The first volume of eighth grade mathematics linear function knowledge 1
Knowledge point 1 concepts of linear function and proportional function
If the relationship between two variables X and Y can be expressed as Y = KX+B (where K and B are constants and k≠0), then Y is a linear function of X (where X is an independent variable), especially when b=0, Y is a proportional function of X. 。
Knowledge point 2 function diagram
Because two points determine a straight line, two special points are generally selected: the intersection of the straight line and the Y axis, and the intersection of the straight line and the X axis. These two special points need not be selected.
When you draw an image with the proportional function y=kx, you only need to trace points (0,0) and (1, k).
Knowledge point 3 Properties of linear function y=kx+b(k, b is constant, k≠0)
The sign of (1)k determines the inclination direction of the straight line;
①k & gt； The values of 0 and y increase with the increase of x value;
② When k￠o, the value of y decreases with the increase of the value of x..
(2) The size of | k | determines the inclination of the straight line, that is, the greater the | k |.
(1) when b >; 0, the straight line intersects the Y axis on the positive semi-axis;
② when b
③ When b=0, the straight line passes through the origin, which is a proportional function.
(4) Because the symbols of K and B are different, the quadrants that the straight line passes through are also different;
(1) As shown in the figure, when k>0, b>0, the straight line passes through the first, second and third quadrants (the straight line does not pass through the fourth quadrant);
② As shown in the figure, when k>0, B
③ As shown in the figure, when k￠O, b>0, the straight line passes through the first, second and fourth quadrants (the straight line does not pass through the third quadrant);
④ As shown in the figure, when k￠O and b￠O, the straight line passes through the second, third and fourth quadrants (the straight line does not pass through the first quadrant).
(5) Because |k| determines the size of the acute angle where the straight line intersects with the X axis, the same k means that the two acute angles are equal in size and are the same angle, so they are parallel. In addition, they can also be analyzed from the perspective of translation. For example, the straight line y=x+ 1 can be regarded as a proportional function y=x that translates upward by one unit.
The first volume of eighth grade mathematics linear function knowledge II
Knowledge Point 4 Properties of Proportional Function y=kx(k≠0)
(1) The image of the proportional function y=kx must pass through the origin;
(2) when k >; 0, the image passes through the first and third quadrants, and y increases with the increase of x;
(3) When k < 0, the image passes through the second and fourth quadrants, and y decreases with the increase of x 。
The relationship between knowledge point 5 P(x0, y0) and the image with straight line y = kx+b.
(1) If point P(x0, y0) is on the image of straight line y=kx+b, then the values of x0 and y0 must satisfy the analytical formula y = kx+b;
(2) If x0 and y0 are a pair of corresponding values satisfying the resolution function, then the point p (1, 2) with x0 and y0 as coordinates must be on the image of the function.
For example, if the point P (1, 2) satisfies the straight line y=x+ 1, that is, when x= 1, y=2, then the point P (1, 2) is on the image of the straight line y = x+ 1 The point p ′ (2, 1) does not satisfy the analytical formula y=x+ 1, because when x=2, y=3, so the point p ′ (2, 1) is not on the image of the straight line y = x+L. 。
Knowledge point 6 Conditions for determining expressions of proportional function and linear function
(1) Because there is only one undetermined coefficient k in the proportional function y=kx(k≠0), the value of k can be obtained by only one condition (such as a pair of values of x and y or a point).
(2) Because there are two undetermined coefficients k and b in the linear function y=kx+b(k≠0), two independent conditions are needed to determine the two equations about k and b, and the values of k and b can be obtained. These two conditions are usually two points or two pairs of x and y values.
Knowledge point 7 undetermined coefficient method
Set the relationship between the functions to be solved (including unknown constant coefficients), and then list the equations (or equations) according to the conditions to find the unknown coefficients, so as to get the results. The method is called undetermined coefficient method, in which the unknown coefficients are also called undetermined coefficients. For example, in the function y=kx+b, k and b are undetermined coefficients.
Mathematics in Grade 8 Book 1 Knowledge of Linear Functions 3
Knowledge point 8 General steps to determine linear function expression by undetermined coefficient method
(1) Let the function expression be y = kx+b;
(2) Substituting the coordinates of known points into the function expression to solve the equation (group);
(3) Find the values of k and b to get the function expression.
Summary of thought and method (1) function method. (2) Number-shape combination method.
The influence of (1) constants k and b on the position of straight line y=kx+b(k≠0).
(1) when b >; 0, the straight line intersects the positive semi-axis of the Y axis;
When b=0, the straight line passes through the origin;
When b￠0, the straight line intersects the negative semi-axis of the Y axis.
(2) When k and b are different signs, the straight line intersects the positive semi-axis of the X axis;
When b=0, the straight line passes through the origin;
When K and B have the same sign, the straight line intersects the negative semi-axis of the X axis.
③ When k >; O, b>o, the image passes through the first, second and third quadrants;
When k>0, b=0, the image passes through the first and third quadrants;
When b>o, b
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