(1) Find the analytical expressions of inverse proportional function and linear function;
(2) Find the coordinates of point B;
(3) Find the point E on the X axis so that △ACE is a right triangle (directly write the coordinates of the point E).
Test center: inverse proportional function synthesis problem.
Special topic: comprehensive questions.
Analysis: (1) Pass point A as AD? The x axis is in D. according to the coordinates of a and c, AD=6, CD=n+2, tan? ACO=2, the value of n can be obtained, and the inverse proportional function and the primary resolution function can be obtained by substituting the coordinates of points into the analytical formula;
(2) Find another intersection point between the inverse proportional function and the linear function;
(3) There are two situations: ①AE? X axis, ②EA? AC, just write the coordinates of e separately.
Solution: Solution: (1) Take point A as an AD transmission? The x axis is on the d axis,
The coordinate of ∵C is (∵ 2,0), and the coordinate of A is (n,6).
? AD=6，CD=n+2，
Tan? ACO=2，
? = =2,
Solution: n= 1,
So a (1, 6),
? m= 1？ 6=6,
? The expression of the inverse proportional function is y=,
And ∵ points a and c are on a straight line y=kx+b,
? ,
Solution:
? The expression of linear function is: y = 2x+4;
(2) From: =2x+4,
Solution: x= 1 or x =-3,
∫A( 1，6)，
? b(﹣3,﹣2)；
(3) There are two situations: ① When AE? On the x axis,
That is to say, point e coincides with point d,
At this time, e 1 (1, 0);
2 When is EA? Communication,
At this time △ADE∽△CDA,
Then =,
DE= = 12，
The coordinate of ∵D is (1, 0),
? E2( 13，0)。
Comments: This topic examines the synthesis problem of inverse proportional function, involving the solution of the coordinates of points and the knowledge of solving the resolution function by undetermined coefficient method, mainly examining the students' computing ability and the ability to observe graphics.
(20 13? Jiaxing) as shown in the figure, the linear function y=kx+ 1(k? 0) and inverse proportional function y=(m? 0) has a common point A (1, 2). Straight line l? The x axis is at point n (3,0), and it intersects with the images of linear function and inverse proportional function at point b and point c respectively.
(1) Find the analytical expressions of linear function and inverse proportional function;
(2) Find the area of △ABC?
Test site: the intersection of inverse proportional function and linear function.
Special topic: calculation problems.
Analysis: (1) Substitute the A coordinate into the principal resolution function to find the value of k, determine the principal resolution function, and substitute the A coordinate into the inverse proportional resolution function to find the value of m, thus determining the inverse proportional analytical formula;
(2) Let the intersection of the linear function and the X axis be point D, let A be AE perpendicular to the X axis, and the triangular ABC area = triangular BDN area-trapezoidal AECN area under the three-port arrangement, and then get.
Solution: Solution: (1) Substitute A (1, 2) into the resolution function once to get: k+ 1=2, that is, k= 1.
? The primary resolution function is y = x+1;
Substitute a (1, 2) into the inverse proportional analytical formula: m=2,
? The analytical formula of inverse ratio is y =；;
(2) Let the linear function intersect with the X axis at point D, make y=0, and find X =- 1, that is, OD= 1.
? A( 1，2)，
? AE=2，OE= 1，
∫N(3，0)，
? The abscissa of b is 3,
Substitute x=3 into the linear function: y=4, and x=3 into the inverse proportional analytical formula: y=,
? B (3 3,4), that is, ON=3, BN=4, C (3 3,4), that is, CN=,
So s △ ABC = s △ BDN-s △ ade-s trapezoidal AECN=? 4? 4﹣? 2? 2﹣? (+2)? 2= .
Comments: This question examines the intersection of a function and an inverse proportional function. The knowledge involved includes: the nature of coordinates and figures, the undetermined coefficient method for solving the resolution function, the triangle and trapezoid area method, and mastering the undetermined coefficient method skillfully is the key to solve this problem.
(20 13? Ziyang) as shown in the figure, it is known that the straight line L intersects the X axis and the Y axis at two points A and B respectively, and the hyperbola y= (a? 0, x>0) intersect at d and e respectively.
(1) If the coordinate of point D is (4, 1), the coordinate of point E is (1, 4):
① The analytical expressions of straight line L and hyperbola are obtained respectively;
(2) If the straight line L is shifted downward by m (m > 0) unit, when m is what value, the straight line L and hyperbola have only one intersection?
(2) Assuming that the coordinates of point A are (a, 0), point B is (0, b) and point D is the bisector of line segment AB, please write the value of b directly.
Test center: inverse proportional function synthesis problem.
Analysis: (1)① Analytic expressions of straight line L and hyperbola can be obtained by using undetermined coefficient method;
② The straight line L is translated downward by m (m > 0) unit gets y=﹣x=5﹣m, according to the meaning of the question, when the equation group has only one solution, it is transformed into an equation about x, and x2+(5﹣m)x+4=0, then △ = (m ﹣ 5) 2 ﹣. 4=0, the solution is m 1= 1, m2=9. When m=9, the common point is not in the first quadrant, so m =1;
(2) do DF? On the X-axis, we can get △ADF∽△ABO from df∨ob, get AF= and DF= according to the similarity ratio, and the coordinate of point D is (a,). Then we can get the value of b by substituting the coordinate of point D into the inverse proportional resolution function.
Solution: (1)① substitute D (4, 1) into y= to get a= 1? 4=4,
So the inverse proportional decomposition function is y = (x >; 0);
Let the analytical formula of straight line l be y=kx+t,
Substitute d (4, 1) and e (1, 4),
Solve.
Therefore, the analytical formula of the straight line L is y =-x+5;