renforcéDuré en heuresTarifdécourverte 9AH9Modèle 12x10263,5Modèle 2 E.9285 The Blanche-Neige ski resort proposes the following rates for the 2004-2005 season : rate A : each ski day costs 20 euros ; tariff B : by joining the sports club, whose annual mem-bership fee is 60 euros, there is a discount of 30 % on the price of each day at 20 euros. 1 Copy and complete the following table : Number of days of ski pour the 2004 season-2005 5 8 Cost with the A (in euros) 100 220 Cost with tariff B (in euros) 130 2 We call x the number of ski days during the 2004-2005 season. Express as a function of x : a the annual cost C A in euros for a user who has chosen the A rate; b the annual cost C B in euros for a user who has chosen the B rate. 3 Below are represented in a marker the curves C f and C g representative of the functions f and g defined by: f ( x )=20 x ; g ( x )=14 x +60 The questions will be answered using the graph (show the necessary lines on the graph) . a Lea needs to come skiing for twelve days during the 2004-2005 season. What is the most attractive rate for her? What is the corresponding price? b In studying the season’s rates. Chloe finds that, for her stay, the rates A and B are equal. How many days of skiing does she plan to do? What is the corresponding price? E.11656 Juliette désire apprendre la planche à voile, elle prend des renseignements auprès d’un club qui propose trois tarifs mensuels. Le tarif découverte à 1 600 F par heure de cours. Le tarif personnalisé qui comprend une carte d’adhérent à 4 800 F et un prix fixe de 600 F par heure de cours. Le tarif renforcé à 9 600 F pour un nombre illimité d’heures de cours. 1 Calculer le prix à payer pour 4 heures de cours avec le tarif découverte. 2 a Montrer que 4 heures de cours avec le tarif person-nalisé coûtent 7 200 F. b Calculer le prix à payer pour 10 heures de cours avec le tarif personnalisé. On désigne par x le nombre d’heures de cours. On note P ( x ) le prix à payer en francs avec le tarif personnalisé. c Exprimer P ( x ) en fonction de x . Les fonctions donnant les prix à payer avec les tarifs dé-couverte et renforcé sont représentées sur l’annexe. 3 a Pour combien d’heures de cours ces deux tarifs sont-ils égaux ? b Tracer la représentation graphique de la fonction P définie par P ( x ) = 600 x + 4800 sur l’annexe. c Quel est le tarif le plus économique pour Juliette si elle décide de prendre 7 heures de cours ? Justifier la réponse. 4 Pour combien d’heures de cours Juliette paie-t-elle le même prix avec le tarif personnalisé et le tarif renforcé ? 14. Problems and geometry E.9288 https://chingmath.fr chapExoCorrec/9285 sacados/9285 Nombredejours012345678910111213Coût(eneuros)102030405060708090100110120130140150160170180190200210220230240250260CfCg chapExoCorrec/11656 sacados/11656 0123456789101112131415162000400060008000100001200014000Tarif en FTarif renforcéDuré en heuresTarifdécourverte chapExoCorrec/9288 sacados/9288 9AH9Modèle 12x10263,5Modèle 2
x32;5x2;5 x00.511.5y123456789101112131415161718 The unit of length is the centimeter. The diagrams above represent two clipboards. Model 1 is a regular pyramid whose height [ AH ] mea-sures x and whose base is a square with side 9. Model 2 consists of two parallelepipeds are 2, 6, 10. The dimensions of the other are x , 2 and 3.5 1 In this question x =8 . Then calculate the volume of each paperweight (the unit of volume is the cubic cenitmeter) . 2 We coniderate the two functions f and g defined : f ( x ) = 27 x ; g ( x ) = 7 x + 120 In the plane related to a reference frame where : On the x-axis, 1 cm will represent one unit. On the ordinate axis, 1 cm will represent ving units. the x-axis and y-axis are perpendicular. Graph the representative curves C f and C g of the func-tions f and g . 3 Graphically, determine the value of x for which the two clipboards of model 1 and model 2 have the same volume? Then give this volume, justifying your approach. E.9287 In order to collect rainwater from his roof, Lucas decides to install a water collector in the soil of his garden. The depth he has available is 2.5 m . A manufacturer then offers him the two tank models dia-grammed below. Dimensions are in meters. The first model has the shape of a straight block, the second is cylindrical: in each case, x can vary between 0.5 m and 1.5 m . 