Grade 10 / Affine functions and first degree equation 93 exercises (100% corrected)

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6x24x3 x2x2x−35×2−3 ChingQuizz : 8 exercises available for Quizz assessment : 1. Reminders: first degree equation E.8050 The adjacent figure shows a bal-ance in equilibrium: the left and right pans have the same weight. On each tray are weights whose mass is indicated on their front face où x represents the same positive number on both trays. What is the value of the number x achieving this equilibrium situation? E.8051 Consider the following calculation program : Choose a starting number; Multiply the number by 2 ; Subtract 3 ; Write the final result. 1 Give the number returned when the starting number has the value : 5 ; 0 ; − 2 2 a It is assumed that the number obtained is 5 . This situation is illustrated by the diagram below : Determine the starting number used in this case. b Determine the value of the start in the case où the final result is : 7 ; 1 ; 4 E.8055 Shown below are the manipulations performed on a balance to determine the unconfirmed value x . 1 For each step, indicate the manipulation that was on each of the scales. 2 Deduce the value of the unknown x . https://chingmath.fr chapExoCorrec/8050 sacados/8050 6x24x3 chapExoCorrec/8051 sacados/8051 x2x2x−35×2−3 chapExoCorrec/8055 sacados/8055
4x33x5 5x12x10 8x24x14 3x2x6 ABCDMNQR5cm x-4-3-2-101234y-11234 E.8052 Determine, for each question, the value of x achieving equilibrium of the balance : a b c d E.8053 Solve the following equations : a 3 x + 5 = 5 x + 8 b 5 − 3 x = 2 x + 13 c 6 x − 2 = x − 6 d − 8 x − 3 = − 3 x − 6 E.8054 Three identical equilateral triangles are cut from the corners of an equilateral triangle of side 6 cm . The sum of the perimeters of the three small triangles is equal to the perimeter of the remaining gray hexagon. What is the measure of the side of the small triangles? Any trace of research, even unsuccessful, will appear on the copy and will be taken into account in the scoring. E.8056 Consider the figure opposite where ABCD is a square, AMND and MQRB are two rectangles where M and N belong to the segments [ AB ] and [ CD ] respectively, R is the mid-dle of segment [ BC ] and CD =5 cm . Note x the length of segment [ AM ] . Determine the value of x for which the rectangles AMND and BMQR have the same area. 2. Representation of affine functions E.2802 Definition: An affine function f is a function admitting an expression of the form : f ( x ) = m · x + p m; p ∈ R The number m is called the coefficient directeur and the number p is called the ordinate at origine . In the plane provided with a reference frame : 1 Consider the straight line (Δ) representative of the affine function : f ( x )= 2 3 x − 1 Which of the points below belong to the line (Δ) ? a A ( − 3 ; 0) b B (6 ; 3) c C (2 ; 2) d D (0 ; − 1) 2 Consider the line ( d ) through the points E (6 ; 6) and F ( − 9 ; − 4) . The line ( d ) is the representation of an affine function whose expression is : a g ( x ) = 2 3 x + 2 b h ( x ) = − 1 3 x − 7 c j ( x ) = 1 3 x − 2 d k ( x ) = 4 3 x − 2 E.1802 Proposition: for any affine function f , the representative curve of the function f is a straight line. Method : to draw the representative line of an affine func-tion, it is sufficient to know two points belonging to it. Consider the three affine functions below : f ( x ) = 1.5 x + 1 ; g ( x ) = − 1 2 x + 2 ; h ( x ) = 3 1 Complete the tables of values below : x − 1 2 f ( x ) x − 1 2 2 g ( x ) x 0 2.5 h ( x ) 2 Use the tables of values above to draw the representative curves of these three functions in the frame below : https://chingmath.fr chapExoCorrec/8052 sacados/8052 4x33x5 5x12x10 8x24x14 3x2x6 chapExoCorrec/8053 sacados/8053 chapExoCorrec/8054 sacados/8054 chapExoCorrec/8056 sacados/8056 ABCDMNQR5cm chapExoCorrec/2802 sacados/2802 chapExoCorrec/1802 sacados/1802 x-4-3-2-101234y-11234
-5-4-3-2-12345I-3-2-123JO(d3 x-4-3-2-101234y-2-1123AB xxyyOCf(d−2 E.8269 Proposal: The representative curve of an linear functionis a straight line. Any non-vertical line is the representation of an linear function The y-intercept is the y-intercept of the point of inter-section of the representative line and the y-axis. In the plane provided with a reference frame O ; I ; J , con-sider the line ( d 3 ) shown below : 1 Consider the linear function f defined by the relation: f ( x ) = 3 8 · x + 3 2 a Justify that the straight line ( d 1 ) , representative of the function f , passes through the points with coordi-nates : A ( − 4 ; 0) ; B 2 ; 9 4 b Draw the straight line ( d 1 ) on the graph. 2 Consider the line ( d 2 ) representative of the function g . The y-intercept of the function g is − 1.5 . Thus, its ex-pression is of the form : g ( x ) = m · x − 1.5 where m ∈ R a Assume that the straight line ( d 2 ) passes through the point with coordinates C (2 ; 0) . Determine the slope of the function g . b Draw the line ( d 2 ) on the graph. 3 The straight line ( d 3 ) passes through the points D − 2 ; 1 2 and E (1 ; − 1) is representative of the linear function h . Determine the expression of the function h . E.6638 Consider the linear functiondefined by the relation: f ( x ) = − 3 2 x + 1 4 In the reference frame below, consider the two points A and B represented below and note C f the representative line of the function f : 1 a Justify that the points A and B belong to the line C f . b Draw the line C f representative of the function f . 2 a Give the abscissa of the single point C of C f having − 1 2 as its ordinate. b Justify algebraically that the antecedent of the number − 1 2 is 1 2 . 