Grade 9 / Introduction to functions 69 exercises (100% corrected)

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ChingQuizz : 21 exercises available for Quizz assessment : 1. Reading coordinates on the map E.11305 In a plane equipped with a coor-dinate system, consider the two points A and B : Reading coordinates: A point in the plane is identified using the scale on the two half-lines. We will perform the following two readings in this order : we read the number on the horizontal half-line (x-axis) we read the number on the vertical half-line (y-axis) Point A is identified by the number 14 , then the number 4 . Vocabulary : We say that the point A has coordinates (14 ; 4) L’ x-coordinate of the point A has a value of 14 . L’ ordinate of point A has a value of 4 . 1 Give the coordinates of point B . 2 Place the two points C and D with coordinates : C (2 ; 1) ; D (10 ; 3) Note: Check that points A , B , C , and D are aligned. E.11306 Consider the coordinate system in the diagram below : 1 Determine the coordinates of points A , B , C , D , and E located in the coordinate system above. 2 Name the (s) point (s) with strictly negative x-coordinates. 3 Name the (s) point (s) with strictly negative y-coordinates. E.11307 Consider the following coordinate system in the plane : 1 Determine the coordinates of points B and C . 2 Place the points : A ( − 4 ; 1) ; D ( − 3 ; − 2) . 3 What conjecture can be made about the nature of the quadrilateral ABCD ? E.11311 Consider the grid below : 1 Given that point B has coordinates ( − 3 ; − 1) , correctly place the origin and the missing axes of the coordinate system. 2 Place points A , C , and D : A ( − 2 ; 2) ; C (7 ; − 1) ; D (6 ; − 4) 3 What type of quadrilateral is ABCD ? 2. Introduction E.11310 Consider the following calculation program : https://chingmath.fr chapExoCorrec/11305 sacados/11305 012468101214124xyAB chapExoCorrec/11306 sacados/11306 -7-6-5-4-3-2-1O123456-3-2-112ABCDE chapExoCorrec/11307 sacados/11307 xxyy-6-5-4-3-2-1O12345-3-2-112BC chapExoCorrec/11311 sacados/11311 B chapExoCorrec/11310 sacados/11310 Choisir un nombreAjouter 4Multiplier par 3Soustraire 12A`cher le résultat
1 Give the values displayed by this calculation program when the following values are entered : a the number 0 , b the number 2 , c the number 5 , d the number 10 2 What conjecture can be made about the number dis-played by this calculation program? E.11309 Consider the following calculation program : 1 Give the output number of this calculation program when the following input numbers are chosen : a the number 0 , b the number 2 , c the number 4 , d the number 5 2 What conjecture can be made about the result of this calculation program? E.7989 Consider the calculation pro-gram below in which x , Step 1 , Step 2 and Résultat are four variables : 1 a Julie ran this program choosing the number 5 . Check that what it says at the end is : ˇ I finally get 20 ı. b What does the program say if Julie makes it work by initially choosing the number 7 ? 2 Complete the table of values below : Number value choisi 5 7 1 4 8 10 Value retournée par l’algorithme 3 If we call x the number chosen at the start, check then jus-tify that the value returned by the algorithm is 3 × x +5 . https://chingmath.fr chapExoCorrec/11309 sacados/11309 Choisir un nombreSoustraire 1Ajouter 1Multiplier les deux nombresAjouter 1 chapExoCorrec/7989 sacados/7989 quandest cliquédemanderChoisir un nombreet attendremettrexàrépondredireJe multiplie le nombre par 6pendant2secondesmettreEtape 1à6*xdireJ’ajoute 10 au résultatpendant2secondesmettreEtape 2àEtape 1+10direJe divise par 2.pendant2secondesmettreRésultatàEtape 2/2direregroupeJ’obtiens ∏nalementRésultat
E.11312 Consider the figure below, which shows the three stages of producing a pattern : 1 Construct the next stage ( E 5 ) . 2 Enter the perimeter and area of each of these figures in the table below : Stage 2 3 4 5 Perimeter Area 3 Can we guess the perimeter and area for step 6 ? Step 7 ? E.11308 Consider the figure below, which represents three stages in the production of a pattern : 1 Complete the figure obtained in stage 5 . 2 Enter the perimeter and area of each of these figures in the table below : Step 2 3 4 5 Perimeter Area 3 Can we guess the perimeter and area for step 6 ? For step 7 ? E.4076 The company sources supplies for all of its projects from two wholesalers. The prices offered for packs of 10 tiles are: Wholesaler A : 48 e per pack, free delivery. Wholesaler B : 42 e per pack, delivery 45 e regardless of the number of packs. Note: n the number of packs purchased. the price p A in euros for an order of n packages from wholesaler A ; the price p B in euros for an order of n packages from wholesaler B . Complete the table below : n 2 5 9 p A p B E.11418 Consider the sequence of figures shown below : Let n be the number of the step. Give the expression for the function f that gives the perimeter of the figure in step n . 3. Problems and graph reading E.7990 To bake macaroons, the oven temperature must absolutely be 150 o C . For some time now, the store manager has not been satisfied with the baking of his pastries. So he decided to check the reliability of his oven by setting it to 150 o C and regularly taking the temperature with a probe. Here is the curve showing the change in temperature of his oven as a function of time. https://chingmath.fr chapExoCorrec/11312 sacados/11312 E2E3E4 chapExoCorrec/11308 sacados/11308 E2E3E4E5 chapExoCorrec/4076 sacados/4076 chapExoCorrec/11418 sacados/11418 E2E3E4E5 chapExoCorrec/7990 sacados/7990
