1. Because the three points of ABC are all points on a parabola, substitute the coordinates into the analytical formula and the ternary linear equation.
0=-a-b+c a=- 1
0=9a+3b+c, the simultaneous solution is: b=2, and the analytical formula is: y=-x? +2x+3。
3=c c=3
D is the vertex. According to the maximum formula of quadratic function, the abscissa of d can be calculated as -b/2a= 1. Substituting the abscissa into the analytical formula, the corresponding ordinate can be obtained. -( 1)? +2* 1+3=4, so d (1, 4)
The area of 2.2. PMAC =AOC+PMOC, AOC is easy to find = 1*3/2=3/2.
Because the point P is on the straight line of DB, P satisfies the equation of straight line DB. Given the two coordinates of DB, we can find the primary resolution function of DB by substitution: y=-2x+6. The coordinates of point P always satisfy (x, -2x+6).
So the area of trapezoidal PMOC is S=3/2+(-2x+6+3)*x/2=-x? +9/2x+3/2。
Therefore, according to the maximum formula, the value and maximum area of X, that is, the values of abscissa and ordinate at this time, that is, the coordinates of P, P(9/2, 105/ 16), and the maximum value is 105/ 16.
3. if PQAC is a parallelogram, then PC must be parallel to the x axis, so p can be any point on the straight line y=3. Therefore, since the problem does not need to be solved, you can go to any point, such as P (1 3), p (2,3 3) P (10086,3) p. ....
If it is necessary for PQC to be trapezoidal, it should be calculated. You should do as I say: the direction extending from C to P intersects the X axis at a certain point N. At this time, ACN is an isosceles triangle (because the two bottom angles of the isosceles trapezoid PQC are equal), so the length of An = the length of CN, and the coordinate of N is (m, 0). Because CON is a right triangle, there is an equation m? +3? =(m+ 1)？ , the solution is m=4, so n (4,0). According to two points N and C, we can get a resolution function y=-(3/4)x+3, so P can be any point on the line segment CN. Pick any point. Remember that you can't take points C and N, otherwise the figure is not trapezoidal. You can take P (1, 9/).