1.(08 Putian, Fujian) 26. (14) As shown in the figure, the parabola passes through three points: A (-3,0), B (0 0,4) and C (4 4,0).
(1) Find the analytical formula of parabola.
(2) It is known that AD = AB(D is on the line segment AC), and a moving point P moves from point A along the line segment AC at a speed of/kloc-0 per second; At the same time, another moving point Q moves from point B along BC line at a certain speed. After moving for t seconds, divide the line PQ vertically by BD to find the value of t;
(3) In the case of (2), is there a point m on the parabola axis of symmetry that minimizes the value of MQ+MC? If it exists, request the coordinates of point m; If it does not exist, please explain why.
(Note: Parabolic symmetry axis is)
(08 Fujian Putian 26 questions analysis) 26( 1) Solution 1: Let the analytical formula of parabola be y = a (x +3 )(x-4).
Because B (0 0,4) is on a parabola, 4 = a (0+3) (0-4) is solved to get a=-1/3.
So the parabolic analytical formula is
Solution 2: Let the analytical formula of parabola be,
According to the meaning of the question: c=4 and solve it.
So the analytical formula of parabola is
(2) connect DQ, in rt delta AOB,
So AD=AB= 5, AC=AD+CD=3+4 = 7, CD = AC-AD = 7-5 = 2.
Because BD vertically divides PQ, PD=QD, PQ⊥BD, so ∠PDB=∠QDB.
Because AD=AB, ∠ABD=∠ADB, ∠ABD=∠QDB, DQ ∠ AB.
So ∠CQD=∠CBA. ∠CDQ =∠ cab, so △CDQ∽△ cab.
that is
So AP = ad-DP = ad-dq = 5 -=–=,
So the value of t is
(3) There is a point m on the symmetry axis that minimizes the value of MQ+MC.
Reason: Because the symmetry axis of parabola is
Therefore, A (-3,0) and C (4 4,0) are symmetrical about a straight line.
If the intersection line connecting AQ is at point M, the value of MQ+MC is the smallest.
Q is QE⊥x axis, at point E, so ∠QED=∠BOA=900.
DQ‖AB，∠ BAO=∠QDE，△DQE ∽△ABO
that is
So QE=, Germany =, so OE = OD+ Germany =2+ =, so Q (,).
Let the analytical formula of straight line AQ be
Then the next step is
Therefore, the analytical formula of straight line AQ is simultaneous.
Therefore, m
Then: there is a point m on the axis of symmetry that minimizes the value of MQ+MC.
2.(08 Gansu Baiyin and other 9 cities) 28. (12 minute) As shown in Figure 20, in the plane rectangular coordinate system, the quadrilateral OABC is a right angle, and the coordinate of point B is (4,3). The straight line M parallel to the diagonal AC starts from the origin O and moves at the speed of 1 unit length per second along the positive direction of the X axis. The two sides of the straight line M and the right angle OABC are set respectively.
(1) The coordinates of point A are _ _ _ _ _ _ _, and the coordinates of point C are _ _ _ _ _ _ _ _ _;
(2) When t= seconds or seconds, MN = AC；;
(3) Let the area of △OMN be S, and find the functional relationship between S and T;
(4) Does the function S obtained in (3) have a maximum value? If yes, find the maximum value; If not, explain why.
(08 analysis of 28 questions in 9 cities such as Baiyin, Gansu) 28. The full score of this small question is 12.
Solution: (1) (4,0), (0,3); 2 points
(2) 2,6; 4 points
(3) When 0 < t ≤ 4, OM = t. 。
From △OMN∽△OAC
∴ In =, s = .6 points
When 4 < t < 8,
As shown in the figure, od = t, ∴ ad = t-4.
Method 1:
From △DAM∽△AOC, we can get AM=, ∴ BM = 6-.7 points.
From △BMN∽△BAC, BN= =8-t, you can get △ CN = t-4.8.
S= rectangular OABC area -Rt area △OAM-Rt area △MBN-Rt area △ NCO.
= 12- - (8-t)(6- )-
=. 10 point
Method 2:
It is easy to know that the quadrilateral ADNC is a parallelogram, ∴ CN=AD=t-4, bn = 8-t.7 points.
From △BMN∽△BAC, BM= =6-, you can get ∴ AM = .8 points.
The following is the same as method 1.
(4) There is a maximum value.
