A linear function is one of the functions, and its general form is y=kx+b (k, b is a constant, k ≠ 0), where x isindependent variable, y is the dependent variable.In particular, when b=0, y=kx (k is a constant, k ≠ 0), y is called x'sPositive proportional function（direct proportion function）。
The linear function and its graph areJunior high school algebraIt is also the cornerstone of analytic geometry in high school, and it is also the key content of the high school entrance examination.
The graph of a linear function is a straight line.
The term "function" was originally used by German mathematiciansLeibnizFirst adopted in the 17th century, Leibniz used the word "function" to express the power of variable x, that is, xtwo，xthreeNext, Leibniz used the word "function" to mean abscissa, ordinate, length of tangent, length of vertical line and other variables related to points on the curve, so the word "function" gradually prevailed.
In China, people in ancient times used the word "letter" and the word "contain" in common, both of which have the meaning of "contain". Mathematicians, astronomers, translators and educators in the Qing Dynasty, and pioneers of modern scienceLi ShanlanThe definition given is: "Where there is heaven in a formula, it is a function of heaven." The ancient Chinese also used the words "heaven, earth, people, and things" to express four different unknowns or variables. Obviously, the meaning in Li Shanlan's definition is that "where a formula contains a variable x, the formula is called a function of x." Thus, in China, "function"It means that the formula contains variables.
Swiss mathematician Jacques Bernoulli gave the same function definition as Leibniz.In 1718, John Bernoulli, the younger brother of Jacques Bernoulli, gave the following definition of function: a quantity consisting of any form of any variable and constant is called a function of this variable. In other words, any matrix consisting of x and constant can be called a function of x.
In 1775,EulerThe function is defined as: "If some variables depend on other variables in a certain way, that is, when the latter variables change, the former variables also change. We call the former variables functions of the latter variables." It can be seen from this that the concept of function introduced from Leibniz to Oura is still the same as that of analytic expressionConcepts such as curve expression are intertwined.
The leading French mathematicianCauchyA new definition of function is introduced: "There is a certain relationship between some variables. When the value of one variable is given, the value of other variables can also be determined accordingly, then the original variable is called 'independent variable', and other variables are called 'functions'.In Cauchy's definition, the word "independent variable" first appeared.
In 1834, Russian mathematicianLobachevskyIt further proposes the definition of function: "The function of x is such a number that it has a certain value for each x and changes with x. The value of the function can be given by the analytic expression or by a condition, which provides a method to find all corresponding values. This dependency of the function can exist, but is still unknown."This definition indicates the correspondence.That is, the necessity of the condition. Use this relationship to find the corresponding value of each x.
1837 German mathematicianDirichletHe thought that how to establish the correspondence between x and y was irrelevant, so his definition was: "If for every value of x, y always has a completely certain value corresponding to it, then y is a function of x."
German mathematicianRiemannA new definition of function is introduced: "For every value of x, y always has a completely determined value corresponding to it. Regardless of the method of establishing the corresponding between x and y, y is called a function of x."
From the evolution of the above function concept, we can know that the definition of function must grasp the essential attribute of function. The variable y is called the function of x, and only one rule exists, so that every value in the value range of this function has a certain value of y corresponding to it, regardless of whether this rule is a formula or a graph or a table or other forms.
Thus, we have the definition of function in our textbooks: generally, in a change process, if there are two variables x and y, and for each determined value of x, y has a unique determined value corresponding to it, then we say that x is an independent variable and y is a function of x.[1]
Representation method
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Image method
There are three ways to express the primary function, as follows:
The method of tabulating the function relationship represented by the function value y corresponding to a series of x values is called tabulation method.