1 What is the temperature reached after 3 minutes? No justification is required. 2 By how many degrees Celsius did the temperature in-crease between the second and seventh minute? 3 After how long is the temperature of 150 o C needed to bake the macaroons? E.9252 A factory in Moorea makes fruit juice. Let C be a function that, to a quantity of juice manufactured in litre (s) associates the manufacturing cost in F . Shown below is the function C for a quantity of juice between 0 and 130 liters. By showing your construction lines on the figure found above, answer the following questions : 1 a Give the cost of making 100 liters of juice. b For quelle (s) quantité (s) of juice, is the manufacturing cost greater than 550 F ? 2 a Give the image of 85 by the function C . b Read C (75) 4. Problems and expression of a function E.11319 Consider the following calculation program : 1 Complete the first three columns of the table below : Input number 1 2 3 100 Output number 2 If we denote the value of the input number by x , one of the calculations below expresses the value of the output number. Which one? a 3 × x b 3 × + 5 c 2 × x d 2 × + 5 3 Complete the table. E.11318 Consider the following calculation program : 1 Complete the first three columns of the table below : Input number 3 4 5 10 Output number 2 If we denote the value of the input number by x , one of the calculations below expresses the value of the output number. Which one? a x 2 b x 2 + 1 c 2 × x 2 d 2 × x 2 + 1 3 Complete the table. https://chingmath.fr Temps (en minutes)024681012141618Température (enoC)20406080100120140160180 chapExoCorrec/9252 sacados/9252 0102030405060708090100110120130100200300400500600700 chapExoCorrec/11319 sacados/11319 Choisir un nombreAjouter 3Multiplier par 2Soustraire 6A`cher le résultat chapExoCorrec/11318 sacados/11318 Choisir un nombreSoustraire 3Ajouter 3Multiplier les deux nombresAjouter 10
E.11317 Consider building a house of cards : 1 Complete the first three columns of the table below : Step 1 2 3 10 Number of cards 2 Noting n as the number of steps in building the house of cards, what expression gives the number of cards needed? a 2 n + 7 b 50 − 3 n c n 2 + n − 5 2 d 3 n 2 + n 2 3 Complete the table using the formula from the previous question. 5. Functional ratings E.11321 Consider the function g defined by the calculation program below : Note: Let x be the input number and f ( x ) be the output number. Complete the dotted lines : a f (0) = : : : b f (1) = : : : c f (2) = : : : E.11320 Consider the function f represented by the calculation program below : Note: where x is the input number and f ( x ) is the output number. Complete the dotted lines : a f (2) = : : : b f (3) = : : : c f (4) = : : : E.11354 Here are excerpts from a computer program that takes an input value and returns a return value : Let f be the function associated with this algorithm. Com-plete the dotted lines below : a f (1) = : : : b f (6) = : : : c f (7) = : : : E.11352 Below are the first regular poly-gons : Let f be the function that associates the number of sides with the angle at the center between two consecutive vertices : Complete the dotted lines : a f (3) = 120 b f (4) = : : : c f (5) = : : : d f (6) = : : : e f (7) = : : : f f (8) = : : : https://chingmath.fr chapExoCorrec/11317 sacados/11317 Étape1Étape2Etape3 chapExoCorrec/11321 sacados/11321 Choisir un nombreAjouter 2Multiplier par 3Soustraire 1A`cher le résultat chapExoCorrec/11320 sacados/11320 Choisir un nombreAjouter 1Ajouter 2Multiplier les deux nombresAjouter 3 chapExoCorrec/11354 sacados/11354 Valeursd’entréesValeursde sorties1346723589f chapExoCorrec/11352 sacados/11352 ¸oOTriangleéquilatéral¸oOCarré¸oOPentagone¸oOHexagone¸oOHeptagone¸oOOctogone
