(1) When k= 1, the analytical expressions of parabola and straight line are simultaneous, and the coordinates of point A and point B are obtained by solving the equations;

(2) As shown in Figure 2, as an auxiliary line, find the expression of △ABP area, and then use the properties of quadratic function to find the maximum value and the coordinates of point P;

(3) The meaning of "there is a unique point Q, so that ∠ OQC = 90" means that the circle with a diameter of OC and the straight line AB are tangent to the point Q. According to the theorem of circumferential angle, at this time ∠ OQC = 90 and the point Q is unique. On this basis, the similar triangles is constructed, and the value of k is obtained by using the equation of proportional formula.

Solution:

This topic is the finale of quadratic function, which comprehensively examines the important knowledge points of quadratic function and linear function, such as images and properties, solving equations, Pythagorean theorem, the positional relationship between straight lines and circles, similarity and so on. And it is difficult. In question (3), the key to solving the problem is to understand the meaning of "there is a unique point Q that makes ∠ OQC = 90".

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