Problems

1. Three quadratic functions \(y=ax^2 + bx + c\), \(y=bx^2 +cx + a\), and \(y=cx^2+ax+b\) are drawn in Figure 1. Do you think they were correctly drawn? Why?

2. What is the minimum value of function \(y=\sqrt{x^2 + 2x + 2} + \sqrt{x^2 - 4x + 8}\)?

3. Assume that the vertex of a parabola \(y=ax^2 + bx + c\) is \((d, 0)\) and \(d<0\). The parabola passes the first quadrant. Which of the following statements is true?

A. a, b, and c are positive;

B. a, b, and c are negative;

C. \(a > 0\), \(b > 0\), and \(c < 0\);

D. \(a > 0\), \(b < 0\), and \(c > 0\).

4. Assume the vertex of a parabola \(y=ax^2 + bx + c (a\not= 0)\) is in the first quadrant. It passes the points (0, 1) and (-1, 0). Which of the following statements is true for \(S=a+b+c\)?

A. \(0 < S < 1\);

B. \(0 < S < 2\);

C. \(1 < S < 2\);

D. \(-1 < S < 1\).

5. Let \(x\) be a positive real number. What is the minimum value of function \(y=x^2 -x + \frac{1}{x}\)?

Solutions

1. No

2. 3\(\sqrt{2}\)

3. A

4. B

5. 1