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Problems

  1. Assume a quadratic curve \(y = x^2 - 2ax + b\) passes (1, 1) and has only one cross point with x-axis. what is the vertex of the parabola?

  2. In Figure 1, the side length of a square is 4. \(AF=2\) and \(FB=1\). Let the coordinates of D be (0,0), x-axis align with DC, and y-axis align with DE. Find the coordinates of a point P in \(AB\) so that the rectangle \(PNDM\) has maximum area.


Figure 1: Figure 1

  1. Given a quadratic function \(y(x) = 2mx^2 + (1-m)x - (m + 1)\). Choose the right statements.

A. (1)(2)(3)(4)

B. (1)(2)(4)

C. (1)(3)(4)

D. (2)(4)

\((1)\) If \(m=-3\), the vertex of the x-y curve is \((1/3, 8/3)\).

\((2)\) If \(m>0\), the x-y curve crosses x-axis at A and B, and \(|B-A|>3/2\).

\((3)\) If \(m<0\), y decreases when x increases in the range \(x>1/4\).

\((4)\) For all \(m\not=0\), all quadratic curves pass one common point.

  1. In Figure 2, a football goalkeeper kicks off a ball at A (0, 1). A player stands at B(6, 0). The vertex of the first parabola is M(6, 4). The ball falls at C and bounces off the ground to D following the same parabolic shape as the first one. The maximum height of the second parabola is 2.

\((1)\) Write two quadratic functions to represent the two parabolas.

\((2)\) What is the distance between D and B?


Figure 2: Figure 2

Solutions

  1. (0, 0), (2, 0)

  2. 12

  3. B

  4. \(y = -\frac{1}{12} (x-6)^2 + 4\); \(y = -\frac{1}{12} (x-6-4\sqrt{3}-2\sqrt{6})^2 + 2\)

Set5