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Problems

  1. The minimum value of a quadratic function \(y = a x^2 + b x + c\) is -10. The minimum value is achieved at \(x=-1\). The parabola passes a point \((2, 8)\). What are a, b, and c?

  2. A quadratic function \(y = a x^2 + b x + c\), \((a\not= 0)\) is shown in Figure 1. Choose all the true statements: (1) \(a+b+c<0\); (2) \(a-b+c<0\); (3) \(b+2a<0\); (4) \(abc>0\).


Figure 1: Figure 1

  1. In Figure 2, \(AB=2\), \(\angle BAC=30^{\circ}\), \(\angle ABC=135^{\circ}\). What is \(AC\)?


Figure 2: Figure 2

  1. In Figure 3, \(BC\perp AC\), \(DA\perp AC\), \(AD=10\), \(\angle BAC=60^{\circ}\), \(\angle ADB=120^{\circ}\). What is \(BC\)?


Figure 3: Figure 3

  1. In Figure 4, \(AC=2\sqrt{3}\), \(\tan B = \frac{\sqrt{3}}{2}\), \(\angle A=30^{\circ}\), What is \(AB\)?


Figure 4: Figure 4

Hints



Solutions

  1. \(a=2, b=4, c=-8\)

  2. ( 2 ) and ( 3 )

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

  4. 15

  5. 5

Set2