1 Complete the table below. Longueur x (en m ) 0.5 1.5 Volume du réservoir R 1 (en m 3 ) Volume of réservoir R 2 (en m 3 ) Valeur exacte Valeur arrondie à 0.1 m 3 Note: The details of the exact value calculations will need to be included on your copie 2 Consider the function f 2 : x ↦−→ 2.5 ıx 2 whose represen-tative curve is given belowor for values of x between 0.5 and 1.5 : Consider the function f 1 : x ↦−→ 7.5 x . Plot the function f 1 in the graph above. 3 Answer the following questions : Hint: will be answered with approximate values and construction lines will be shown to allow reading on the graph. a What is the volume of the tank R 2 for x =0.8 m ? b What is the radius of the tank R 2 so that it has a capacity of 10 m 3 ? c What is the antecedent of 9 by the function f 1 ? Inter-pret this number concretely. d For what value of x are the volumes of the two tanks equal? e For what values of x is the volume of R 1 greater than that of R 2 ? https://chingmath.fr chapExoCorrec/9287 sacados/9287 x32;5x2;5 x00.511.5y123456789101112131415161718
ABCDMNPRR1R2x1;5cm5cm2;5cm x0510y51015 ABCx5x E.954 ABCD is a rectangle: DC = 5 cm ; BC =2.5 cm . N is the point on segment [ AD ] such that : AN =1.5 cm . M is a point of the segment [ AB ] . Note x the length of segment [ AM ] expressed in centimetres ( x is between 0 and 5) . AMPN and MBCR are rectangles noted R 1 and R 2 respec-tively. 1 a Express, as a function of x , the perimeter of R 1 . b Express, as a function of x , the perimeter of R 2 . 2 Solve equation : 2 x +3= − 2 x +15 . 3 On the following reference frame, for 0 x 5 , graph the two affine functions : x ↦−→ 2 x + 3 ; x ↦−→ − 2 x + 15 4 What are the values of AM for which the perimeter of R 2 is greater than or equal to the perimeter of R 1 ? (No justification is expected.) E.11114 Consider the triangle ABC shown opposite, where x is an integer: 1 a Calculate AB and AC for x =4 . b Is triangle ABC a right triangle for x =4 ? 2 Let f be the function that associates the number x with the value AB 2 − AC 2 . a Expand the expressions : x + 7 2 ; x + 8 2 Establish the equality: f ( x ) = 2 x + 15 b Determine the antecedent of the number 25 using the function f . 3 Give the value of x so that triangle ABC is a right trian-gle at C . What are the dimensions of triangle ABC ? 15. Share E.3276 This exercise is a multiple-choice questionnaire (QCM) . No justification is required. For each of the following questions, three answers are proposed, only one is correct. For each question, indicate on the copy its number and copy the correct answer. Let f be the function defined by f ( x )= − 2 x +3 1 f ( x ) is of the form ax + b . The value of a is 3 − 2 2 2 The image of 0 by f is : 1 1.5 3 3 The straight line repre-senting the function f passes through the point A ( − 1; 1) B ( − 1; 5) C (1; − 18) 4 The antecedent of 4 by the function f is : − 5 7 2 − 1 2 5 The straight line repre-senting the function f intersects the ordinate axis en D (1.5; 0) E (0; 3) F (0; 2) 16. Unclassified exercises https://chingmath.fr chapExoCorrec/954 sacados/954 Groupe Sud-Ouest - Septembre 2002 ABCDMNPRR1R2x1;5cm5cm2;5cm x0510y51015 chapExoCorrec/11114 sacados/11114 ABCx5x chapExoCorrec/3276 sacados/3276 Brevet juin 2010 - 5 points
Temps (en s)020406080100Vitessederotation(ennombredetoursparseconde)510152025Inspiréde:https://www.sciencesetavenir.frlfondamental/combien-de-temps-peut-tourner-votre-hand-spinner-112808 E.7987 The ˇ hand spinner ı is a type of flat spinning top that rotates on its own axis. The hand spinner was given an initial rotation speed at time t =0 , then, over time, its rotation speed decreased until the hand spinner came to a complete stop. Its rotational speed is then equal to 0 . Using a measuring device, the rotational speed was recorded in terms of the number of revolutions per second. The graph below shows this speed as a function of time in seconds : 1 Are the time and rotational speed of the ˇ hand spinner ı proportional? Justify your answer. 