Since the variable x and dependent variable y has the following relationship: y = kx + b Is now known as x y is a function. When b = 0,, y is a function of x in direct proportion. Ie: y = kx (k is a constant, k ≠ 0)
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The nature of a function
一次函数的性质
一次函数的性质
一次函数的性质
一次函数的性质
1.y changes in the value of the change and the corresponding value of x is proportional to the ratio for the k Ie: y = kx + b (k is any nonzero real number b to take any real number) 2. When x = 0,, b as a function of the y-axis intercept.
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A function of the image and nature of
一次函数的图像及性质
一次函数的图像及性质
一次函数的图像及性质
一次函数的图像及性质
一次函数的图像及性质
1. Practices and graphics: The following 3 steps (1) list [generally take two points, two points determine a straight line according to]; (2) described point; (3) the connection can be made a function of the image - a straight line. Therefore, as a function of the image just need to know 2 points, and even into a straight line can be. (Usually to find function for image and x-axis and y-axis) 2. Properties: (1) at a function on any point p (x, y), satisfies the equation: y = kx + b (k ≠ 0). (2) a function of the coordinates of the intersection with the y-axis is always (0, b), and the x-axis is always intersect at (-b / k, 0) is always directly proportional to the function of the image through the origin. 3. k, b and the function of the quadrant image: When k> 0, the line will be through the first and third quadrants, y with the increase of x; When k <0, the line will be by two, four-quadrant, y with x, increases. When b> 0, the line will be through one or two quadrants; When b = 0, the line will pass through the origin. When b <0, the line will be through three or four quadrants. In particular, when b = 0, the straight line through the origin o (0,0) function that is directly proportional to the image. Then, when k> 0, the straight-line only through the first and third quadrants; when k <0, the straight-line only through the second and fourth quadrants. 4, a special position relationship When the plane rectangular coordinate system in two parallel lines, its function is analytic in the k value (ie a coefficient) is equal When the plane rectangular coordinate system, the two vertical lines, its function is analytic in the k value of the negative reciprocal of each other (ie, the value of the product of two k -1)
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To determine the expression of a function
确定一次函数的表达式
确定一次函数的表达式
确定一次函数的表达式
Known point a (x1, y1); b (x2, y2), make sure that through the point a, b of a function expression. (1) Let a function of expression (also called analytic) as y = kx + b. (2) because any point on a function p (x, y), satisfies the equation y = kx + b. So can list two equations: y1 = kx1 + b ... ... ① and y2 = kx2 + b ... ... ② (3) a solution of this binary equation, k, b values. (4) Finally, be a function of the expression.
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A function application in life
一次函数在生活中的应用
一次函数在生活中的应用
一次函数在生活中的应用
1. When the time t fixed, s is the velocity v from a function. s = vt. 2. When the pool pump speed f must, pool, water pumping time t g is a function. Let the original pool of water s. g = s-ft.