3. Image method
Represented by imagesFunctional relationThis method is called image method.[1]
Analytic expression
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The analytic expression of the primary function is:
Where m isSlope, cannot be 0;X represents the independent variable, and b represents the y-intercept.And m and b are bothconstant。First, the analytic expression of the function is set, and then the unknown slope in the analytic expression is determined according to the conditions, thus the analytic expression is obtained.The analytic expression is similar to the oblique cut expression in the linear equation.[1]
Basic properties
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1. Method and figure: through the following three steps:
(1) List: each time a value of the independent variable x is determined, a value of the dependent variable y is calculated and listed;
(2) Tracing points: generally, take two points, and trace the points corresponding to the values in the table according to the principle of "two points determine a straight line", that is, in the plane rectangular coordinate system, take the value of the independent variable as the abscissa, and the corresponding function value as the ordinate.
In general, the image of y=kx+b (k ≠ 0) can be drawn through (0, b) and (- b/k, 0).
The image of the positive proportional function y=kx (k ≠ 0) is a straight line passing through the origin of coordinates, which is generally drawn by taking two points (0, 0) and (1, k).
(3) Wires: an image that can be used to make a primary function - onestraight line。Therefore, as a linear functionimageJust know 2 points and connect them into a straight line.
2. Properties: (1) Any point P (x, y) on a linear function satisfies the equation: y=kx+b (k ≠ 0).(2) The coordinates of the intersection point of a linear function and the y-axis are always (0, b), and the intersection point with the x-axis is always (- b/k, 0)Positive proportional functionThe images of are all across the origin.
3. Function is not a number, it refers to the relationship between two variables in a change process.
When y=kx (that is, b is equal to 0, y is proportional to x, and the image is a straight line passing through the origin)
When k>0, the straight line must pass through the first and third quadrants, and y increases with the increase of x;
When k<0, the straight line must pass through the second and fourth quadrants, and y decreases with the increase of x.
When y=kx+b (k, b is a constant, k ≠ 0):
When k>0, b>0, the image of this function passes through one, two, and three quadrants;
When k>0, b<0, the image of this function passes through one, three, and four quadrants;
When k<0 and b>0, the image of this function passes through the first, second and fourth quadrants;
When k<0, b<0, the image of this function passes through two, three, and four quadrants.
When b>0, the straight line must pass through 1 and 2quadrant；
When b<0, the straight line must pass through the three and four quadrants.
In particular, when b=0, the line passing through the origin O (0, 0) represents the image of a positive proportional function.
At this time, when k>0, the straight line only passes through the first and third quadrants, not the second and fourth quadrants.When k<0, the straight line only passes throughQuadrant, will not pass the first and third quadrants.
5. When x=0, b is the intersection of the function on the y-axis, and the coordinates are (0, b).
When y=0, the intersection coordinate of the function image on the x-axis is (- b/k, 0).
6. The relationship between the image and property of the straight line y=kx+b and k, b is shown in the following table:
k> 0, b>0: passing through the first, second and thirdquadrant
k> 0, b<0: passing through the first, third and fourth quadrants
k> 0, b=0: passing the first and third quadrants (passing the origin)
Conclusion: when k>0, the image rises from left to right, and y increases with the increase of x.
K<0, b>0: passing through the first, second and fourth quadrants
K<0, b<0: passing the second, third and fourth quadrants
K<0, b=0: passing the second and fourth quadrants (passing the origin)
Conclusion: when k<0, the image decreases from left to right, and y decreases with the increase of x.
7. Move the function up to n lattice, the function analytic formula is y=kx+b+n, and move the function down to n lattice, the function analytic formula is y=kx+b-n, move the function left to n lattice, the function analytic formula is y=k (x+n)+b, and move the function right to n lattice, the function analytic formula is y=k (x-n)+b.
8. K is the linear function y=kx+bSlope,K=tan θ (angle θ is the angle between the linear function image and the positive direction of the x-axis, θ≠ 90 °).