E.11353 Below is a step-by-step guide to constructing the pattern : 1 Let f be the function that associates the number of white squares in the figure with its number. Complete the dotted lines : a f (1) = : : : b f (2) = : : : c f (3) = : : : d f (4) = : : : f f (5) = : : : 2 We denote by g the function that associates the number of gray squares that compose it with the number of the figure. Complete the dotted lines : a g (1) = : : : b g (2) = : : : c g (3) = : : : d g (4) = : : : f g (5) = : : : E.11351 1 Consider the function f defined by: f : x ↦−→ 3 x − 2 Determine the following images : a f (1) = : : : b f (3) = : : : c f (5) = : : : 2 Consider the function g defined by: g : x ↦−→ x 2 + 3 Determine the following images : a g (1) = : : : b g (5) = : : : c g (6) = : : : 6. Graphic reading of images E.5118 Consider the function f whose rep-resentative curve C f is given in the frame below : 1 Which of the following points belong to the representa-tive curve of the function f : A ( − 3 ; 0) ; B ( − 1 ; 2) ; C (0 ; 1.5) ; D (2 ; 1) 2 Deduce the value of the following images : f ( − 3) ; f ( − 1) ; f (2) E.972 The graph below shows the represen-tative curve of the function f . Let f ( x ) be the image of the number x under the function f . 1 Using the graph, complete the following table : x 0 3 6 8 10 f ( x ) 2 Graphically, determine a rounded value of the image of 9 under the function f ; that is, the rounded value of f (9) . https://chingmath.fr chapExoCorrec/11353 sacados/11353 Fig.1Fig.2Fig.3Fig.4Fig.5 chapExoCorrec/11351 sacados/11351 chapExoCorrec/5118 sacados/5118 x-4-3-2-101234y-2-1123Cf chapExoCorrec/972 sacados/972 x0246810y2468Cf
E.4050 Consider the two functions f and g defined at − 4 ; 5 whose representative curves C f and C g are given in the marker below : Determine the following images : f ( − 2) ; g ( − 1) ; f (2) ; g (4) 7. Graphical reading of priors E.9251 In the reference frame below, the representative curve of the function f is plotted. Let f ( x ) be the image of the number x by the function f . 1 a Complete the following table : x 0 3 6 8 10 f ( x ) b Give an antecedent of the number 6 by the function f . 2 Give an antecedent of the number 2 by the function f . 3 Determine an approximate value of the antecedent of 4; that is, the approximate value of a number x verifying f ( x ) = 4 . E.9253 Consider the function f whose rep-resentative curve C f is given in the frame below : 1 a Give the coordinates of the two points on the curve with 2 as their ordinates. b What are the antecedents of the number 2 by the func-tion f ? 2 a How many antecedents by the function does the number 1 admit? Justify your answer. b Give the set of antecedents of the number 1 by the function f . 3 Give the set of priors of the number − 1 . https://chingmath.fr chapExoCorrec/4050 sacados/4050 x-4-3-2-1012345y-2-11234CgCf chapExoCorrec/9251 sacados/9251 x0246810y2468Cf chapExoCorrec/9253 sacados/9253 x-4-3-2-101234y-2-1123Cf
E.9254 Consider the two functions f and g defined at − 4 ; 5 whose representative curves C f and C g are given in the marker below : 1 Determine, by the function f , the priors of the following numbers : 3 ; 2.5 ; 0 ; − 1.5 2 a Determine the set of antecedents of the number − 1.5 by the function g . b Determine the set of antecedents of the number 2 by the function g . c Determine the set of antecedents of the number 1 by the function g . E.5182 A factory in Moorea makes fruit juice. Let C be a function that, to a quantity of juice manufactured in litre (s) associates the manufacturing cost in F . Below is a representation of the function C for a quantity of juice between 0 and 130 liters. With your construction lines showing in the figure above, an-swer the following questions : Give le (s) antécédent (s) of 600 by the function C . 8. Graphical reading of images and history E.11414 Consider the function f whose rep-resentative curve C f is given below : 1 Determine the images : f ( − 3 ; 5) ; f (0) ; f (2) 2 Determine all antecedents of the number 3 by the func-tion f . E.11487 Consider the function f whose graph is shown below : 1 Give the values of f (3) and f ( − 0 ; 25) . 2 Give the set of antecedents of 1 ; 25 by the function f . https://chingmath.fr chapExoCorrec/9254 sacados/9254 x-4-3-2-1012345y-2-11234CgCf chapExoCorrec/5182 sacados/5182 0102030405060708090100110120130100200300400500600700 chapExoCorrec/11414 sacados/11414 x-6-5-4-3-2-10123456y-2-11234Cf chapExoCorrec/11487 sacados/11487 xxyy-2-10123412Cf
E.11502 Consider the function f whose graph is shown below : 1 Give the image of − 3 ; 5 by the function f . 2 a Give the three antecedents of the number 0 ; 5 by the function f . b Give the set of antecedents of the number − 0 ; 5 by the function f . E.5137 Consider the function f whose graph-ical representation C f is given in the frame below : All of the questions in this exercise will be answered graphi-cally. 1 Determine the images of the following numbers by the function f : − 4 ; 0 ; 3 2 a Give the set of antecedents of the number 3 by the function f . b Give the set of antecedents of the number − 1 by the function f . c How many antecedents does the number 2 have by the function f . 3 Give the approximate values of the coordinates of the points of intersection of the curve C f and the x-axis. 9. Images and priors from a table of values E.7991 We consider a function f whose only knowledge is the table of values below : x − 2 0 1 3 6 7 f ( x ) 6 1 2 3 0 − 2 Copy and complete each of the sentences below correctly: 1 The image of the number − 2 by the function f is . . . . 2 The number 0 is . . . . . . of 6 by the function f . 3 An antecedent of the number . . . by the function f is 0 . 