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Common formula (incomplete, I hope it was added)
一次函数 常用公式（不全，希望有人补充）
一次函数 常用公式（不全，希望有人补充）
一次函数 常用公式（不全，希望有人补充）
1. The image of the k value of a function: (y1-y2) / (x1-x2) 2. Find x-axis parallel to the line with the midpoint: | x1-x2 | / 2 3. Request line with the y-axis parallel to the midpoint: | y1-y2 | / 2 4. Find any segment of the length: √ (x1-x2) ^ 2 + (y1-y2) ^ 2 (Note: The root of the next (x1-x2) and (y1-y2) and the square) 5. Find the intersection of two coordinates of the first functional image: Solutions to two functional
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Applications
一次函数 应用
一次函数 应用
A function y = kx + b is the nature of: (1) When k> 0 时, y with the increase of x; (2) when k <0 时, y with x, increases. A function of the nature of the use of the following issues to resolve. First, determine the range of letters coefficient Example 1. Known to directly proportional function, the time when m =______________, y with x, increases. Solution: According to the definition of the function and nature of direct proportion, too, and m <0, that is, and, so. Second, compare the value of x value or the size of y Example 2. Known point p1 (x1, y1), p2 (x2, y2) is a function y = 3x +4 two points on the image, and y1> y2, x1 and x2 are the size of relations ( ) a. x1> x2 b. x1 <x2 c. x1 = x2 d. not sure Solution: According to the meaning of the questions, knowing k = 3> 0, and y1> y2. According to the nature of a function "when k> 0 时, y increases with x," have x1> x2. Was chosen a. Third, determine the location of the image function Example 3. A function y = kx + b satisfy the kb> 0, and y increases as x decreases, the image of this function without () a. first b. second quadrant quadrant c. third d. fourth quadrant quadrant Solution: the kb> 0, known k, b the same number. As y decreases with x, so k <0. So b <0. Therefore, a function y = kx + b image after the second, third and fourth quadrants, without first quadrant. Was chosen a
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Mathematical terms
【】 Yī cì hán shù pronunciation 【English】 linear function 【】 Function to explain the basic concept: In general, in a process of change, there are two variables x and y, if given an X value, corresponding to only one determined value of X corresponding to Y, then we call X Y is a function (function). where X is the independent variable, Y is the dependent variable, ie Y is a function of X. When x = a, the function is called when the value of the function when x = a value. Direct proportion function is a function of the particular circumstances.
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Definition and the definition of style
Since the variable x and dependent variable y has the following relationship: y = kx (k is any non-zero real number) Or y = kx + b (k is an arbitrary non-zero constant, b is an arbitrary constant) Is now known as x y is a function. In particular, when b = 0,, y is a function of x in direct proportion. Ie: y = kx (k is an arbitrary constant) Proportional function of the image pass through the origin Domain: the range of variables, the values of variables that function should be meaningful; to be consistent with the actual. When x is only a certain amount of time corresponding to the y and x.
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The nature of a function
一次函数的性质
一次函数的性质
1.y changes in the value of the change and the corresponding value of x is proportional to the ratio for the k Ie: y = kx + b (k ≠ 0) (k is not equal to 0, and k, b are constants) 2. When x = 0,, b as a function of the y-axis, the coordinates (0, b). 3.k as a function of y = kx + b the slope, k = tanΘ (angle Θ as a function of the image with the x-axis is the direction angle, Θ ≠ 90 °) Shape. Take. Like. Pay. Less 4. When b = 0, the first function of the image into a proportional function, a direct proportion function is a special function. 5. Function Image properties: the same as k and b are not equal, the image parallel; when k is different and b are equal, the image intersection; when k negative reciprocal of each other, the two vertical lines; when k, b are the same, the two straight line overlap.
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A function of the image and nature of
一次函数的图像及性质
1. Practices and graphics: The following 3 steps (1) list [generally take two points, two points determine a straight line according to]; (2) described point; (3) the connection can be made a function of the image - a straight line. Therefore, as a function of the image just need to know 2 points, and even into a straight line can be. (Usually to find function for image and x-axis and y-axis) 2. Properties: (1) at a function on any point P (x, y), satisfies the equation: y = kx + b (k ≠ 0). (2) a function of the coordinates of the intersection with the y-axis is always (0, b), and the x-axis is always intersect at (-b / k, 0) is directly proportional to the function of the image through the origin. 3. Function is not a number, it refers to a process of change in the relationship between two variables. 4. k, b and the function of the quadrant image: y = kx (ie, b is equal to 0, y and x is proportional) When k> 0, the line will be through the first and third quadrants, y with the increase of x; When k <0, the line will be by two, four-quadrant, y with x, increases. y = kx + b when: When k> 0, b> 0, then the image of this function after one, two, three quadrants. When k> 0, b <0, then the image of this function after one, three, four quadrant. When k <0, b> 0, then the image of this function after one, two, four quadrants. When k <0, b <0, then the image of this function through two, three, four quadrants. When b> 0, the line will be through one or two quadrants; When b <0, the line will be through three or four quadrants. In particular, when b = 0, the straight line through the origin O (0,0) function that is directly proportional to the image. Then, when k> 0, the straight-line only through the first and third quadrants, not through the second and fourth quadrant. When k <0, the straight-line only through the second and fourth quadrants, not through the first and third quadrants. 4, a special position relationship When the plane rectangular coordinate system in two parallel lines, its function is analytic in K value (ie a coefficient) is equal When the plane rectangular coordinate system, the two vertical lines, its function is analytic in each other the negative reciprocal of K values (ie, the value of the product of two K -1)
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To determine the expression of a function
确定一次函数的表达式
Known points A (X1, y1); B (X2, y2), make sure that over points A, B expression of a function. (1) Let a function of expression (also called analytic) as y = kx + b. (2) because any point on a function P (x, y), satisfies the equation y = kx + b. So can list two equations: y1 = kx1 + b ... ... ① and y2 = kx2 + b ... ... ② (3) a solution of this binary equation, k, b values. (4) be a function of the last expression.