When two straight lines are perpendicular in the plane rectangular coordinate system, the product of their function slopes is - 1.[1]
Proof of special location relationship
The proof that the slope of two straight lines is negative reciprocal to each other when they are perpendicular in the plane rectangular coordinate system:
As shown in the figure, these two functions are perpendicular to each other, but if it is directly proved, it is difficult to understand. If the plane rectangular coordinate system is translated, so that the intersection point of these two functions intersects the origin, it will be simpler.Just like this, we can set the expressions of these two functions as;
y=ax，y=bx。
Take a point (z, 0) on the positive half axis of x (convenient for calculation), and make a straight line parallel to the y-axis, as shown in the figure. It can be seen that OC=z, AC=a * z, BC=b * zPythagorean theoremWe can get:
OA=√z^2+（a*z）^2
OB=√z^2+（b*z）^2
OA ^ 2+OB ^ 2=AB ^ 2
Z ^ 2+(az) ^ 2+z ^ 2+(bz) ^ 2=(az bz) ^ 2 (because b is less than 0, it is az bz) is reduced to:
z^2+a^2*z^2+z^2+b^2*z^2=a^2*z^2-2ab*z^2+b^2*z^2
2z^2=-2ab*z^2
ab=-1
That is, k=- 1
So the product of two K values is - 1.
Note: The straight line parallel to the y-axis has no analytic formula of function, and the analytic formula of the straight line parallel to the x-axis is a constant function, so these two lines are excluded from the above properties.[1]
Two linear functions are parallel
Biquadratic function vertical
learning method
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Key points of knowledge
1. Understand the meaning of the function.
2. Understanding of function image in connection with practice.
2. There are mistaken ideas about the image and nature of the linear function;
3. Ignore the value range of the independent variable of the primary function;(Sometimes x ∈ Z, whose image is shown as a set of discontinuous points)
4. In the primary function, consider the independent variable as not equal to zero.
Similarities and differences between and equations
1. A linear function has a similar expression to a linear equation with one variable.
2. A linear function represents the relationship between a pair of (x, y), and it has countless pairs of solutions;The unary linear equation represents the value of the unknown number x, with only one value at most.
3. The abscissa of the intersection of a linear function and the x-axis is correspondingUnary linear equationThe root of.
4. Withlinear equation in two unknownsThe image composed of points whose solution of group ax+by=c is coordinate is the same as the image of linear function y=(- a/b) x+c/b.
5. Binary linear equations aonex+boney=cone，atwox+btwoy=ctwoThe solution of can be regarded as two linear functions y=(- aone/bone）x+cone/doneAnd y=(- atwo/btwo）x+ctwo/dtwoThe intersection point of the image.
And inequality relations
From a functional point of view, the solutionInequalityThe method of is to seek the value range of the independent variable x that makes the value of the primary function y=kx+b greater (or less) than 0;
From the perspective of function image, it is the set formed by determining the abscissa of all points on the upper (or lower) part of the x axis of the line y=kx+b.
The corresponding linear function y=kx+b, its intersection with the x-axis is (- b/k, 0).
When k>0, the solution of inequality kx+b>0 is: x>- b/k, and the solution of inequality kx+b<0 is: x<- b/k;
When the solution of k<0 is: the solution of inequality kx+b>0 is: x<- b/k, and the solution of inequality kx+b<0 is: x>- b/k.[1]
(2) Clarifying the meaning of the question is the key to using piecewise functions to solve problems.
Common formula
1. Find the k value of the function image: (yone-ytwo）/（xone-xtwo）, i.e. k=tan α (α is the included angle between the straight line and the positive direction of the x-axis)
3. Find the midpoint of the line segment parallel to the y-axis: (yone+ytwo）/2
4. Find the length of any line segment: √ [(xone-xtwo）two+（yone-ytwo）two]
5. Finding the coordinates of intersection points of two linear function images: solving the two function equations
Two linear functions yone=konex+bone，ytwo=ktwox+btwo, let yone=ytwo, get konex+bone=ktwox+btwo。Let the solved x=xzeroValue back to yone=konex+bone，ytwo=ktwox+btwoAny one of the two formulas, and get y=yzero, then (xzero，yzero）Is yone=konex+boneAnd ytwo=ktwox+btwoCoordinates of the intersection point.
