Given that the fixed-point coordinates of the square + K image of the quadratic function y = (x + m) are m (1, - 4), the focal coordinates of the function image and the x-axis are obtained

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• 1. Given that f (x) is a quadratic function, if y = (x) obtains the minimum value of - 4 at x = 2, and the image of y = f (x) passes through the origin, the function y is obtained
• 2. If the quadratic function y = (M + 1) x square + m square - 2m-3 passes through the origin, find M
• 3. As shown in the figure, in the right angle trapezoid oabc, AB / / OC, O are the origin of the coordinate graph, point a is on the positive half axis of Y axis, and point C is on the positive half axis of X axis
• 4. As shown in the figure, in the right angle trapezoid oabc, CB ‖ OA, CB = 8, OC = 8, ∠ OAB = 45 ° (1) calculate the coordinates of points a, B and C; (2) calculate the area of △ ABC
• 5. In the rectangular trapezoid oabc, CB / / OA, CB = 10, OC = = 10, OAB = 45 °. Find the coordinates of a, B, C
• 6. In the rectangular trapezoid oabc, CB ‖ OA, CB = 8, OC = 8, ∠ oaba = 45 ° (1) calculate the coordinates of points a, B and C; (2) calculate the area of △ ABC
• 7. In trapezoidal ABCO, OC ∥ AB establishes a plane rectangular coordinate system with o as the origin. The coordinates of a, B and C are a (8,0), B (8,10) and C respectively In trapezoidal ABCO, OC ‖ The coordinates of a, B and C are a (8,0), B (8,10) and C (0,4) respectively. Point d (4,7) is the midpoint of line BC, and the moving point P starts from point O and moves along the route of broken line OAB at the speed of 1 unit per second. The moving time is T seconds (1) Find the analytical formula of the straight line BC; (2) Let the area of △ OPD be s, the functional relationship between S and t is obtained, and the value range of independent variable t is pointed out; (3) When t is, the area of △ OPD is the same as that of trapezoidal oabc
• 8. As shown in the figure, in the rectangular coordinate system, one side of an acute triangle AOB coincides with the positive half axis of the X axis, and the other side OA intersects the image of the function y = 1 / X at point P. take point P as the center of the circle, take 2PO length as the radius, draw an arc to intersect the image of y = 1 / X at point R, and make parallel lines of the X axis and Y axis through points P and R respectively to obtain the rectangular pqrm, connecting OM
• 9. As shown in the figure, the two right angle sides OA and ob of RT △ ABC are respectively on the positive half axis of X axis and the negative half axis of Y axis, and C is a point on OA As shown in the figure, the two right angle sides OA and ob of RT △ ABC are respectively on the positive half axis of X axis and the negative half axis of Y axis, C is a point on OA, OC = ob, and the parabola y = (X-2) (x-m) - (P-2) (P-M) (where m and P are constants, and M + 2 ≥ 2p > 0) passes through two points a and C (1) It is proved that: (P, 0) is on a parabola; (2) The length of OA and OC is represented by M and P respectively; (3) When m and P satisfy what relation, the area of △ AOB is the largest
• 10. As shown in the figure, it is known that there is a straight line and a curve in the rectangular coordinate system. The straight line and the positive half axis of x-axis and y-axis intersect at point a and point B respectively, and OA = ob = 1. This curve is a branch of the image of function y = 12x in the first quadrant. Point P is any point on this curve, and its coordinates are (a, b). The perpendicular lines PM and PN made from point P to x-axis and y-axis are m, n, Line AB intersects PM and PN at points E and f respectively (1) Calculate the coordinates of points E and f respectively (use the algebraic expression of a to represent the coordinates of point E, and use the algebraic expression of B to represent the coordinates of point F, only need to write the results, not the calculation process); (2) Calculate the area of △ OEF (the result is expressed by the algebraic formula containing a and b); (3) The values of AF and be are calculated respectively (the results are expressed by algebraic expressions containing a and b); (4) Please prove whether △ AOF and △ BOE are necessarily similar; if they are not necessarily similar or not, briefly explain the reasons. Solutions: (1) point E (a, 1-A), point F (1-B, b); (2 points) (2) S △ EOF = s rectangular monp-s △ emo-s △ fno-s △ EPF, =ab-12a(1-a)-12b(1-b)-12(a+b-1)2, =12 (a + B-1); (4 points) （3）BE=a2+(1-1+a)2=2a, AF = (1-1 + b) 2 + B2 = 2B; (6 points) (4) (7 points) It is proved that: OA = ob = 1, ∴∠FAO=∠EBO； ∵ point P (a, b) is a point on the curve y = 12x, 2 ab = 1, that is AF &; be = 1; OA and ob = 1, ∴AFOB=OABE； Why is 2Ab = 1, that is AF ﹥ 8226; be = 1; Why is 2Ab = 1?
• 11. Given the quadratic function y = (x + m) square + K image The vertex coordinates are (3 / 2, - 1 / 4) Finding function relation The intersection coordinates of image and x-axis When x is what value, Y > 0, = 0, < 0?
• 12. When does the image of quadratic function pass through the origin?
• 13. The quadratic function y = x ^ - 4x + 5 is a curve about the symmetric movement of x-axis, Y-axis and origin
• 14. Quadratic function y = xsquare - 4x + 1, find the function analytic formula of the image about the Y axis, the image analytic formula of the X axis, and the image analytic formula of the origin centrosymmetry
• 15. As shown in the figure, in the parallelogram ABCO, the coordinates of C and D are C (- 3 radical 3,0), B (- 3 radical 3. - 3 radical), and (1) find the coordinates of point a As shown in the figure, in the parallelogram ABCD, the coordinates of C and D are C (- 3 radical 3,0), B (- 3 radical 3. - 3 radical), (1) find the coordinates of point a (2) The parallelogram aboc is translated to the right by two root signs and three unit lengths, and the coordinates of four vertices of the parallelogram a'b'o'c 'are obtained (3) Finding the area of a parallelogram aboc There's no mistake. It's the coordinates of A
• 16. As shown in the figure, the coordinates of a, B and C are (3,0), (1, root 3), (0,1) respectively, so as to find the area of the quadrilateral oabc
• 17. As shown in the figure, in the parallelogram ABCO, it is known that the coordinates of two points a and C are a (root 3, root 3) respectively Title: 1. Find the coordinates of point B 2. Translate the parallelogram oabc to the left by 3 unit lengths of the root, and get the coordinates of the four vertices of the parallelogram 3. Find the area of parallelogram oabc (it's not necessary to solve the root sign with the knowledge of grade one in junior high school. Thank you. Hurry up ~ ~~~~~~~~~~~~~~~~~
• 18. As shown in the figure, in the parallelogram oabc, it is known that the coordinates of O and a are o (0,0) a (5 radicals 3,0), and the angle AOC is 60 degrees AB = 2 root sign 3, the distance from C point to X axis is 3 (1) Finding the coordinates of B and C (2) Finding the area of parallelogram oabc (3) Translate the parallelogram to the right by three root signs and three unit lengths, and get the four vertex coordinates of the parallelogram
• 19. If the vertex B of the square oabc and the vertex e of the square ADEF are both on the image of the function y = 1 / X (x > 0), what are the coordinates of the point e
• 20. The vertex A.D.C of square oabc rectangle ADEF on the coordinate axis point F on AB point b.e area of ad = 0.5adef on graph with function y = 1 / X (x > 0)