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Common formula
1. The image of the k value of a function: (y1-y2) / (x1-x2) 2. Find x-axis parallel to the line with the midpoint: | x1-x2 | / 2 3. Request line with the y-axis parallel to the midpoint: | y1-y2 | / 2 4. Find any segment of the length: √ (x1-x2) ^ 2 + (y1-y2) ^ 2 (Note: The root of the next (x1-x2) and (y1-y2) and the square) 5. Find the intersection of two coordinates of the first functional image: Solutions to two functional Two time functions y1 = k1x + b1 y2 = k2x + b2 so y1 = y2 have k1x + b1 = k2x + b2 to solve for the x = x0 value on behalf of the back y1 = k1x + b1 y2 = k2x + b2 any type of two-type then get y = y0 (x0, y0) is the y1 = k1x + b1 and y2 = k2x + b2 intersection coordinates 6. Find any connection point 2 of the midpoint of segment coordinates: [(x1 + x2) / 2, (y1 + y2) / 2] 7. Find any 2 points of a function analytic connection: (X-x1) / (x1-x2) = (Y-y1) / (y1-y2) (where the denominator is 0, then the numerator is 0) xy + + In a quadrant + - In the four-quadrant - + In the second quadrant - - In three quadrants 8. If the two lines y1 = k1x + b1 ∥ y2 = k2x + b2, then k1 = k2, b1 ≠ b2 9. If two lines y1 = k1x + b1 ⊥ y2 = k2x + b2, then k1 × k2 =- 1 10. y = k (x + n) + b n units is to the left shift y = k (xn) + b n units is the right translation Formulas: Add right-minus left (only for the change in x) y = kx + b + n is the upward shift n units y = kx + bn n units is down shift Formulas: on the plus next cut (only for change b)
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Applications
一次函数 应用
A function y = kx + b is the nature of: (1) When k> 0 时, y with the increase of x; (2) when k <0 时, y with x, increases. A function of the nature of the use of the following issues to resolve. First, determine the range of letters coefficient Example 1. Known to directly proportional to the function, when k <0 时, y with x, increases. Solution: According to the definition of the function and nature of direct proportion, too, and m <0, that is, and, so. Second, compare the value of x value or the size of y Example 2. Known point P1 (x1, y1), P2 (x2, y2) is a function y = 3x +4 two points on the image, and y1> y2, x1 and x2 are the size of relations ( ) A. x1> x2 B. x1 <x2 C. x1 = x2 D. can not be determined Solution: According to the meaning of the questions, knowing k = 3> 0, and y1> y2. According to the nature of a function "when k> 0 时, y increases with x," have x1> x2. Was chosen A. Third, determine the location of the image function Example 3. A function y = kx + b satisfy the kb> 0, and y increases as x decreases, the image of this function without () The second quadrant of the first quadrant B. A. C. third D. fourth quadrant quadrant Solution: the kb> 0, known k, b the same number. As y decreases with x, so k <0. So b <0. Therefore, a function y = kx + b image after the second, third and fourth quadrants, without first quadrant. Was chosen A. Typical example: Example 1. A spring, do not hang objects length 12cm, hanging objects will stretch, stretch the length of the link with the mass of the object in direct proportion. If the hang 3kg object, the spring length is 13.5cm, find the spring length is y (cm) linked to the object with the mass x (kg) as a function of relationship. If the spring is the maximum length of 23cm, find the range of the independent variable x. Analysis: This question from the physical problem into mathematical qualitative quantitative problem, but also practical issues, its core is the length of the spring-load and load length is the length of and after the elongation, while the range of variables