6. Find the midpoint coordinates of the line segment connected by any two points: ((xone+xtwo）/2，（yone+ytwo）/2）
7. Find the analytic expression of the linear function of the line of any two points: (x-xone）/（xone-xtwo）=（y-yone）/（yone-ytwo）(If the denominator is 0, the numerator is 0)
When the positive and negative of (x, y) is+,+(positive, positive), the point is in the first quadrant
When the positive and negative of (x, y) is -,+(negative, positive), the point is in the second quadrant
When the positive and negative of (x, y) is -, - (negative, negative), the point is in the third quadrant
When the positive and negative of (x, y) is+, - (positive, negative), the point is in the fourth quadrant
8. If two straight lines yone=konex+bone，ytwo=ktwox+btwoIf they are parallel to each other, then kone=ktwo，bone≠btwo
9. For example, two straight lines yone=konex+bone，ytwo=ktwox+btwoeach othervertical, then kone×ktwo=-1
Y=f (x-n)=k (x-n)+b is the straight line to the righttranslationN units
Y=f (x+n)=k (x+n)+b means that the straight line shifts n units to the left
Y=f (x)+n=kx+b+n is to shift n units upwards
Y=f (x) - n=kx+b-n is the downward translation of n units
Formula: Left plus right minus is relative to X, and top plus bottom minus is relative to b.
11. Intersection point of straight line y=kx+b and x-axis: (- b/k, 0), intersection point with y-axis: (0, b)
Application in life
1. When time t is constant, distance s is a function of velocity v.s=vt。
2. If the pumping speed f of the pool is fixed, the water volume g in the pool is a linear function of the pumping time t.The original water volume of the pool is S.g=S-ft。
3. When the original length b of the spring (the length when the weight is not hung) is fixed, the length y of the spring after the weight is hung is a linear function of the weight x of the weight, that is, y=kx+b (k is any positive number).
Common question types
The primary function and its image of common questions areJunior high school algebraIt is also the cornerstone of analytic geometry in high school, and it is also the key content of the high school entrance examination.
Among them, finding the analytic expression of the primary function is a common type of questions.Taking some high school entrance examination questions as examples, this paper introduces several common question types for solving the analytic expression of the primary function.I hope it will be helpful to your study.
1、 Definitional
Example 1. Known functions
Is a linear function, find its analytic expression.
Note: when using the definition to find the analytic expression of the linear function y=kx+b, it is necessary to ensure that k ≠ 0.In this example, m-3 ≠ 0 should be guaranteed.
2、 Point oblique type
Example 2. Given the graph crossing point (2, - 1) of the linear function y=kx-3, find the analytic expression of this function.
Solution: The image of a linear function crosses point (2, - 1), that is, k=1.Therefore, the analytic expression of this linear function is y=x-3.
Variant method: Given the primary function y=kx-3, when x=2, y=- 1, find the analytic expression of this function.
3、 Two-point type
Example 3. If the coordinates of the intersection point of the image of a certain linear function with the x-axis and the y-axis are (- 2,0) and (0,4) respectively, the analytic expression of this function is _____.
Solution: Let the analytic expression of the primary function be y=kx+b
According to the meaning of the question,
Therefore, the analytic expression of this linear function is y=2x+4.
Figure 1
4、 Image type
Example 4. If the image of a certain linear function is known as shown in Figure 1, the analytic expression of the function is __________.
Solution: Let the analytic expression of the primary function be y=kx+b, and it can be seen from the figure that the image crossing points (1, 0), (0, 2) of the primary function have
So k=- 2
b=2
Therefore, the analytic expression of this linear function is y=- 2x+2.
