Problems

1. If a quadratic function \(y =a x^2 + b \;\; (ab\not= 0)\) has the same y-value when \(x=x_1\) and \(x=x_2\), \((x_1 \not= x_2)\). What is the y-value when \(x=x_1 + x_2\)?

2. If a quadratic function \(y =a x^2 + bx \;\; (a\not= 0)\) has the same y-value when \(x=x_1\) and \(x=x_2\), \((x_1 \not= x_2)\). What is the y-value when \(x=x_1 + x_2\)?

3. If a quadratic curve \(y=(m+1)x^2+m^2 -2m -3\) passes \((x=0, y=0)\). What is \(m\)?

4. If \(y\) is proportional to \((x+5)^2\), and \(y=200\) when \(x=5\). Write down the function of y about x.

5. Let \(y = y_1 + y_2\). Assume that \(y_1\) is proportional to \((x+1)^2\), and \(y_2\) is proportional to \(x\), \(y = -1\) when \(x=-2\); \(y=14\) when \(x=1\). Write down the function of y about x.

6. Figure 1 shows the cross section of a tunnel composed of a half circle and a rectangle. The total area is \(S\). Write down the equation of the outer perimeter \(y=DA + AB + BC + \text{half circle}\) as a function of the length of \(AB\).

7. A supermarket sells a product whose manufactory cost is $16 per piece. If the supermarket sells it at the price of $20 per piece, 360 pieces can be sold in a month. If the supermarket sells it the the price of $25 per piece, 210 pieces can be sold in a month. Assume that the sold number of pieces \(y\) in a month is a linear function of the sale price \(x\) per piece,

\((i)\) Write down the linear equation of y about x;

\((ii)\) Write down the equation of profit \(p\) the supermarket makes per month about \(x\).

Solutions

1. \(b\)

2. 0

3. 3

4. \(y = 2 (x+5)^2\)

5. \(y = 3 (x+5)^2 + 2x\)

6. \(y=\frac{2}{x}(S-\frac{\pi}{8}x^2) + x + \frac{\pi}{2}x\)

7. \(y = -30x + 960\); \(p = (-30x + 960)(x-16)\);