3. Representation of affine and tangent functions E.4714 Definition: let f be a function on an interval I and x 0 be a number from I . We denote A the point of abscissa x 0 of the representative curve of the function f . We say that the line ( d ) is tangent to the curve C f at the point of abscissa x 0 if : the line ( d ) passes through the point A ; in the vicinity of the point A , the straight line ( d ) has the same ˇ direction ı as the curve C f . Example: below, the straight line ( d ) is the tangent to the curve C at the point of abscissa − 2 : The straight line ( d ) is the tangent to the curve C f at the point of abscissa − 2 . Consider the function f defined by the relation is : f ( x ) = 1 4 x 2 − 1 2 x − 2 In the plane provided with the reference frame represented https://chingmath.fr chapExoCorrec/8269 sacados/8269 -5-4-3-2-12345I-3-2-123JO(d3 chapExoCorrec/6638 sacados/6638 x-4-3-2-101234y-2-1123AB chapExoCorrec/4714 sacados/4714 xxyyOCf(d−2
x-3-2-1012345y-4-3-2-112 x-4-3-2-10123y-4-3-2-11234Cf x-4-3-2-101234y-2-11234Cf below, we note C f the representative curve of the function f : 1 a Draw the straight line ( d ) representative of the linear function g defined by: g ( x )= 1 2 x − 3 b What is the name of the straight line ( d ) relative to the curve C f ? 2 a Draw the straight line (Δ) representative of the lin-ear function h defined by: h ( x )= − 3 2 x − 3 b What is the name of the straight line (Δ) relative to the curve C f ? E.4716 Consider the function f defined by the relation is : f ( x ) = x + 4 × 1 2 x 2 − 2 In the plane provided with the reference frame given below, we note C f the representative curve of the function f : 1 a Draw the line ( d ) representative of the linear function g defined : g ( x )= − 1 2 x − 4 b What is the name of the straight line ( d ) relative to the curve C f ? 2 a Draw the straight line (Δ) representative of the lin-ear function h defined : h ( x )= − 7 4 x − 11 4 b What is the name of the straight line (Δ) relative to the curve C f ? E.4718 Consider the function f defined by the relation is : f ( x ) = 4 x + 2 2 x 2 + x + 1 In the plane provided with an orthonormal reference frame O ; I ; J , note C f the representative curve of the function f : 1 a Draw the line ( d 1 ) representative of the function g defined by: g ( x ) = 1 2 x − 1 2 b What is the name of the straight line ( d 1 ) relative to the curve C f . 2 a Draw the line ( d 2 ) representative of the linear function h defined by: h ( x ) = 2 x + 2 b What is the name of the straight line ( d 2 ) relative to the curve C f . 3 a Draw the line ( d 3 ) representative of the function j defined by: j ( x ) = − x + 5 2 b What is the name of the straight line ( d 3 ) relative to the curve C f . 4. Introduction to the governing coefficient E.2123 In the reference frame below, con-sider the straight line ( d ) : https://chingmath.fr x-3-2-1012345y-4-3-2-112 chapExoCorrec/4716 sacados/4716 x-4-3-2-10123y-4-3-2-11234Cf chapExoCorrec/4718 sacados/4718 x-4-3-2-101234y-2-11234Cf chapExoCorrec/2123 sacados/2123
(d)x-4-3-2-1012345y-2-112MNP x-6-4-2024y-224ABCMNCg x-3-2-10123y12CfCgABCD -5-4-3-2-12345I-12JOCfBA 1 Give the coordinates of the points M , N and P . 2 Using the coordinates obtained in the previous question, complete the following table : y N − y M x N − x M y M − y P x M − x P y P − y N x P − x N y M − y N x M − x N 3 What can we say? Is there an explanation for this? E.543 In the reference frame below, we have plotted the representative line C g of an affine function g . 1 Determine the coordinates of points A , B and C . 2 Place the points M and N of coordinates : respectively ( x B ; y A ) ; ( x C ; y B ) . 3 Give, without justification, the nature of triangles AMB and BCN . 4 Determine the exact values of the trigonometric ratios : tan BAM ; tan CBN Justify that these two values are equal. E.9360 Let m and p be two given numbers. Consider the linear functionsuch that the image of x is given by the relation: f ( x ) = m × x + p Show that, whatever the values of a and b , we always have equality: f ( b ) − f ( a ) b − a = m 5. Steering coefficient E.9359 Definition-proposition: the slope of the linear functionis the rate of increase of its representative line. For A and B two points on this line, it has the value : m = y B − y A x B − x A Consider two linear functions f and g verifying the following equalities: f ( − 2)=0.8 ; f (0)=1.6 ; g ( − 1)=0.4 ; g (0)=0.2 In the plane provided with a reference frame, consider the straight lines C f and C g representative of the functions f and g : For each of these lines, two of their points have been high-lighted, along with a vector indicating the direction of the line. 1 Determine the slope of the function f . 2 Determine the slope of the function g . E.11393 1 The graph below shows the curve C f representing the function f . Determine the slope of the function f . 2 Consider the linear function g satisfying : g ( − 2) = 4 ; g (1) = − 2 Determine the slope of the function g . https://chingmath.fr (d)x-4-3-2-1012345y-2-112MNP chapExoCorrec/543 sacados/543 x-6-4-2024y-224ABCMNCg chapExoCorrec/9360 sacados/9360 chapExoCorrec/9359 sacados/9359 x-3-2-10123y12CfCgABCD chapExoCorrec/11393 sacados/11393 -5-4-3-2-12345I-12JOCfBA