5、 Oblique section
Example 5. Given that the straight line y=kx+b is parallel to the straight line y=- 2x, and the intercept on the y-axis is 2, the analytic expression of the straight line is ___________.
Resolution: two straight lines;.When kone=ktwo，bone≠btwoWhen,
The line y=kx+b is parallel to the line y=- 2x,.The intercept of the straight line y=kx+b on the y-axis is 2, so the analytic expression of the straight line is y=- 2x+2 or y=- 2x-2.
6、 Translational type
Example 6: Translate the straight line y=2x+1 downward for 2 units to obtain the image analytic expression ___________.
Resolution: setFunction analytic expressionIs y=kx+b, the straight line y=kx+b is parallel to the straight line y=2x+1 by translating down 2 units
The intercept of the line y=kx+b on the y-axis is b=1-2=- 1.
7、 Practical application type
Example 7. There is 20 liters of oil in a certain oil tank, and the oil flows out of the pipeline at a constant speed. The flow rate is 0.2 liter/minute. Then the functional relationship between the amount of oil left in the oil tank Q (liters) and the outflow time t (minutes) is ___________.
Solution: According to the meaning of the question, Q=20-0.2t, that is, Q=-0.2t+20
Note: The value range of the independent variable should be written in order to find the function relationship of practical application problems. Don't forget to consider the case that the variable is equal to 0.
8、 Areal type
Example 8. If the area of the triangle formed by the known straight line y=kx-4 and the two coordinate axes is equal to 4, thenLinear analytic expressionIs __________.
Solution: It is easy to find the intersection point between the straight line and the x-axis is, so
, so | k |=2, that is
Therefore, the linear analytic expression is y=2x-4 or y=- 2x-4.
(2) If the y axis is symmetric, the analytical formula of the straight line is y=- kx+b;
(3) If the straight line y=x is symmetric, the analytical formula of the straight line is;x = ky + b
(4) If the line y=- x is symmetric, the analytical formula of the line is;x = -ky + b
(5) If the origin is symmetric, the analytical expression of the line is y=kx-b.
Example 9. If the line l and the line y=2x-1 are symmetrical about the y-axis, the analytical formula of the line l is ____________.
Solution: According to (2), the analytical expression of line l is y=- 2x-1.[2]
10、 Open
Example 10. If the image of a known function crosses points A (1, 4) and B (2, 2), please write two different analytic expressions of the function that meet the above conditions, and briefly explain the solution process.
Solution:
(1) If the function image passing through A and B points is a straight line, it is easy to get y=- 2x+6 from the two-point formula
(2) Since the product of the abscissa and ordinate of A and B points is equal to 4, the function image passing through A and B points can also be hyperbola.
11、 Geometric
Example 11. As shown in Figure 2, in the plane rectangular coordinate system, A and B are two points on the x-axis, and the semicircle with AO and BO as diameters intersects AC and BC at two points E and F respectively. If the coordinates of point C are (0, 3).(1) Finding the quadratic function of an image passing through A, B, CAnalytic expression, and find its symmetry axis;(2) Find the analytic expression of the linear function of the image passing points E and F.
Solution: (1) Byright triangleA (- 3 √ 3,0), B (√ 3,0)Undetermined coefficient methodThe analytic expression of the quadratic function can be obtained as, the axis of symmetry is x=- √ 3 (2), connecting OE and OF, then.Through E and F, make the perpendicular lines of the x and y axes respectively, and the perpendicular feet are M, N, P, G, so that E and F can be easily obtained, and the analytic expression of the primary function can be obtained by the method of undetermined coefficient.
Figure 2
12、 Square type
Example 12. If equation xtwo+The two 3x+1=0 are, respectively, to find the analytic expression of the linear function image passing through points P and Q
Solution: obtained from the relationship between root and coefficient
Points P (11, 3), Q (- 11, 11)
Let the analytic expression of the linear function passing through points P and Q be y=kx+b