Method 1:
When 0 < t ≤ 4,
The opening of parabola S= is upward, and on the right side of symmetry axis t=0, s increases with the increase of t,
When t=4, the maximum value of s = 6; 1 1 min
When 4 < t < 8,
∵ The opening of parabola S= is downward, and its vertex is (4,6), ∴ s < 6.
To sum up, when t=4, the maximum value of S is 6. 12 points.
Method 2:
∫S =
∴ When 0 < t < 8, draw the image of the functional relationship between s and t, as shown in the figure. 1 1.
Obviously, when t=4, the maximum value of S is 6. 12 points.
Note: Only when the answer to question (3) is correct and the answer to question (4) is only "maximum" and there are no other steps, the score can be1; Otherwise, no points will be given.
3.(08 Guangzhou, Guangdong) 25, (2008 Guangzhou) (14 points) as shown in figure 1 1, in trapezoidal ABCD, AD‖BC, AB=AD=DC=2cm, BC=4cm, at the isosceles △ pq. If the isosceles △PQR moves at a uniform speed 1cm/ s along the direction indicated by the arrow of the straight line L, the area of the overlapping part between the trapezoidal ABCD and the isosceles △PQR in t seconds is recorded as s square centimeters.
(1) When t=4, find the value of s.
(2) If the functional relationship between S and T is found, the maximum value of S is found.
(08 Analysis of 25 Questions in Guangzhou, Guangdong) 25. (1) When t = 4, q and b coincide, and p and d coincide.
Overlap =
4.(08 Shenzhen, Guangdong) 22. As shown in fig. 9, in the plane rectangular coordinate system, the vertex of the image of the quadratic function is point D, which intersects with the Y axis at point C, and intersects with the X axis at points A and B. Point A is on the left side of the origin, and the coordinate of point B is (3,0).
OB=OC，tan∠ACO=。
(1) Find the expression of this quadratic function.
(2) The straight line passing through points C and D intersects the X axis at point E. Is there such a point F on this parabola, and the quadrilateral with points A, C, E and F as its vertices is a parallelogram? If it exists, request the coordinates of point f; If it does not exist, please explain why.
(3) If the straight line parallel to the X axis intersects the parabola at two points, M and N, and the circle with the diameter of MN is tangent to the X axis, find the length of the radius of the circle.
(4) As shown in figure 10, if point G(2, y) is a point on the parabola and point P is a moving point on the parabola below the straight line AG, when point P moves to what position, what is the maximum area of △APG? Find the coordinates of point P and the maximum area of △APG at this time.
(08 Analysis of 22 Questions in Shenzhen, Guangdong) 22. (1) Method 1: From the known: C(0, -3), a (- 1, 0)... 1.
Substitute the coordinates of point A, point B and point C to get 2 points.
Solution: 3 points.
So the expression of this quadratic function is
Method 2: From the known: C(0, -3), a (- 1, 0) ............................1min.
Let this expression be
Substitute the coordinates of point C.
So the expression of this quadratic function is
(Note: The final result of the expression will not be deducted in any of the three forms)
(2) Method 1: exists, and the coordinates of point F are (2, -3)4 points.
Because: D( 1, -4) is easy to get, the analytical formula of linear CD is:
∴ The coordinate of point E is (-3,0) ........................... 4 points.
From the coordinates of a, c, e and f, AE = cf = 2, AE ‖ cf.
A quadrilateral with vertices a, c, e and f is a parallelogram.
∴ There is point F, and the coordinates are (2,-3). .............................................................................. is 5 points.
Method 2: D( 1, -4) is easy to obtain, so the analytical formula of linear CD is:
∴ The coordinate of point E is (-3,0) ............................ 4 points.
A quadrilateral with vertices A, C, E and F is a parallelogram.
∴ The coordinates of point F are (2, -3) or (-2, -3) or (-4, 3).
Only (2, -3) satisfies the parabolic expression test.
∴ There is point F, and the coordinates are (2, -3). ...................................................................................................................................................
(3) As shown in the figure, ① When the straight line MN is above the X axis, let the radius of the circle be r (r >; 0), then N(R+ 1, r),
Substituting into the expression of parabola, the solution is
② When the straight line MN is below the X axis, let the radius of the circle be r (r >; 0),
Then N(r+ 1, -r),
Substitute the parabola expression and you get ... 7 points.