4 The number . . . is an antecedent of − 2 by f . E.11415 Consider a function f for which we only have the table of values below : x − 3 − 2 − 1 0 1 2 3 f ( x ) 1 2 − 3 − 1 3 − 2 0 Copy and complete each of the sentences below correctly: 1 The image of the number − 1 by the function f is . . . . 2 The antecedent of the number − 3 by the function f is . . . . 3 The number 0 is . . . . . . of 3 by the function f . 4 The number − 3 is . . . . . . of 1 by the function f . E.9256 A spreadsheet was used to obtain images of different values of x by a function f . Here is a copy of the resulting screen : 1 What is the image of − 1 by the function f ? 2 Give an ’antecedent of the number 5 by the function f ? 10. Images from the algebraic expression E.9257 Consider the function which, to any number x , associates the triple of its square : 1 Give the expression for the image f ( x ) of the number x by the function f . 2 Complete the table of values below : https://chingmath.fr chapExoCorrec/11502 sacados/11502 xxyy-5-4-3-2-1012-11Cf chapExoCorrec/5137 sacados/5137 x-6-5-4-3-2-10123456y-2-11234Cf chapExoCorrec/7991 sacados/7991 chapExoCorrec/11415 sacados/11415 chapExoCorrec/9256 sacados/9256 fx12ABCDEFGHx-2-101234f(x-10-7-4-1258B2×B1−4 chapExoCorrec/9257 sacados/9257
x − 2 − 1 0 1 2 f ( x ) E.5138 Consider the two functions f and g whose images of a number x are defined by the relations : f ( x ) = 2 x − 1 ; g ( x ) = x 2 − 2 x + 3 Copy and complete the table of values below : x − 2 1 4 f ( x ) x − 1 0 2 g ( x ) E.11416 1 Let f be the function defined by: f ( x )=2 x +3 Determine the values of f (2) , f ( − 2) , and f 2 3 2 Determine the antecedent of 9 ; 8 using the function f . E.9262 The following calculation pro-gram is given : Choose a number Add 1 Raise the result by the square Subtract from the result the square of the starting number We denote f the function that, to any number x chosen at the input of the program, associates the result obtained at the end of this calculation program. 1 Give the expression for the image f ( x ) of the number x . 2 Show that : f ( x ) = 2 x + 1 3 Complete the table of values for the function f : x − 1 0 1 2 f ( x ) E.9272 Consider the rectangle ABCD where AB =3 cm and AD =1 cm , and a point E belonging to seg-ment [ DC ] but distinct from the point D . We note x the length of segment [ CE ] . We call F the point of intersection of the lines ( AE ) and ( CB ) . We denote f the function that associates the number x with the area of triangle ABF . 1 For what reason does the statement state that the point E cannot be confused with the point D . 2 Establish that the function f admits the expression : f ( x ) = 3 x 6 − 2 x + 3 2 . 3 Complete the table of values below : x 0 1 4 1 2 3 4 1 5 4 3 2 7 4 2 9 4 5 2 11 4 f ( x ) 3 2 18 11 9 5 18 7 3 18 5 9 2 6 9 18 4 Below is shown the representative curve of the function f : What can be said about the area of the triangle ABF when the point E approaches the point D . 11. Antecedents and equations E.2561 Consider the function f whose im-age of a number x is defined by the relation: f ( x ) = 3 x − 4 1 Calculate images by f of numbers : − 3 ; − 1 ; 2.5 ; 10 2 Using an equation, determine the antecedents of the num-bers 5 and − 10 by the function f . E.11486 Consider the function f whose im-age of a number x is defined by: f ( x ) = 4 x − 1 1 Calculate the images by f of the numbers : − 2 ; 3 2 Using an equation, determine the antecedent of the num-bers 6 by the function f . E.11503 Consider the function f whose image of the number x is de-fined by the relation: f ( x ) = 7 x + 1 1 Calculate the images by f of the numbers : − 4 ; 2 2 Using an equation, determine the antecedent of the num-ber 4 by the function f . https://chingmath.fr chapExoCorrec/5138 sacados/5138 chapExoCorrec/11416 sacados/11416 chapExoCorrec/9262 sacados/9262 chapExoCorrec/9272 sacados/9272 ABCDEF3cm1cmx x0123y102030Cf chapExoCorrec/2561 sacados/2561 chapExoCorrec/11486 sacados/11486 chapExoCorrec/11503 sacados/11503
E.11417 Consider the following calculation program : Choose a number Subtract 3 Multiply by 2 Add 4 We call f the function that associates the result of the calcu-lation program with the chosen number. 1 Show that the function f can be expressed as : f ( x ) = 2 x − 2 2 a Determine the image of 3 ; 2 by the function f . b Determine the antecedent of the number − 0 ; 4 by the function f . E.5720 We consider the following calcula-tion program : Choose a number Add 5 Multiply by 3 Subtract 2 We call f the function that, to the chosen number, associates the result of the calculation program. 1 Give the expression for the image f ( x ) in terms of x . 2 a Determine the image of 4 by the function f . b Determine the antecedent of the noble 4 by the func-tion f . 12. Antecedents and product equations E.9255 We consider the calculation program : Choose a number Take the square of that number Add three times the starting number. Add 2 1 We call x the starting number, express the result of the program in terms of x . 