may be the → General → maximum elongation of the largest maximum quality and practical ideas to deal with. Solution: find the meaning of the questions _set_ by the function y = kx +12 Is 13.5 = 3k +12, was k = 0.5 ∴ analytic function of the demand for the y = 0.5x +12 Get by the 23 = 0.5x +12: x = 22 ∴ The range of the independent variable x is 0 ≤ x ≤ 22 Example 2 A school must burn some CD-ROMs, computer company if the burn, take 8 yuan each, if a school from the moment, in addition to hiring outside burners 120, each needs to cost 4 yuan, and asked the computer company is to burn the disc, carved their own cost, or more provincial schools? The scope of issues to consider X Solution: Let the total cost, Y Yuan, X Zhang burning Computer: Y1 = 8X School: Y2 = 4X +120 When X = 30 when, Y1 = Y2 When X> 30 when, Y1> Y2 When X <30 when, Y1 <Y2 】 【Test center means an The definition of a function, image and character description in the test point is C level knowledge, in particular the conditions under question and the use of a function analytic method of undetermined coefficients of a function analytic in D-level exam is the knowledge that point. It is often associated with inverse functions, quadratic functions and equations, equations, inequalities combined, the multiple choice, fill in the blank to answer questions and other kinds of questions in the exam, about 8 minutes or so plays. to solve such problems common to Category discussion Shuoxingjiege, equations and conversion of mathematical thinking. Example 3. If a function y = kx + b in the range of x is -2 ≤ x ≤ 6, the corresponding function value in the range -11 ≤ y ≤ 9. Find the analytical expressions of this function. Solution: (1) If k> 0, then the equations can be out-2k + b =- 11 6k + b = 9 Solutions have k = 2.5 b =- 6, then the relationship at this time as a function of y = 2.5x-6 (2) If k <0, then the equations can be out-2k + b = 9 6k + b =- 11 Solutions have k =- 2.5 b = 4, then the analytic function at this time is y =- 2.5x +4 】 【Test center means an This problem mainly on the students understanding of the nature of the function, if k> 0, then y increases as x increases; if k <0, then y increases as x decreases. A function of several types of analytic ① ax + by + c = 0 [general formula] ② y = kx + b [inclined cutting] (K as the slope of the line, b for the vertical intercept of a straight line, directly proportional to the function b = 0) ③ y-y1 = k (x-x1) [point slope form] (K as the slope of the line, (x1, y1) off the line by a point) ④ (y-y1) / (y2-y1) = (x-x1) / (x2-x1) [Two-Point] ((X1, y1) and (x2, y2) for the two points on a straight line) ⑤ x / ay / b = 0 [intercept-type] (A, b are linear in x, y axis intercept) Analytic expression of limitations: ① conditions required more (3); ②, ③ can not express the slope of the straight line is not (a straight line parallel to the x-axis); ④ many parameters, calculation is too cumbersome; ⑤ not express a straight line parallel to the axis and over-dot lines. Tilt angle: x-axis to the line angle (a straight line with the x-axis direction as being the angle) is called the line angle. _Set_ the tilt angle of a straight line a, then the slope of the line k = tg (a)
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Related Phrases
math
case number theory
function
direct proportion function
beeline
quadratic function
Containing Phrases
bout function temperament
bout function denaturalisation
Bout function and quadratic function
Bout function of Image to reach temperament
ascertain Bout function of expression
Bout function Cast (in) one's lot with [throw in one's lot with] center of appliance