2 Using graph reading , , answer the following questions : a What is the initial rotation speed of the ˇ hand spinner ı (in revolutions per second) ? b What is the rotational speed of the ˇ hand spinner ı (in revolutions per second) after one minute and twenty seconds? c How long will it take for the ˇ hand spinner ı to stop? 3 To calculate the rotation speed of the ˇ hand spinner ı as a function of time t , denoted V ( t ) , we use the following function V ( t ) = − 0.214 × t + V initiale t is the time (expressed in s ) that has elapsed since the start of rotation of the ˇ hand spinner ı ; V initiale is the rotation speed at which the ˇ hand spin-ner ı was launched at the start. a The ˇ hand spinner ı is launched at an initial speed of 20 revolutions per second. Its rotation speed is there-fore given by the formula : V ( t ) = − 0.214 × t + 20 Calculate its rotation speed after 30 s . b How long will it take for the ˇ hand spinner ı to stop? Justify your answer with a calculation. c Is it true that, generally speaking, if you spin the ˇ hand spinner ı twice as fast at the start, it will spin twice as long? Justify your answer. E.4133 A theater offers two ticket prices : price A or ˇ full price ı : 20 euros per ticket ; B rate or ˇ reduced rate ı : including a 60 e subscription and offering a 30 % discount on the price of each ticket at 20 euros. 1 Yann is a member of the theater. Given that he has al-ready paid for his subscription, explain why he will have to pay 14 euros per ticket. 2 Copy and complete the following table : Number of tickets purchased 5 8 Cost with the A rate (in euros) 100 220 Cost with the B rate (in euros) 130 3 Let x be the number of tickets purchased. Express in terms of x : a the annual cost C A in euros for a person who has cho-sen the A rate; b the annual cost C B in euros for a person who has cho-sen the B rate. 4 Knowing that Yann, a theater subscriber, spent a total of 242 euros, how many tickets did he purchase? 5 On graph paper, draw the following graph : on the x-axis: 1 cm for 1 tickets ; on the y-axis: 1 cm for 10 euros ; The x-axis and y-axis are perpendicular The origin of the coordinate system will be placed at the bottom left of the sheet, with the x-axis drawn on the short side of the sheet. Plot the graphs of the linear functions f and g defined by: f ( x ) = 20 x ; g ( x ) = 14 x + 60 6 In this section, we will answer various questions using the graph (draw the necessary lines on the graph) . a Lea wants to attend 12 shows. What is the best price for her? What is the corresponding price? b Looking at the season’s prices, Chloe notices that, for the number of tickets she wants, prices A and B are the same. How many shows does she plan to see? What is the corresponding price? https://chingmath.fr chapExoCorrec/7987 sacados/7987 Temps (en s)020406080100Vitessederotation(ennombredetoursparseconde)510152025Inspiréde:https://www.sciencesetavenir.frlfondamental/combien-de-temps-peut-tourner-votre-hand-spinner-112808 chapExoCorrec/4133 sacados/4133
E.968 A videocassette rental agency offers its customers a choice of two rates. Tariff 1: a monthly subscription of 15 e and 0.70 e per cassette rented. Tariff 2: a monthly subscription of 11 e and 1.50 e per cassette rented. 1 Complete the following table : Number of cassettes louées 0 1 2 6 10 Price paid with tariff 1 Price paid with tariff 2 2 We call x the number of cassettes rented by a customer in one month. Express, as a function of x : a the price paid with tariff 1, denoted P 1 ( x ) ; b the price paid with tariff 2, noted P 2 ( x ) . 3 Graphically represent affine functions. a P 1 : x ↦→ P 1 ( x ) = 0.7 x + 15 . b P 2 : x ↦→ P 2 ( x ) = 1.5 x + 11 We’ll take 1 cm on the x-axis for a cassette and 1 cm on the y-axis for 2 e . 4 a Solve the equation : 0.7 x +15=1.5 x +11 . Interpret the result. b Check this solution graphically, showing the useful dot-ted lines. 5 Using the graph, how many cassettes need to be rented in a month for rate 1 to be more attractive than rate 2? 6 Mr. Avent chose rate 2 and paid 29 e for the month. Use the graph to determine the number of cassettes he rented in the month. Show the useful dotted lines. 7 Mr. Comic chose rate 1 and paid 19.90 e for the month. a Find by calculation the number of cassettes he rented in the month. b In this case, what is the average price of renting a cas-sette? Give the result to the euro cent. 8 The agency decides to offer its customers a third rate: a monthly price of 23 e regardless of the number of cas-settes rented in the month. a Show on the same graph, the price P 3 paid with tariff 3. b How many cassettes must be rented for this new tariff to be more advantageous than the others? E.4332 In physics, the voltage U across a ˇresistorı is proportional to the intensity I of the current flowing through it, i.e.: U = R × I where R (value of the resistance) is the coefficient of pro-portionality. Remember that the unit of current is the ampere and the unit of voltage is the volt. L’intensité I (en ampères) 0.02 0.03 0.04 0.08 Voltage U (in volts) 3 4.5 6 12 1 a Check that this table is a proportionality table. b What is the coefficient of proportionality? c Calculate the voltage U if the current I is 0.07 am-peres. We call f the function that gives the voltage U as a function of the current I . 