IJO220;5-1-10;5(d1(d2(d3 x-4-3-2-101234y-11234CfCg IJO(d1(d2(d3(d4 x-4-3-2-101234y-1123CfCgAB E.2690 We provide the plane with a refer-ence frame O ; I ; J orthonormal. Consider the three lines ( d 1 ) , ( d 2 ) and ( d 3 ) shown below : The coordinates of the intersection points of these lines are given on the representation. Determine the slopes of the linear functions associated with each of these lines. E.1818 Consider the reference frame given belowbelow where are represented the two straight lines ( C f ) and ( C g ) representing respectively the affine functions f and g . Determine, in the form of irreducible fractions, the directing coefficients of the affine functions f and g . E.1965 In the plane provided with a refer-ence frame O ; I ; J , consider the four lines shown below : These lines admit the following numbers as their slopes : − 3 2 ; − 1 5 ; 1 2 ; 2 Associate each of the lines with its slope. 6. Determine the reduced equation by algebraic calculation E.8113 Consider the two linear functions f and g having respectively C f and C g for representative lines are given below : 1 The points A (2 ; 1.5) and B ( − 2 ; − 0.5) belong to the curve C f . a Justify that the function f admits for an expression of the form : f ( x ) = 0.5 · x + p where p ∈ R b Using the coordinates of the point A , determine the value of the number p . We’ll give the expression of the function f . 2 a Justify that the function g admits an expression of the form : g ( x ) = − 0.4 · x + p where p ∈ R b Determine the expression of the function g . https://chingmath.fr chapExoCorrec/2690 sacados/2690 IJO220;5-1-10;5(d1(d2(d3 chapExoCorrec/1818 sacados/1818 x-4-3-2-101234y-11234CfCg chapExoCorrec/1965 sacados/1965 IJO(d1(d2(d3(d4 chapExoCorrec/8113 sacados/8113 x-4-3-2-101234y-1123CfCgAB
-4-3-2-1234I-12JO -5-4-3-2-12345I-3-2-123JO(d1(d2(d3(d4 IJO(d1(d2(d3-1;(- xxyy-6-5-4-3-2-10123456-2-112(d1(d2ABCD E.540 In a reference frame, consider the line (Δ) passing through the points with coordinates A (1 ; 5) and B (5 ; 8) . Let f be the affine function f that admits as its representative curve the straight line (Δ) . 1 a Determine the directing coefficient of the affine func-tion f . b Among the two expressions proposed with p ∈ R , which is the one representing the function f : f ( x ) = 0.75 × x + p ; f ( x ) = p × x + 0.75 2 Using the coordinates of point A , determine the complete expression of function f . E.7088 Consider the function f affine whose image of two numbers is known : f ( − 0.4) = 1.6 ; f (2.4) = − 0.5 Determine the expression of the function f . E.9613 In the plane provided with an orthonormal reference frame O ; I ; J , consider the linear function f whose representative line ( d ) passes through the two points A (2.4 ; − 2.5) and B ( − 1.8 ; 2.75) . 1 Determine the expression of the function f . 2 Draw precisely the straight line ( d ) . E.542 In the plane provided with a reference frame, consider the points A (2 ; − 1) and B (5 ; 4) . Determine the expression of the affine function f admitting the line ( AB ) as its representative line. E.8484 In the plane provided with a reference frame O ; I ; J , consider the function f affine whose representative line C f passes through the two points : A ( − 1 ; 1.5) ; B (2 ; − 0.5) Determine the expression of the linear function f . E.2192 In the plane provided with a refer-ence frame O ; I ; J , consider the four straight lines shown below : Determine the reduced equations of these four straight lines. E.2129 In the plane provided with a refer-ence frame O ; I ; J orthonormal, consider the straight lines ( d 1 ) , ( d 2 ) and ( d 3 ) shown below : The coordinates of some points on these lines are given on the diagram. 1 The directing coefficient of the line ( d 1 ) has the value 7 4 . Determine the reduced equation of the line ( d 1 ) . 2 The y-intercept of the line ( d 2 ) has the value 1 . Deter-mine the reduced equation of the line ( d 2 ) . 3 Determine the reduced equation of the line ( d 3 ) . 7. Reduced equation by graphic reading E.9358 Proposition: The ordinate at the origin is the ordinate of the point of intersection of the representative line and the ordinate axis. We provide the plane with a reference frame and consider the straight lines ( d 1 ) and ( d 2 ) representative of the linear functions f and g : For each of these straight lines, two points on the grid belong-ing to them are indicated, and the direction is highlighted by a vector. https://chingmath.fr chapExoCorrec/540 sacados/540 chapExoCorrec/7088 sacados/7088 chapExoCorrec/9613 sacados/9613 -4-3-2-1234I-12JO chapExoCorrec/542 sacados/542 chapExoCorrec/8484 sacados/8484 chapExoCorrec/2192 sacados/2192 -5-4-3-2-12345I-3-2-123JO(d1(d2(d3(d4 chapExoCorrec/2129 sacados/2129 IJO(d1(d2(d3-1;(- chapExoCorrec/9358 sacados/9358 xxyy-6-5-4-3-2-10123456-2-112(d1(d2ABCD
x-5-4-3-2-1012345y-2-1123(d1(d3(d2 x-4-3-2-101234y-2-11234(d1(d2(d3(d4(d5 IJO(d1(d2(d3(d4(d5(d6 xxyy-6-5-4-3-2-10123456-3-2-112345(d1(d2(d3 1 a Using the vector −−→ AB , determine the slope of the function f . b Without justification, give the expression of the func-tion f . 