2 Show that this result can also be written as ( x +2)( x +1) for all values of x . 3 We denote f the function that has any number x asso-ciates with the return value of the calculation program. We have the value array below : a From the table of values, give two priors of the number 0 by the function f . b Justify that the function f admits only two an-tecedents of the number 0 . E.5919 The screenshot below shows the work Camille did using a spreadsheet about the functions g and h defined by: g ( x ) = 5 x 2 + x − 7 ; h ( x ) = 2 x − 7 She copied the formulas she had entered in cells B2 and B3 to the right. 1 Give a number that has image − 1 by the function g . 2 Write the calculations showing that : g ( − 2)=11 3 What formula did Camille enter in cell B3 ? 4 a Deduce from the table a solution to the equation : 5 x 2 + x − 7 = 2 x − 7 b Does this equation have a solution other than the one found through the spreadsheet? E.9259 We consider the following cal-culation program : Choose a number Add 5 Take the square of this sum We call f the function that associates the result of the calcu-lation program with the chosen number. 1 a Which of the following functions is f ? x ↦−→ x 2 + 25 x ↦−→ ( x + 5) 2 x ↦−→ x 2 + 5 x ↦−→ 2( x + 5) b Is it true that − 2 is an antecedent of 9 ? 2 a Solve the equation : ( x +5) 2 =25 . b Deduce all the numbers that can be chosen to get 25 to this calculation program. https://chingmath.fr chapExoCorrec/11417 sacados/11417 chapExoCorrec/5720 sacados/5720 chapExoCorrec/9255 sacados/9255 12ABCDEFGHIJx-4-3-2-191234(xx620026122030 chapExoCorrec/5919 sacados/5919 fx123ABCDEFx-2-1012g(x5x2x−711-3-7-115h(x2x−7-11-9-7-5-3B2∗B1∗B1B1−7 chapExoCorrec/9259 sacados/9259 Asie Juin 2013 7 points
E.9258 Let g be the function defined by: g : x ↦−→ x 2 + 1 1 Calculate the following numbers : g (2) ; g ( − 5) ; g ( − 1) . 2 Determine by the function g the two priors of the number 5 . 3 Determine by the function g the unique antecedent of the number 1 . 4 Justify that the number 0 admits no antecedent by the function g . E.9261 We consider the following cal-culation program : Choose a number; Add 7 to that number; Subtract 7 from the number chosen at the beginning ; Multiply the two previous results ; Add 50 ; Consider the function f that associates with any number x the result of the calculation program when the chosen number is x . 1 Express the image f ( x ) of the number x by the focntion f . 2 Determine the antecedent of the number 17 by the func-tion f . 13. Background, equations, double distributivity E.9263 We consider the two calculation pro-grams below : We denote f (resp. g ) the function that associates the result of the program A (resp. program B ) with the chosen number x . 1 Give the expressions for the images f ( x ) and g ( x ) as functions of x . 2 Determine the number chosen so that these two calcula-tion programs yield the same number. E.9266 Program A Choose a number Subtract 3 Calculate the square of the result obtained Program B Choose a number Calculate the square of this number Add triple the starting number Add 7 We denote f (resp. g ) the function which to the chosen num- ber x associates for image the number returned by the pro-gram A (resp. program B ) . 1 a Give the expression of f ( x ) the image of the number x by the function f . b Give the expression of g ( x ) the image of the number x by the function f . 2 Determine the value(s) of x returning the same value for the calculation programs A and B . E.11378 Consider the following calculation pro-gram: Let f be the function that associates the input number with the output number of this algorithm. 1 a Give the expression of the function f . b Expand and reduce the expression of the function f . 2 Complete the table of values below : x − 4 − 1 2 5 f ( x ) 14. Graphical representation E.2562 1 Construct a unit marker 1 cm defined as follows : The x-axis and y-axis are perpendicular ; the x-axis must represent all numbers from − 3 to 3 ; the y-axis must represent any number from − 4 to 4 . 2 Consider the function f defined by the relationship : f : x ↦−→ 1.5 x + 1 a Complete the table below : https://chingmath.fr chapExoCorrec/9258 sacados/9258 chapExoCorrec/9261 sacados/9261 chapExoCorrec/9263 sacados/9263 Choisir un nombreAjouter3Soustraire3Multiplier lesdeux nombresChoisir un nombreMettreau carréMultipler par 2AdditionerSoustraire 4ProgrammeAProgrammeB chapExoCorrec/9266 sacados/9266 chapExoCorrec/11378 sacados/11378 Choisir un nombreSoustraire 1Ajouter 1Multiplier les deux nombresAjouter 1 chapExoCorrec/2562 sacados/2562
