Problems

1. The size length of a regular hexagon ABCDEF shown in Figure 1 is \(2 cm\). The center of the hexagon is at the origin of the coordinates. (1) What are the coordinates of A, D and E? (2) Write down the quadratic function passing the three points A, D and E.

2. A gate has a parabola shape shown in Figure 2. \(AB=4\) m, The distance of the vertex C from AB is 4.4 m. The width and height of a truck is 2.4 m, 2.8 m, respectively. Can the truck pass through the gate?

3. In the quadratic function \(y=x^2 + mx +n\), \(m-n=0\). Which point must the parabola pass?

4. The quadratic function \(y=ax^2 + bx + c\) passes \((c, 2)\). \(a|a|+b|b|=0\). The inequality \(ax^2 + bx + c -2 > 0\) has no solutions. What are a, b and c?

5. What are the minimum and maximum value of a function \(y=|-2x^2 + 8x -6|\) in the region \(3 \le x \le \frac{17}{5}\)?

Solutions

1. \(a = \frac{2}{3}\sqrt{3}\), \(b = \sqrt{3}\), \(c = -\frac{2}{3}\sqrt{3}\)

2. (1.2, 2.816) yes

3. (-1, 1)

4. a = -6, b = 6, c=1/2

5. 2, 0; \(-2(x-2)^2+2\)