2 a Using the vector −−→ CD , determine the slope of the function g . b Without justification, give the expression of the func-tion g . E.545 In the reference frame below, three straight lines ( d 1 ) , ( d 2 ) and ( d 3 ) are represented. By graphi-cal reading, determine the algebraic expressions of the three affine functions represented by these lines : E.1803 In the reference frame below are represented five straight lines. Graphically, determine the expression of the linear functionas-sociated with each of these lines. E.2691 In the plane provided with a O ; I ; J orthonormal coordinate system, consider the six straight lines shown below : Each line represents one of the following six functions : f : x ↦−→ 3 2 · x + 1 ; g : x ↦−→ − x − 1 ; h : x ↦−→ 2 5 · x + 1 j : x ↦−→ − 2 5 · x − 1 ; k : x ↦−→ − 3 4 · x +1 ; ‘ : x ↦−→ − 1 Associate, by reasoning and without calculations, the curve representation to each function. E.550 Three straight lines are shown in the frame below : Associate each of these lines with the affine function it repre-sents, from among : f : x ↦−→ 0.75 x + 1.25 g : x ↦−→ 0.75 x + 1.15 h : x ↦−→ − 2 7 x + 20 7 j : x ↦−→ − 2 7 x + 19 7 k : x ↦−→ 1 6 x − 11 6 ‘ : x ↦−→ 1 6 x − 10 6 Hints: the directing coefficient of a straight line can be read graphically, then the coordinates of one of the points on this line can be used. https://chingmath.fr chapExoCorrec/545 sacados/545 x-5-4-3-2-1012345y-2-1123(d1(d3(d2 chapExoCorrec/1803 sacados/1803 x-4-3-2-101234y-2-11234(d1(d2(d3(d4(d5 chapExoCorrec/2691 sacados/2691 IJO(d1(d2(d3(d4(d5(d6 chapExoCorrec/550 sacados/550 xxyy-6-5-4-3-2-10123456-3-2-112345(d1(d2(d3
xxyy-6-5-4-3-2-10123456-2-112(d1(d2ABCD -1I234JOCf x-1-0,500,511,52y0,51Cf x-2-1,5-1-0,500,51y0,51Cf E.11631 Proposition: The ordinate at the origin is the ordinate of the point of intersection of the representative line and the ordinate axis. We provide the plane with a reference frame and consider the straight lines ( d 1 ) and ( d 2 ) representative of the linear functions f and g : For each of these straight lines, two points on the grid belong-ing to them are indicated, and the direction is highlighted by a vector. 1 a Using the vector −−→ AB , determine the slope of the function f . b Without justification, give the expression of the func-tion f . 2 a Using the vector −−→ CD , determine the slope of the function g . b Without justification, give the expression of the func-tion g . 8. Equations of the first degree: antecedents E.8491 Consider the linear function f whose im-age of x is expressed as : f ( x ) = 3 x + 2 In the reference frame O ; I ; J oppo-site, is given the straight line C f repre-sentative of the function f . 1 Graphically, give the antecedent of the number 0.5 by the function f . 2 a Solve the equation : f ( x )=8 b Deduce the antecedent of the num-ber 8 by the function f E.8066 In the reference frame given be-low, consider the straight line C f representative of the linear function f defined by: f ( x ) = − 0.3 x + 0.4 1 Graphically and rounded to the nearest tenth, give the value of the antecedent of the number 0.5 by the function f . 2 a Solve equation : f ( x )=1.6 b Deduce the antecedent of the number 1.6 by the func-tion f . E.8114 In the reference frame given be-low, consider the straight line C f representative of the linear function f defined by: f ( x ) = 0.7 x + 0.9 1 a Graphically and rounded to the nearest tenth, give the value of the antecedent of the number 1 by the function f . b Solve equation : f ( x ) = 1 2 Determine the antecedent of the number 0 by the func-tion f . https://chingmath.fr chapExoCorrec/11631 sacados/11631 xxyy-6-5-4-3-2-10123456-2-112(d1(d2ABCD chapExoCorrec/8491 sacados/8491 -1I234JOCf chapExoCorrec/8066 sacados/8066 x-1-0,500,511,52y0,51Cf chapExoCorrec/8114 sacados/8114 x-2-1,5-1-0,500,51y0,51Cf
xxyy-3-2-1012345-112Cf x-1-0,500,511,52y0,51Cf -8-7-6-5-4-3-2-12345678I-1234JOABCD E.9351 Consider the linear function f de-fined on R by the expression : f ( x ) = − 2 3 · x + 3 4 Below, the straight line C f representative of the function f is given in the plane provided with a reference frame : Determine the set of antecedents of the number 5 6 . E.11632 In the reference frame given below, con- sider the straight line C f representative of the linear function f defined by: f ( x ) = − 0.3 x + 0.4 1 Graphically and rounded to the nearest tenth, give the value of the antecedent of the number 0.5 by the function f . 2 a Solve equation : f ( x )=1.6 b Deduce the antecedent of the number 1.6 by the func-tion f . 9. Parallelism and alignment of points E.8492 Proposition: Two straight lines representative of linear functions are par-allel if, and only if, their slopes are equal. In the plane provided with a reference frame, consider the four points below : A (0 ; 1) ; B (3 ; 8) ; C (1 ; 1) ; D (7 ; 15) Show that the straight lines ( AB ) and ( CD ) are parallel. E.1819 In the plane with a reference frame, consider the three points : A ( − 104 ; − 22) ; B (56 ; 18) ; C (82 ; 24) Are the points A , B , C aligned? Justify your answer. E.1615 In the plane provided with a refer-ence frame, we consider the four points : A ( − 1 ; 3) ; B (1 ; 6) ; C (2 ; 4) ; D ( − 2 ; − 2) 1 Show that the lines ( AB ) and ( DC ) are parallel. 2 a Determine the coordinates of the points K , L , M middles of the segments [ AD ] , [ BC ] , and [ AC ] respec-tively. b Show that the points K , L , and M are aligned. E.2829 In a reference frame, consider the following three points : A (1 ; 6) ; B (5 ; 16.4) ; C 1 2 ; 47 10 Justify whether or not these three points are aligned. 10. Secant lines and intersection E.8117 In the plane provided with a refer-ence frame, consider the four points : A ( − 2 ; 3) ; B (0 ; − 1) ; C ( − 3 ; − 1) ; D (2 ; 4) 1 Show that the straight lines ( AB ) and ( CD ) are secant. To do this, we will use the proposition below : Proposition: Let f and g be two linear functions. If the slopes of the functions f and g are distinct then their representative lines are secant. 2 We admit that the straight line ( AB ) is the representa-tive straight line of the linear function f admitting for expression : f ( x ) = − 2 x − 1 a Determine the expression of the linear function g ad-mitting as representative line the line ( CD ) . b Determine the coordinates of the point of intersection of the straight lines ( AB ) and ( CD ) . To do this, we use the proposition below. https://chingmath.fr chapExoCorrec/9351 sacados/9351 xxyy-3-2-1012345-112Cf chapExoCorrec/11632 sacados/11632 x-1-0,500,511,52y0,51Cf chapExoCorrec/8492 sacados/8492 chapExoCorrec/1819 sacados/1819 chapExoCorrec/1615 sacados/1615 chapExoCorrec/2829 sacados/2829 chapExoCorrec/8117 sacados/8117 -8-7-6-5-4-3-2-12345678I-1234JOABCD