The radius of a circle is 7 points or ........................
(4) When the Y axis intersects with AG at point Q, the parallel line passing through point P,
G(2, -3) is easily obtained, and the straight line AG is ........................................................................................................................................................
Let P(x,), then Q(x, -x- 1), pq.
9 points ... 9 points.
When △APG has the largest area.
At this point, the coordinates of point p are, ...................................................................................................................................................................
5.(08 Guiyang, Guizhou) 25. (The full mark of this question is 12) (There is no answer to this question)
The hotel housekeeping department has 60 rooms for tourists to live in. When the price of each room is 200 yuan per day, the room will be full. Each room will be increased by 65,438+00 yuan per day, and one room will be given as a gift. There are rooms for tourists, and the hotel needs to pay various fees for each room in 20 yuan every day.
Let the daily price of each room increase by RMB. Q:
(1) The daily occupancy of the room (room) is a function of (yuan). (3 points)
(2) The daily room rate of the hotel (yuan) is a function of (yuan). (3 points)
(3) the functional relationship between the daily profit (yuan) and (yuan) of the housekeeping department of this hotel; When the price of each room is several yuan per day, there is a maximum. What is the maximum value? (6 points)
6.(08 Enshi, Hubei) VI. (The full score of this big question is 12)
24. As shown in figure 1 1, put two isosceles right triangles ABC and AFG together on the same plane, where A is the common vertex, ∠ BAC = ∠ AGF = 90, and their hypotenuse length is 2. What if? ABC is fixed. AFG rotates around point A, and the intersections of AF, AG and BC are D and E respectively (point D and point B do not coincide, and point E does not coincide with point C). Let BE=m and CD = n.
(1) Please find two pairs of similar but unequal triangles in the diagram and choose one pair to prove it.
(2) Find the functional relationship between m and n, and write the range of independent variable n directly.
(3) same? The straight line on the hypotenuse BC of ABC is the X axis, and the straight line on the height of BC is the Y axis, thus establishing a plane rectangular coordinate system (as shown in figure 12). Find a point D on the side of BC so that BD=CE, find out the coordinates of point D, and verify BD +CE =DE through calculation.
(4) Whether the equivalence relation BD +CE =DE in (3) always holds in the process of rotation, if so, please prove it, if not, please explain the reasons.
(08 Hubei Enshi 24 questions analysis) VI. (The full score of this big question is 12)
24. Solution: (1)? Abe? DAE？ Abe? DCA 1 min
∠∠BAE =∠BAD+45，∠CDA=∠BAD+45
∴∠BAE=∠CDA
∠ b =∠ c = 45。
∴? Abe? DCA 3 points
(2)∵? Abe? Defense Communications Agency
∴
According to the meaning of the question, CA=BA=
∴
M = 5 points
The range of independent variable n is 1
(3) BD=CE, BE=CD, that is, m = n.
∫m =
∴m=n=
∫OB = OC = BC = 1
∴OE=OD= - 1
∴ d (1-0) 7 points
∴bd=ob-od= 1-(- 1)= 2-= ce，DE=BC-2BD=2-2(2- )=2 -2
∫BD+CE = 2bd = 2(2-)= 12-8，DE =(2 -2) = 12-8
∴ BD+CE = DE 8 points
49 points.
Proof: as shown in the picture, will it? ACE rotates 90 clockwise around point a to? The position of ABH, CE=HB, AE=AH,
∠ ABH =∠ C = 45, and the rotation angle ∠ EAH = 90.
Connect HD, at? EAD and? Hade clock
AE = AH，∠HAD=∠EAH-∠FAG=45 =∠EAD，AD=AD。
∴? EAD？ have
∴DH=DE
And < hbd = < abh+< Abd = 90.
∴bd+ hemoglobin =DH
That is BD+CE = DE 12.
7.(08 Jingmen, Hubei) 28. (The full score of this small question is 12)
It is known that the vertex A of the parabola y=ax2+bx+c is on the X axis, and the intersection with the Y axis is b (0, 1), and b =-4ac.
(1) Find the analytical formula of parabola;
(2) Is there a point C on the parabola that makes the circle with diameter BC pass through the vertex A of the parabola? If there is no explanation; If it exists, find the coordinates of point C and the coordinates of the center point P of the circle at this time;
(3) According to the conclusion of (2), what is the relationship between the abscissa and ordinate of B, P and C?