2 Specify the nature of the function f and give the alge-braic expression of f ( I ) . 3 In the coordinate system in the appendix, plot the graph of the function f . 4 Read the intensity graphically when U =10 volts (give an approximate value with the accuracy allowed by the graph) . Determine by calculation the exact value of the current when U =10 volts. Part B In physics, the power P of the ˇresistanceı is the product of the voltage U across its terminals and the current I flowing through it, i.e.: P = U × I Remember that the unit of power is the watt 1 Using the expression obtained in question 3 in part A , justify that : P = 150 × I 2 We call g the function that gives power P as a function of intensity I . 2 Calculate the image of 7.5 par the function g . The representative curve of the function g is given below. 3 Graphically read the power P when I =5 ampères (the construction lines that made the reading possible will be shown on the graph) . 4 Graphically read an antecedent of 2 500 by the function g (the construction lines that made the reading possible will be shown on the graph) . 5 Is the power P proportional to the intensity I ? Justify the answer. https://chingmath.fr chapExoCorrec/968 sacados/968 Groupe Nord - Septembre 2002 chapExoCorrec/4332 sacados/4332 Asie Juin 2011
I00,010,020,030,040,050,060,070,080,090,1U246810121416Partie A: représentation de la fonctionf I012345678910P100020003000400050006000700080009000Partie B: représentation de la fonctiong x-10-8-6-4-20246810y-8-6-4-22468 E.7988 Here is a calculation program : Choose a number; Add 1 to this number; Calculate the square of the result ; Subtract the square of the starting number from the previous result ; Write the result. 1 We choose 4 as the starting number. Prove by calculation that the result obtained with the program is 9 . 2 Let x be the number chosen. a Express the result of the program as a function x . b Prove that this result equals 2 x +1 . 3 Let f be the function defined by: f ( x ) = 2 x + 1 a Calculate the image of 0 by f . b Determine by calculation the antecedent of 5 by f . c Below, plot the representative line of the function f . d By graphical reading, determine the result obtained by choosing − 3 as the starting number in the calculation program. (leave construction lines visible) . E.952 A customer wants to buy a cell phone from a telecommunications company, which offers two subscription rates. Tariff 1: 0.30 e per minute and free mobile. Tariff 2: 0.18 e per minute and 108 e d mobile purchase. 1 Complete the following tables : Tariff 1: Duration in min : x 0 300 600 Price to be paid in e y 1 180 360 Tariff 2: Duration in min : x 0 300 900 1200 Price to be paid in e y 2 2 Express the price to be paid y 1 as a function of the call duration x for tariff 1. Express the price to be paid y 2 as a function of the call duration x for tariff 2. 3 Show in the same frame the prices to be paid y 1 and y 2 as a function of call duration ; the following scale will be used : 1 cm for 50 e ; 1 cm for 100 min communication. 4 Determine graphically (leave construction lines visible): a according to tariff 1 , the price to be paid for 500 min-utes of communication. b according to tariff 2 , the call time corresponding to an amount of 180 e . c the coordinates of the point for which the amount to be paid is identical for both tariffs. d For a duration of more than 900 minutes, which tariff is the most advantageous? https://chingmath.fr I00,010,020,030,040,050,060,070,080,090,1U246810121416Partie A: représentation de la fonctionf I012345678910P100020003000400050006000700080009000Partie B: représentation de la fonctiong chapExoCorrec/7988 sacados/7988 x-10-8-6-4-20246810y-8-6-4-22468 chapExoCorrec/952 sacados/952