x-3-2-101234y-2-1123(d1(d2 x-4-3-2-10123y-2-11(d1(d2 x-3-2-101234y123(d1(d2 x-3-2-101234y-1123A(d Proposition: Let f and g be two linear functions whose director coefficients are distinct. The abscissa of the point of intersection of their represen-tative lines is the unique solution of the equation : f ( x )= g ( x ) E.8068 In the reference frame below, are given the two straight lines ( d 1 ) and ( d 2 ) respectively of the functions f and g defined by: f ( x ) = 0.3 x + 1 ; g ( x ) = − 0.7 x + 1.4 1 Solve the equation : f ( x )= g ( x ) 2 Give the coordinates of the point M intersection of the lines ( d 1 ) and ( d 2 ) . E.8115 In the reference frame below, are given the two straight lines ( d 1 ) and ( d 2 ) representative re-spectively of the functions f and g defined by: f ( x ) = 0.4 x − 1.2 ; g ( x ) = 1.4 x − 0.8 1 Solve the equation : f ( x )= g ( x ) 2 Give the coordinates of the point M intersection of the lines ( d 1 ) and ( d 2 ) . E.8067 In the reference frame below, are given the two straight lines ( d 1 ) and ( d 2 ) respectively of the functions f and g defined by: f ( x ) = 0.4 x + 1 ; g ( x ) = − 0.5 x + 2 1 Solve the equation : f ( x )= g ( x ) 2 Give the coordinates of the point M intersection of the lines ( d 1 ) and ( d 2 ) . E.2912 In the plane provided with a refer-ence frame, consider the line ( d ) representative of the affine function f defined by: f ( x ) = 1 3 x + 2 3 Consider the line ( d ) passing through the two points : A ( − 3 ; 0) ; B (1 ; − 2) 1 Determine the expression of the affine function g repre-sented by the straight line ( d ) . 2 a Solve the following equation : f ( x )= g ( x ) b Give the coordinates of the point of intersection of the straight lines ( d ) and ( d ) . E.4734 In a plane with a reference point, consider the two points : A ( − 3 ; 2) ; B (2 ; − 1) Note f the linear functionadmitting the straight line ( AB ) as its representation in this frame of reference. 1 Determine the algebraic expression of the function f . 2 Consider the straight line ( d ) representative of the linear function g defined by: g ( x ) = 1 2 x − 5 2 a Justify that the straight lines ( d ) and ( AB ) intercept at a point. b Determine the coordinates of the point of intersection. E.6654 In the plane provided with the refer-ence frame below, we give the straight line ( d ) representative of the linear function f : 1 Graphically, determine the expression of the linear function f . 2 Consider the linear function g defined by: g ( x ) = − 1 2 x + 3 4 Let (Δ) be the representative line of the function g . a Justify that the point with coordinates A 3 ; − 3 4 be-longs to the line (Δ) . b Draw the straight line (Δ) representative of the func-tion g . 3 Determine the coordinates of the point of intersection of the straight lines ( d ) and (Δ) . 4 Consider the point B with coordinates B − 3 2 ; 1 2 . Determine the algebraic expression of the linear function h represented by the straight line ( AB ) . https://chingmath.fr chapExoCorrec/8068 sacados/8068 x-3-2-101234y-2-1123(d1(d2 chapExoCorrec/8115 sacados/8115 x-4-3-2-10123y-2-11(d1(d2 chapExoCorrec/8067 sacados/8067 x-3-2-101234y123(d1(d2 chapExoCorrec/2912 sacados/2912 chapExoCorrec/4734 sacados/4734 chapExoCorrec/6654 sacados/6654 x-3-2-101234y-1123A(d
−∞0∞−∞:::∞xVariationdef −∞0∞∞:::−∞xVariationdef −∞:::∞−∞0∞xVariationdef −∞:::∞∞0−∞xVariationdef xOyx0(dCoe`cientdirecteurpositif(m>0)xOyx0(dCoe`cientdirecteurnégatif(m<0)xSignedef(x−∞∞x00−xSignedef(x−∞∞x00− x-2-10123y-112(d1(d2 11. Perpendicular lines E.9614 Consider the two functions f and g defined by: f ( x ) = 2 3 · x − 7 6 ; g ( x ) = − 3 2 · x + 1 In the plane provided with an orthonormal frame of refer-ence, note ( d ) and ( d ) the representative lines of the linear functions f and g : 1 a Justify that the two straight lines ( d ) and ( d ) are perpendicular. b Determine the coordinates of the point A of intersec-tion of the straight lines ( d ) and ( d ) . 2 Give the coordinates of two points B and C belonging respectively to the straight lines ( d ) and ( d ) so that the triangle ABC is rectangular at A . 12. Variations E.8493 In a plane with a reference point, consider the two points : A ( − 3 ; 2) ; B (2 ; − 1) Note f the linear functionadmitting the straight line ( AB ) as its representation in this frame of reference. 1 Determine the algebraic expression of the function f . 2 a What is the direction of variation of the function f ? Justify your statement. b Draw up the table of variations of the function f . E.9353 Consider the linear function f de-fined on R whose expression is given by: f ( x ) = 3 x − 1 Which of the two tables of variations shown below corresponds to that of the function f ? Then complete the table of varia- tions. E.9354 Consider the linear function f de-fined on R whose expression is given by: f ( x ) = − 3 x − 1 Which of the two tables of variations shown below corresponds to that of the function f ? Then complete the table of varia-tions. 13. Table of signs E.9783 Proposal: 1 Consider the linear function f defined by: f ( x ) = 2 x + 3 Determine the sign table of the function f . 