x − 3 − 2 − 1 0 1 2 f ( x ) b In the frame of reference of question 1 , place the six points with coordinates ( x ; f ( x )) obtained in the previous table. c What do we notice about the positions of these points? 3 Consider the function g whose image of a number x is defined by: g ( x ) = 0.5 · x 2 − 3 a Complete the table below : x − 3 − 2 − 1 0 1 2 3 g ( x ) b Place the seven points with coordinates ( x ; g ( x )) ob-tained from the previous table in the benchmark and connect them by a curve. E.9267 Consider the function f defined by the relation: f ( x ) = 3 x x 2 + 1 + 2 In the reference frame below, a portion of the curve C f was given. 1 We wish to complete the table of values below to con-struct the missing part of the curve C f . x − 1.5 − 1 − 0.5 0 0.5 1 f ( x ) 2 Place the missing points in the graph, then complete the curve. 15. Problems and values table E.9268 During a motocross race, after crossing a ramp, Gaëtan made a record jump on his motorcy-cle. The jump begins as soon as Gaëtan leaves the ramp. We note t the duration (in seconds) of this jump. The height (in meters) is determined as a function of the du-ration t by the following function h : h : t ↦−→ − 5 t − 1.35 t − 3.7 Here is the representative curve of this function h . Are the following statements true or false? Justify using ei-ther the graph or calculations. 1 Expanding and reducing the expression for h gives : h ( t ) = − 5 t 2 + 17.15 t + 4.995 2 When he leaves the ramp, Gaëtan is 3.8 m high. 3 Gaëtan’s jump takes less than 4 seconds. 4 The number 3.5 is an antecedent of the number 3.77 by the function h . 5 Gaëtan got the maximum height before 1.5 second. https://chingmath.fr chapExoCorrec/9267 sacados/9267 x-4-3-2-101234y-11234Cf chapExoCorrec/9268 sacados/9268 rampehauteurhdistance horizontaledduréet0s 012345101520
E.9260 For his 32 th birthday, Denis bought an exercise bike so he could work out during the win-ter. Heart rate ( HR ) is the number of pulses (or beats) of the heart per minute Denis wants to know his recommended maximum heart rate (FCMC) in order not to exceed it and thus spare his heart. The FCMC of an individual depends on his or her age a , expressed in years, and can be obtained from the following formula {et by Astrand and Ryhming: Fréquence maximum recommended heart rate = 220-age We denote f ( a ) the FCMC as a function of age a , so we have : f ( a ) = 220 − a . 1 a Verify that Denis’ FCMC is equal to 188 pulsations = minute . b Compare Denis’s CMEF with the CMEF of a 15 -year-old. 2 After some research, Denis finds another formula to get his FCMC more accurately. If a denotes an individual’s age, his CMEF can be calculated using the Gellish for-mula : Fréquence maximum recommended heart rate: = 191.5 − 0.007 × âge 2 We note g ( a ) the FCMC as a function of age a , so we have : g ( a ) = 191.5 − 0.007 × a 2 Denis uses a spreadsheet to compare the results obtained using the two formulas : What formula should be inserted into cell C2 and then copied down, so that the column ˇFCMC g ( a ) (Gellish)ı can be completed? E.7622 The diagram below represents Leila’s garden. It is not to scale. [ OB ] and [ OF ] are walls, OB =6 m and OF =4 m . The dotted line BCDEF represents the wire mesh that Leila wants to install to enclose a rectangular enclosure OCDE . She has a roll of 50 m of chicken wire that she wants to use entirely. Leila considers several possibilities for placing the C point. 1 Placing C so that BC =5 m , she gets that FE =15 m . a Verify that it uses 50 m of wire mesh. b Justify that the area A of the enclosure OCDE is 209 m 2 . 2 To get the maximum area, Leila calls on her math teacher neighbor, who, in a bit of a rush, writes on a piece of pa-per: En noting BC = x , we have A ( x )= − x 2 +18 x +144 Check that the neighbor’s formula is consistent with the result in question 1 3 In this section, questions a and b do not require jus-tification . a Leila entered a formula in B2 and then stretched it to cell I2 . Which formula is then written in cell F2 ? b Which of the values in the table will Leila choose for BC in order to obtain a maximum area enclosure? c Give the dimensions of the resulting pen. https://chingmath.fr chapExoCorrec/9260 sacados/9260 fx12345ABCÂgeaFCMCf(a(AstrandetRyhming)FCMCg(a(Gellish)30190185,231189184,77332188184,33233187183,877B2=220-A2 chapExoCorrec/7622 sacados/7622 EnclosOBCDEF fx12ABCDEFGHIx56789101112A(x−x2x209216221224225224221216B2=-B1*B1+18*B1+144
E.6057 In this exercise, consider the rect-angle ABCD opposite such that its perimeter is equal to 31 cm . 1 a If such a rectangle has length 10 cm , what is its width? b Propose another length and find the corresponding width. c We call x the length AB . Using the fact that the perimeter of ABCD is 31 cm , express the width BC as a function of x . d Deduce the area of rectangle ABCD as a function of x . 2 Consider the function f defined by: f ( x ) = x 15.5 − x a Calculate f (4) . b Check that an antecedent of 52.5 is 5 . 3 On the graph below, we have plotted the area of the rectangle ABCD as a function of the value of x . Using this graph, answer the following questions, giving approximate values : a What is the area of the rectangle ABCD when x is 3 cm ? b For what values of x is an area equal to 40 cm 2 ob-tained? c What is the maximum area of this rectangle? For what value of x is it obtained? 