Nombre de voyages01234567891011121314151617Prix à payer2000400060008000100001200014000160001800020000220002400026000EC1C2 Volume enm3203040506070809010Coût en euros200400600800100012001400160018002000220024002600O E.4122 Part A A shipping company provides two boats called CatamaranEx-press and FerryVogue for an inter-island crossing of 17 kilo-meters. 1 The departure of CatamaranExpress is at 5 h 45 min pour an arrival at 6 h 15 min . Calculate its average speed in km = h . 2 FerryVogue’s average speed is 20 km = h . At what time is it scheduled to arrive if it leaves the dock at 6 h ? Part B The graphical representations C 1 and C 2 de of two functions are given in the attachment. One of them is the graphical representation of an affine function g defined by: g ( x ) = 1 000 · x + 6 000 Using the graph, answer the following questions, showing the plots needed to read the graph. 1 Read the coordinates of the point E . 2 What are the x-coordinates of the points of intersection of the two graphs? 3 Which of these graphs is that of g ? Justify your answer. 4 What is the image of 12 by the function g ? Check your answer by calculating. 5 What is the antecedent of 15 000 by the function g ? Find this result by solving an equation. Part C The shipping company offers three rates for a trip regardless of the boat chosen : Rate M : you pay 2 500 for each trip. Fare N : you pay for a monthly pass at 6 000 dollars per trip, plus 1 000 dollars for each trip. Fare P : you pay 3 000 francs per trip up to the seventh trip, then all other trips are free until the end of the month. 1 The prices to be paid depending on the number of trips, with two of these rates represented by the curves C 1 and C 2 . Indicate the associated fare for each curve on your copy. (No justification required) . 2 On the attached document (to be returned with the copy) which includes C 1 and C 2 , construct the graph of the function f defined by: f : x ↦−→ 2 500 · x 3 By reading the graph and plotting the relevant lines on the attached document, find the number of trips for which fare N is more advantageous than the other two. E.962 Mr. Dubois is considering his move. He has commissioned two estimates : 1 The company A provided the attached graph. This rep-resents the cost of the move as a function of the volume to be transported. a What would be the cost for a volume of 20 m 3 ? Leave construction drawings visible. b Is the cost proportional to the volume transported? Justify. Let g be the function that to x , volume to be moved in m 3 , associates the cost of moving with this company. Express g ( x ) as a function of x . 2 The company B gave him a formula : f ( x ) = 10 x + 800 where x is the volume (in m 3 ) to be transported and f ( x ) the price to be paid in ( e ) . a Calculate f (80) . What does the result mean? b Determine by calculation the antecedent of 3 500 by the function f . c Graphically represent the function f on the graph shown below. 3 M Dubois estimates the volume of his move at 60 m 3 . Which company should he choose? Answers will be jus-tified graphically, leaving the plots visible. https://chingmath.fr chapExoCorrec/4122 sacados/4122 Polynesie Juin 2010 Nombre de voyages01234567891011121314151617Prix à payer2000400060008000100001200014000160001800020000220002400026000EC1C2 chapExoCorrec/962 sacados/962 Volume enm3203040506070809010Coût en euros200400600800100012001400160018002000220024002600O
fx12345ABCDEFValeur dex01234Image dex02468Valeur dex00,5124Image dex87640B2×B1 x0123456789101112y1234567891011 E.956 The Blanche-Neige ski area is of-fering the following rates for the 2004-2005 season : tariff A : each day of skiing costs 20 euros ; tarif B : by joining the sports club with an annual mem-bership fee of 60 euros, there is a discount of 30 % on the price of each day at 20 euros. 1 Yann is a member of the resort’s sports club. Knowing that he has already paid his annual membership fee, ex-plain why he will have to pay 14 euros per day of skiing. 2 Reproduce and complete the following table : Number of days of ski pour the 2004 season-2005 5 8 Cost with tariff A (in euros) 100 220 Cost with tariff B (in euros) 130 3 We call x the number of ski days during the 2004-2005 season. Express as a function of x : a the annual cost C A in euros for a user who has chosen the A tariff ; b the annual cost C B in euros for a user who has chosen the B tariff. 