2 Consider the linear function g defined by: g ( x ) = − x + 1 Determine the sign table of the function g . E.2789 In the reference frame below, are represented the two straight lines ( d 1 ) and ( d 2 ) representa-tive respectively of the affine functions f and g defined on R : 1 The following questions will be answered graphically: a Solve the inequation : f ( x ) 0 b Solve the inequation : f ( x ) < 0 c Draw up the sign table for the function f . 2 a Draw up the sign table for the function g . https://chingmath.fr chapExoCorrec/9614 sacados/9614 chapExoCorrec/8493 sacados/8493 chapExoCorrec/9353 sacados/9353 −∞0∞−∞:::∞xVariationdef −∞0∞∞:::−∞xVariationdef chapExoCorrec/9354 sacados/9354 −∞:::∞−∞0∞xVariationdef −∞:::∞∞0−∞xVariationdef chapExoCorrec/9783 sacados/9783 xOyx0(dCoe`cientdirecteurpositif(m>0)xOyx0(dCoe`cientdirecteurnégatif(m<0)xSignedef(x−∞∞x00−xSignedef(x−∞∞x00− chapExoCorrec/2789 sacados/2789 x-2-10123y-112(d1(d2
x-4-3-2-101234y-2-112 x-4-3-2-101234y-2-112 x-1-0,500,511,52y0,5(d xxyy-4-3-2-101234-2-112Cf b Draw up the table of variations of the function g . E.2790 Consider the affine function f de-fined by the relation: f ( x ) = 2 x + 1 1 Solve the inequation : f ( x ) 0 . 2 Deduce the solutions of the inequation : f ( x ) < 0 . 3 Draw up the sign table for the function f . E.8116 Consider the two linear functions f and g defined by: f ( x ) = − 0.5 x + 0.75 ; g ( x ) = 0.4 x + 0.5 In the reference frame below, the straight lines ( d 1 ) and ( d 2 ) representative of the functions f and g respectively, are rep-resented : 1 Draw the straight lines ( d 1 ) and ( d 2 ) in the above refer-ence frame. The coordinates of the points used should be indicated on the copy. 2 Give the direction of variations of the functions f and g . 3 Draw up the sign tables for the functions f and g . E.4703 Consider the two linear functions f and g defined by: f ( x ) = 3 4 x + 1 ; g ( x ) = − 3 5 x + 1 2 The straight lines ( d 1 ) and ( d 2 ) , representing the functions f and g respectively, are shown in the reference frame below : 1 Draw the straight lines ( d 1 ) and ( d 2 ) in the table below. 2 Give the direction of variation of the functions f and g . 3 Draw up the sign tables for the functions f and g . E.9355 Consider the linear function g whose image of x is defined by: g ( x ) = − 1 2 x + 2 3 Draw up the sign table for the function g . 14. First degree equations E.8069 In the plane provided with a refer-ence frame, consider the line ( d ) representative of the function f defined by: f ( x ) = 0.3 x + 0.2 1 Using a graphical reading, give the set of solutions to the inequation : f ( x ) 0.5 2 Solve the inequation : f ( x ) 1 E.9356 Consider the function f affine whose expression is given by: f ( x ) = − 0.3 · x + 0.4 Below, in a reference frame, is given the representative line C f of the function f : 1 Graphically, solve the inequation : f ( x ) < 1 2 Solve the inequation : f ( x ) > − 2 https://chingmath.fr chapExoCorrec/2790 sacados/2790 chapExoCorrec/8116 sacados/8116 x-4-3-2-101234y-2-112 chapExoCorrec/4703 sacados/4703 x-4-3-2-101234y-2-112 chapExoCorrec/9355 sacados/9355 chapExoCorrec/8069 sacados/8069 x-1-0,500,511,52y0,5(d chapExoCorrec/9356 sacados/9356 xxyy-4-3-2-101234-2-112Cf
xxyy-4-3-2-101234-2-112Cf 1234ABCDEFx-20246f(xx2x−7-9-732147g(x4x-35132129h(x951-3-7 E.9357 Consider the f linear functionwhose expression is defined by: f ( x )= − 1 3 · x + 1 8 1 Solve the inequation : f ( x ) > 2 3 2 In a reference frame, we give the curve C f representative of the function f : Plot on the x-axis the set of solutions to the inequation in question 1 15. Affine functions: determine the expression E.958 Consider the affine function f whose graphical representation C f passes through the points A ( − 1 ; 3) and B (3 ; 4) . Determine the algebraic expression of the function f . E.976 Consider the line D through the points A (2 ; 2.5) and B (5 ; 1) 1 Let f be the function with the line representation D . Determine the algebraic writing of the function f . 2 Let C (101 ; − 47) . Are the points A , B , and C aligned? Justify. E.974 In the plane provided with the refer-ence frame O ; I ; J , consider the representative line C f of the affine function f . 1 C f passes through the points A (4 ; 6) and B ( − 4 ; 2) . Determine the algebraic expression of the function f . 2 Let C (102 ; 51) . Are the points A , B and C aligned? E.6307 The screenshot below shows the work Lea did to investigate three functions f , g and h such that : f ( x ) = x 2 + 3 x − 7 g ( x ) = 4 x + 5 h is an linear functionwhose expression Lea forgot to write in cell A4 . 1 Give a number that has image − 7 by the function f . 2 Verify with a detailed calculation that : f (6)=47 . 3 Explain why the table gives a solution to the equation : x 2 + 3 x − 7 = 4 x + 5 What is this solution? 4 Using the table, find the algebraic expression h ( x ) of the linear function h . 16. Affine functions: determine the expression (with rational) E.964 Determine the expression of the func- tion f admitting as representation the straight line passing through the points A (1 ; 3) and B (4 ; 2) . 17. Affine functions and antecedent searches (without rationals) E.11106 In the plane with a reference point, consider the two points : A (3 ; − 4) ; B (6 ; 2) Let f be the linear function whose representative curve is the line ( AB ) . 1 Determine the algebraic expression of the function f . 2 Determine the image of the integer 1 by the function f . 3 Determine the antecedent of the integer 2 by the function f . https://chingmath.fr chapExoCorrec/9357 sacados/9357 xxyy-4-3-2-101234-2-112Cf chapExoCorrec/958 sacados/958 chapExoCorrec/976 sacados/976 chapExoCorrec/974 sacados/974 chapExoCorrec/6307 sacados/6307 1234ABCDEFx-20246f(xx2x−7-9-732147g(x4x-35132129h(x951-3-7 chapExoCorrec/964 sacados/964 chapExoCorrec/11106 sacados/11106
