Mathematics sprint review materials for senior high school entrance examination: quadratic function finale problem
Area grade
1. As shown in the figure, it is known that the parabola passes through points A (- 1, 0), B (3 3,0) and C (0 0,3).
(1) Find the analytical formula of parabola.
(2) The point m is the point on the line segment BC (not coincident with B and C). If the crossing m is a parabola of MN∨y axis, and if the abscissa of point M is M, please use the algebraic expression of M to express the length of MN.
(3) Under the condition of (2), connecting NB and NC, is there M to maximize the area of △BNC? If it exists, find the value of m; If it does not exist, explain why.
Answer:
Solution: (1) Let the analytical formula of parabola be: y = a (x+ 1) (x-3), then:
a(0+ 1)(0﹣3)=3,a=﹣ 1；
Parabolic analytical formula: y = ﹣ (x+ 1) (x ﹣ 3) = ﹣ x2+2x+3.
(2) Let the analytical formula of straight line BC be: y=kx+b, then there is:
, solutions;
So the analytical formula of straight line BC: y =-x+3.
Given that the abscissa of point M is m, Mn∑y, then M(m,-m+3), N(m,-m2+2m+3);
Therefore, Mn =-m2+2m+3-(-m+3) =-m2+3m (0 < m < 3).
(3) As shown in the figure;
∫S△BNC = S△MNC+S△MNB = MN(OD+DB)= MN？ OB，
∴S△BNC=(﹣m2+3m)？ 3=﹣(m﹣)2+(0