4 What can we say about the rectangle ABCD when AB is 7.75 cm ? E.9271 A field is composed of two squares and a right triangle. This field is represented in the figure below : The indeterminate x is a strictly positive number. We denote f the function that, to each value of x , associates the area of the field in m 2 . 1 Give the expression for the image f ( x ) of the number x by the function f . 2 a Determine the antecedent(s) of the number 112 by the function f . Hint: we will use the factorization : x 2 + 3 x − 40 = x − 5 x + 8 b Translate the result of the previous question into a characteristic of the field and its area. 16. Share E.960 In the reference frame below, we give the graphical representations of the functions f , g and h : https://chingmath.fr chapExoCorrec/6057 sacados/6057 ABCD Valeur dex012345678910111213141516Aire deABCD102030405060 chapExoCorrec/9271 sacados/9271 ABCDEFGx6m chapExoCorrec/960 sacados/960 x-8-6-4-202468y-6-4-2246CfCgCh
Say whether the statements below are true or false. 1 The image of 0 by g is − 1 ; 2 0 is the image of 3 by h ; 3 The point (6 ; 2) ∈ C f ; 4 − 5 is an antecedent of the number − 3 by g ; 5 − 3 has image − 5 by f ; 6 The abscissa points 3 of the curves C g and C h have the same ordinate ; 7 By the function h , 1 is the only antecedent of the number − 3 ; 8 By function f , − 6 is the only antecedent of 2 . E.9269 Consider the functions f and g defined by: f ( x ) = 2 x + 1 and g ( x ) = x 2 + 4 x − 5 Lea wants to study the functions f and g using a spreadsheet. So she filled in the formulas and then stretched them out to get the calculation for all the values. Here is a screenshot of her work: 1 What is the image of 3 by the function f ? 2 Calculate the number that should appear in cell C3 . 3 What formula did Lea enter in cell B2 ? 4 Using the screenshot and without justification, give a so-lution to the inequation : 2 x + 1 <x 2 − 4 x − 5 5 Determine an antecedent of 1 by the function f . E.9270 Let the functions f , g and h be defined by: f ( x ) = 6 x ; g ( x ) = 3 x 2 − 9 x − 7 ; h ( x ) = 5 x − 7 Using a spreadsheet, Pauline constructed a table of values for these functions. She stretched to the right the formulas she had entered in cells B2 , B3 , and B4 . 1 Use the spreadsheet to determine the value of h ( − 2) . 2 Write the calculations showing that : g ( − 3) = 47 3 Make a sentence with the word ˇ antecedent ı or the word ˇ image ı to translate the equality: g ( − 3) = 47 4 What formula did Pauline enter in cell B4 ? 5 a Deduce from the table above a solution to the equa-tion below : 3 x 2 − 9 x − 7 = 5 x − 7 b Does this equation have a solution other than the one found using the spreadsheet? Justify the answer. Hint: in this question, any trace of research, even unfinished, will be considered and valorisée 17. Unclassified exercises E.7627 Legionella are bacteria found in drinking water. When the water temperature is between 30 o C and 45 o C, these bacteria proliferate and can reach, in 2 or 3 days, concentrations dangerous to humans. Remember that — m is short for micrometer. One micrometer is equal to one millionth of a meter. 1 The size of a legionella bacterium is 0.8 —m . Express this size in m and give the result in scientifc notation. 2 When the water temperature is 37 o C, this population of Legionella bacteria doubles every quarter of an hour. A population of 100 legionella bacteria is placed in these conditions. The following spreadsheet has been created to give the number of legionella bacteria as a function of the number of quarter-hours elapsed : a In cell B3 , we want to enter a formula that we can stretch down in col-umn B to calculate the number of legionella bac-teria corresponding to the number of quarter-hours elapsed. What is this formula? b What is the number of legionella bacteria after one hour? c Is the number of legionella bacteria proportional to the elapsed time? d After how many quarters of an hour does this popula-tion exceed ten thousand legionella bacteria? 3 We want to test the effectiveness of an antibiotic against legionella bacteria. The antibiotic is introduced into a vessel containing 10 4 legionella bacteria at time t =0 . The grpahic representation, below, gives the number of bacteria in the container as a function of time. https://chingmath.fr chapExoCorrec/9269 sacados/9269 Asie Juin 2016 fx123ABCDEFGHx-3-2-10123f(x-5-3-11357g(x-8-8-50716B3B1∗B14∗B1−5 chapExoCorrec/9270 sacados/9270 fx1234ABCDEFGHx-3-2-10123f(x6x-18-12-6061218g(x3x2−9x−747235-7-13-13-7h(x5x−7-22-17-12-7-238B3×B1×B1−9×B1−7 chapExoCorrec/7627 sacados/7627 12345678910ABNombredequartsd’heure012345678Nombredebactéries100