E.11108 In the plane provided with a reference frame, consider the two points : A ( − 2 ; 1) ; B (1 ; − 8) Note f the affine function admitting the line ( AB ) as its rep-resentative curve. 1 Determine the algebraic expression of the function f . 2 Determine the image of the integer 3 by the function f . 3 Determine the antecedent of the integer 1 by the function f . E.11110 In the plane provided with a reference frame, consider the two points : A ( − 3 ; 1) ; B ( − 1 ; 5) Note f the affine function admitting the line ( AB ) as its rep-resentative curve. 1 Determine the algebraic expression of the function f . 2 Determine the image of the integer 1 by the function f . 3 Determine the antecedent of the integer 6 by the function f . E.11112 In the plane provided with a reference frame, consider the two points : A ( − 1 ; 11) ; B (2 ; − 7) Note f the affine function admitting the line ( AB ) as its rep-resentative curve. 1 Determine the algebraic expression of the function f . 2 Determine the image of the integer 1 by the function f . 3 Determine the antecedent of the integer 3 by the function f . 18. Affine functions and antecedent searches (with rationals) E.11109 In the plane marked with a refer-ence point, consider the two points : A (0 ; 2) ; B (9 ; 0) Let f be the linear function whose representative curve is the line ( AB ) . 1 Determine the algebraic expression of the function f . 2 Determine the image of the integer 2 by the function f . 3 Determine the antecedent of the integer 1 by the function f . E.11107 In the plane provided with a reference frame, consider the two points : A (5 ; − 2) ; B ( − 1 ; 2) Note f the affine function admitting the straight line ( AB ) as its representative curve. 1 Determine the algebraic expression of the function f . 2 Determine the image of the integer − 4 by the function f . 3 Determine the antecedent of the integer 1 by the function f . E.11111 In the plane provided with a reference frame, consider the two points : A ( − 1 ; 2) ; B (6 ; − 1) Note f the affine function admitting the line ( AB ) as its rep-resentative curve. 1 Determine the algebraic expression of the function f . 2 Determine the image of the integer 1 by the function f . 3 Determine the antecedent of the integer 3 by the function f . E.11113 In the plane with a reference point, consider the two points : A ( − 3 ; 2) ; B (6 ; − 2) Let f be the linear function whose representative curve is the line ( AB ) . 1 Determine the algebraic expression of the function f . 2 Determine the image of the integer 1 by the function f . 3 Determine the antecedent of the integer 3 by the function f . 19. Share E.6692 For each question, several answers are possible. We are interested in the functions f and g defined on R by: f ( x ) = 1 − 2 x ; g ( x ) = 2 x + 3 . Note C f and C g their respective graphical representations in a reference frame. 1 The image of ( − 3) by f is : a 7 b 5 2 c − 2 2 The antecedent of ( − 3) by g is : a 3 b 0 c − 3 3 The point A of coordinates (1 ; − 5) belongs to : a C f b C g c ni C f ; ni C g 4 On R : a f est décroissante b f is increasing c g is decreasing d g is increasing 5 Which sign table(s) is/are correct? https://chingmath.fr chapExoCorrec/11108 sacados/11108 chapExoCorrec/11110 sacados/11110 chapExoCorrec/11112 sacados/11112 chapExoCorrec/11109 sacados/11109 chapExoCorrec/11107 sacados/11107 chapExoCorrec/11111 sacados/11111 chapExoCorrec/11113 sacados/11113 chapExoCorrec/6692 sacados/6692
ABCDIM -5-4-3-2-12345I-2-123JO(d1(d3(d2 a x −∞ 2 + ∞ f ( x ) + 0 − b x −∞ 2 + ∞ f ( x ) − 0 + c x −∞ 0.5 + ∞ f ( x ) + 0 − d x −∞ 0.5 + ∞ f ( x ) − 0 + e x −∞ − 1.5 + ∞ g ( x ) + 0 − f x −∞ − 1.5 + ∞ g ( x ) − 0 + 20. Unclassified exercises E.6575 Consider the square ABCD shown opposite with side 1 . Determine the area of the gray S surface. Anytraceofresearch,evenifincomplete,willbe takenintoaccountintheassessment. E.7087 Using a graph, give the slope-intercept formof each of the lines shown below : E.8932 Let the polynomial be : y 2 − x − 2 2 − 2 x − 1 y + x − 2 1 Decompose this polynomial into a product of two factors ; let A and B be these two factors. 2 For what values of x and y do we simultaneously have : A = 0 , B = 0 ? 3 Study and graph the variations of the functions : y = − x + 2 ; y = 3 x − 4 in a rectangular coordinate system ; let ( d ) and ( d ) be the resulting straight lines. What are the coordinates of their I point of intersection? 4 ( d ) intersects ( x x ) at M . ( d ) cuts ( y y ) into N . What are the coordinates of M ? of N ? of the middle P of [ MN ] ? What is the equation of the parallel to the line ( d ) led by P ? E.2726 The directrix m of an affine function f is defined by the quotient : f ( b ) − f ( a ) b − a = m for any real number a and b . 1 Suppose f admits a positive directing coefficient. a Justify that f ( b ) − f ( a ) has the same sign as b − a . b Deduce that the function f is increasing. 2 Determine the direction of variation of the function f in the case where it admits a negative directing coefficient. Justify. E.7866 We define two functions f , g affine by the relation: f ( x ) = 3 x + 4 ; g ( x ) = − x + 2 We will use the sense of variation of these two functions to answer the following questions : 1 Determine the images of the interval [ − 2 ; 5] by each of the functions f and g . 2 Determine the interval I such that its image by f is R + ; that is, verifying the relation: f I = R + 3 Determine the interval J such that its image by g is R + ; i.e. verifying the relation: g J = R + https://chingmath.fr chapExoCorrec/6575 sacados/6575 ABCDIM chapExoCorrec/7087 sacados/7087 -5-4-3-2-12345I-2-123JO(d1(d3(d2 chapExoCorrec/8932 sacados/8932 Brevet Rennes 1962 chapExoCorrec/2726 sacados/2726 chapExoCorrec/7866 sacados/7866