4 Knowing that club member Yann spent a total of 242 euros, how many days did he ski? 5 On graph paper, draw a reference point such that : abscissa : 1 cm for 1 day of skiing; on ordinates : 1 cm for 10 euros. the x-axis and y-axis are perpendicular. Place the origin of the reference frame at the bottom left of the sheet, with the x-axis drawn on the short side of the sheet. In this frame of reference, draw the graphical representa-tions of the affine functions f and g defined by: f ( x ) = 20 x ; g ( x ) = 14 x + 60 6 In this part, the various questions will be answered using the graph (show the necessary lines on the graph) . a Léa needs to come skiing twelve days during the 2004-2005 season. What is the best price for her? What is the corresponding price? b Studying the season’s rates. Chloe finds that, for her stay, the rates A and B are equal. How many days of skiing does she plan to do? What is the corresponding price? E.6288 Using the spreadsheet, we have produced tables of values for two functions whose expressions are: f ( x ) = 2 x ; g ( x ) = − 2 x + 8 1 What is the function ( f or g ) that corresponds to the formula entered in the cell B2 ? 2 Which formula has been entered in cell B5 ? 3 Which of the functions f or g is represented in the marker below? 4 Draw the graphical representation of the second function in the frame below. 5 Give, with justification, the solution to the equation : 2 x = − 2 x + 8 https://chingmath.fr chapExoCorrec/956 sacados/956 Probleme - Aix marseille 2006 chapExoCorrec/6288 sacados/6288 fx12345ABCDEFValeur dex01234Image dex02468Valeur dex00,5124Image dex87640B2×B1 x0123456789101112y1234567891011
x32;5x2;5 x00.511.5y123456789101112131415161718 9AH9Modèle 12x10263,5Modèle 2 E.5147 In order to collect rainwater from his roof, Lucas decides to install a water collector in the ground in his garden. The depth available to him is 2.5 m . A manufacturer offers him two models of tanks, as shown be-low. The dimensions are in meters. The first model is rectangular, while the second is cylindrical: in each case, x can vary between 0.5 m and 1.5 m . 1 Complete the table below. Longueur x (in m ) 0.5 1, 5 Volume du réservoir R 1 (en m 3 ) Volume du reservoir R 2 (en m 3 ) Valeur exacte Valeur arrondie à 0 ; 1 m 3 Details of the calculations of the exact values should ap-pear on your copy . 2 a Monstrate that the expression, as a function of x , of the tank volume R 1 est : 7.5 x b Show that the expression, as a function of x , of the volume of the tank R 2 est : 2.5 ıx 2 3 Consider the function f 1 : x ↦−→ 7.5 x . Specify the nature of this function. 4 For values of x between 0.5 and 1.5 , the function f 2 : x ↦−→ 2.5 ıx 2 is already represented in the graph below : On the same graph, represent the function f 1 . 5 Answer the following questions, represent the function f 1 . Answer with approximate values and show the construc-tion lines used to read the graph. a What is the volume of the tank R 2 for x =0.8 m ? b What is the radius of the tank R 2 so that it has a capacity of 10 m 3 ? c What is the antecedent of 9 by the function f 1 ? Inter-pret this number in concrete terms. d For what value of x are the volumes of the two tanks equal? e For what values of x is the volume of R 1 greater than that of R 2 ? E.977 The unit of length is the centimeter. The diagrams above represent two paperweights. Model 1 is a regular pyramid whose height [ AH ] measures x and whose base is a square with side 9. Model 2 consists of two parallelepipeds are 2, 6, 10. The dimensions of the other are x , 2 and 3.5 1 In this question x =8 . Then calculate the volume of each paperweight (the unit of volume is the cubic cenitmeter) . 2 Express as a function of x , the volume V 1 of model 1 and the volume V 2 of model 2 (the unit of volume is cubic centimeters) . 3 In the plane referred to a reference frame where : On the x-axis, 1 cm will represent one unit. On the ordinate axis, 1 cm will represent twenty units. the x-axis and y-axis are perpendicular. graph the straight lines d 1 and d 2 of respective equations : y = 27 x and y = 7 x + 120 . 4 Calculate the value of x for which the two clipboards of model 1 and model 2 have the same volume? Then cal-culate this volume. What are the coordinates of the point of intersection of d 1 and d 2 ? https://chingmath.fr chapExoCorrec/5147 sacados/5147 x32;5x2;5 x00.511.5y123456789101112131415161718 chapExoCorrec/977 sacados/977 9AH9Modèle 12x10263,5Modèle 2