a After 3 hours, approximately how many legionella bac-teria remain in the container? b After approximately how long does 6 000 legionella bac-teria remain in the container? c It is estimated that an antibiotic will be effective on humans if it succeeds in reducing the initial number of bacteria in the container by 80 % in less than 5 hours. Using the graph as a guide, investigate the efficacy of the antibiotic tested on humans. E.10891 Consider the following calcula-tion program : Part A 1 Show that if 3 is the starting number, the program gives a result equal to 90 . 2 One student chooses 2 as the starting number and an-other student chooses − 2 . Show that they must obtain the same result. 3 If we name x the starting number, show that the result of the program can be written as 10 x 2 . Part B For this part, a student looks for the number(s) (s) they must choose to obtain 30 as the result. 4 To do this, they graph the function f associated with the calculation program defined by: f ( x ) = 10 x 2 They obtain the following curve : Using the graph, determine an approximate value of the antecedents of 30 by the function f . Do not justify. 5 The student wants to find a more accurate value for the positive antecedent found in the previous question. To do this, he uses a spreadsheet, an excerpt of which is given below : a What formula could he have entered in cell B2 before dragging it down? Do not justify. b In this table, what is the starting number that gives the result closest to 30 ? Do not justify. 6 Determine the exact value of the positive number sought by the student. https://chingmath.fr tempstenh012345678910Nombre de bactéries20004000600080001000012000 chapExoCorrec/10891 sacados/10891 Asie juin 2023 CalculerlecarredecenombreMultiplierpar5Ajouter4Multiplierpar2Enlever8NombrededépartRésultat -2-1012-1010203040Cf 123456789101112ABNombrededépartRésultat1,6025,6001,6125,9211,6226,2441,6326,5691,6426,8961,6527,2251,6627,5561,6727,8891,6828,2241,6928,5611,7028,900 123456789101112ABNombrededépartRésultat1,7129,2411,7229,5841,7329,9291,7430,2761,7530,6251,7630,9761,7731,3291,7831,6841,7932,0411,8032,400
E.10892 We consider the function f de-fined by: f ( x ) = x 2 − x − 12 We note C f its representative curve given below in an or-thonomous frame : 1 a Calculate the exact value of f 1 2 . b Graphically determine the antecedents of − 6 by the function f . 2 Consider the function g defined by g ( x ) = 3 x − 7 . A spreadsheet was used to produce a table of values for this function. a What formula did we write in cell B2 before stretching it down? b Below is given the graphical representation of the func-tion g . Give the coordinates of the points A and B . 3 a Assuming that the representation of C g is a straight line, plot the curve C g in graph 1 using points A and B . b Graphically determine the numbers that have the same image by the functions f and g . The construction lines may be left visible on the graph. E.10975 Here are two calculation programs. Program A Choose a number Multiply this number by -2 Add 5 to this result Program B Choose a number Subtract 5 from this number Multiply the result by 3 Add 11 to the result 1 a Show that if you choose ( − 3) as the starting number, the result obtained with program A is 11 . b What result do you get with program B if you choose 5.5 as the starting number? 2 If we designate x as the starting number, we obtain − 2 x +5 as the result with program A . Show that, with the same starting number, the result of program B is equal to 3 x − 4 . 3 Consider the two functions f and g defined by: f ( x ) = − 2 x + 5 ; g ( x ) = 3 x − 4 a Determine the image of the number 3 by the function f . b Determine the image of the number 3 by the function g . 4 The functions f and g are represented in the graph be-low : a Match each line to the corresponding function, giving your reasoning. b By reading the graph, give, as accurately as possible, the number whose image is the same for function f and function g 5 Determine, by calculation, the starting number for which programs A and B give the same result. E.3896 Three curves are shown below. Which curve is not a representation of a function? https://chingmath.fr chapExoCorrec/10892 sacados/10892 -4-3-2-10123456-12-10-8-6-4-22468Graphique 1 12ABCDEFGHIJKLMxg(x)-5-22-4-19-3-16-2-13-1-100-71-42-1324558611 -6-5-4-3-2-101234567-20-16-12-8-448ABCCgGraphique 2 chapExoCorrec/10975 sacados/10975 x01234y-5-4-3-2-1123456(D1(D2 chapExoCorrec/3896 sacados/3896 011Cf 011Cg 011Ch
E.11505 Consider the following calcula-tion program. 1 Show that if 5 is chosen as the starting number, the result of the program is 2 . 2 Choose x as the starting number. a Which of the following expressions can be used to ex-press the result of this calculation program as a func-tion of x ? No justification is required. Expression A x + 4 × x − 2 − x 2 Expression B x + 4 × x − 2 − 2 x Expression C x + 4 × x − 2 − x 2 Expression D x + 4 × x − 2 − 2 x b Show that the calculation program can be written in the form 2 x − 8 . 3 a Determine the result of the program when 4 is the number chosen as the starting number. b What starting number must be chosen for the calcula-tion program to be equal to 100 ? https://chingmath.fr chapExoCorrec/11505 sacados/11505 Nombre choisi au départAjouter 4Soustraire 2Multiplier les deux résultatsSoustraire le carré du nombre de départRésultat ∏nal