x-3-2-10123y-3-2-1123 -4-3-2-1234I-2-123JOCfCg -4-3-2-1234I-2-123JOCh 050100150200500(d1(d2 E.544 Consider the f affine function defined by the algebraic expression : f ( x ) = 1.25 x − 1 . 1 Among the points : A ( − 1 ; − 2.25) ; B (2 ; 1.5) ; C (0 ; 1.25) Which of these points belong to the line C f representa-tive of the function f ? 2 Draw, in the reference frame below, the straight line C f representative of the function f 3 a Place the point M belonging to C f having zero ab-scissa. b What is the ordinate of point M ? What is the y-intercept of the point M relative to the function f called? 4 a Determine the value of the quotient : m = f ( x B ) − f ( x A ) x B − x A b What is the number m called relative to the function f ? E.2847 In the ( O ; I ; J ) orthonormal frame of reference below, consider the three straight lines represent-ing three functions f , g , h Determine, graphically, the slope-intercept forms of the func-tions f and g . E.9804 In the reference frame ( O ; I ; J ) or-thonormal below, consider the line representing the linear function h . 1 Solve the system of equations below : − 2 a + b = − 1 7 4 a + b = 3 2 2 Deduce the slope-intercept formof the function h . Justify your approach. E.4726 Consider the plane provided with a O ; I ; J orthonormal and the two points A (2 ; 4) and B (6 ; − 1) and the line ( d ) of equation : ( d ) : y = − 1 2 · x + 7 2 Show that line ( d ) passes through the middle of segment [ AB ] . E.11674 Une entreprise décide de produire des sacs à main et considère un facteur de qualité noté a ( a est un nombre strictement positif) . En notant x le nombre de sacs produits, il estime que : le coût C de production se détermine par : C ( x ) = a × x + 400 le revenu R de la vente se détermine par : R ( x ) = a + 10 × x 1 L’entreprise veut que sa recette soit le double du coût de production. Exprimer le nombre de sac qu’elle doit vendre en fonction du paramètre a . 2 Après avoir choisi sa politique, voici la représentation de ces deux fonctions : a Associer à chaque fonction sa représentation graphique. b Déterminer la valeur du paramètre a . c Déterminer par le calcul le nombre de sac à vendre. https://chingmath.fr chapExoCorrec/544 sacados/544 x-3-2-10123y-3-2-1123 chapExoCorrec/2847 sacados/2847 -4-3-2-1234I-2-123JOCfCg chapExoCorrec/9804 sacados/9804 -4-3-2-1234I-2-123JOCh chapExoCorrec/4726 sacados/4726 chapExoCorrec/11674 sacados/11674 050100150200500(d1(d2
-4-3-2-1234I-2-123JOABCD -4-3-2-1234I-2-12345JOCDAB E.4061 In the plane provided with the reference frame O ; I ; J orthonormal, consider the points A , B , C and D shown below : 1 Give the coordinates of the points A , B , C , D . 2 Consider the function f whose graphical representation is the straight line ( AB ) : a Show that the directing coefficient of the line ( AB ) is 2 5 . b The function f admits as expression : f ( x ) = 0.4 · x + b where b is a number Using the coordinates of point B , determine the value of b . 3 Consider the function g whose graphical representation is the straight line ( CD ) : a Determine the directing coefficient of the line ( CD ) . b Determine the complete expression of the function g . E.4063 In the reference frame O ; I ; J or-thonormal, consider the points A , B , C , D shown below : 1 Give the coordinates of the points A , B , C and D . 2 a Draw the straight line ( AB ) . b Determine the directing coefficient of the line ( AB ) . c Thus, the affine function f , having the line ( AB ) as its representation, has as its expression : f ( x ) = 1.2 · x + b où b ∈ R . Using the coordinates of the point B , determine the value of the y-intercept of the function f . 3 a Draw the straight line ( CD ) . b Determine the value of the quotient : a = y D − y C x D − y C c Let g be the affine function whose representation is the straight line ( CD ) . Among the following expressions, give the expression of the function g : g ( x ) = 1 2 · x + 1 ; g ( x ) = − 1 2 · x − 1 g ( x ) = − 1 2 · x + 1 ; g ( x ) = − 2 · x + 1 Justify your answer. https://chingmath.fr chapExoCorrec/4061 sacados/4061 -4-3-2-1234I-2-123JOABCD chapExoCorrec/4063 sacados/4063 -4-3-2-1234I-2-12345JOCDAB
-4-3-2-1234I-4-3-2-1234JO(d1(d2(d3(d4 -4-3-2-1234I-2-123JO(d1(d2AB E.2577 Determine the directing coefficients of each of the three straight lines shown below in the reference frame ( O ; I ; J ) : E.4077 In the plane provided with a reference frame O ; I ; J orthonormal, consider the two straight lines ( d 1 ) and ( d 2 ) below : 1 Donner les coordonnées du point d’intersection des droites ( d 1 ) et ( d 2 ) . 2 Let f be the affine function admitting for representation the straight line ( d 1 ) . a Determine the directing coefficient of the function f . b Deduce the complete expression of the function f . 3 Let g be the affine function admitting as its represen-tation the straight line ( d 2 ) . Explaining your approach, determine the expression of the function g . 4 Consider the function h defined, for any real number x , by: h ( x ) = 3 2 × x + 2 In the plane, draw the representative line of the function h . https://chingmath.fr chapExoCorrec/2577 sacados/2577 -4-3-2-1234I-4-3-2-1234JO(d1(d2(d3(d4 chapExoCorrec/4077 sacados/4077 -4-3-2-1234I-2-123JO(d1(d2AB