曲线 CC 的方程为 y=1bx2−2bxy = \frac{1}{\sqrt{bx^2 - 2bx}},其中 b<0b < 0。草绘 CC 的图像,并在适用时用含有 bb 的式子表示出任何渐近线的方程、对称轴方程以及转折点的坐标。

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完整解法

5

1. Determine the domain of the function to identify the vertical asymptotes.

For the curve to be defined, the expression inside the square root in the denominator must be strictly positive:
bx2−2bx>0bx^2 - 2bx > 0
Since it is given that b<0b < 0, dividing the inequality by bb reverses the inequality sign:
x2−2x<0x^2 - 2x < 0
x(x−2)<0x(x - 2) < 0
Hence, the domain of the curve is 0<x<20 < x < 2.
As xx approaches 00 from the right or 22 from the left, the denominator approaches 00, causing yo∞y o \infty.
Thus, the equations of the vertical asymptotes are x=0x = 0 and x=2x = 2.

2. Find the coordinates of the turning point.

Let the expression in the square root be f(x)=bx2−2bxf(x) = bx^2 - 2bx.
Completing the square gives:
f(x)=b(x2−2x)=b[(x−1)2−1]=b(x−1)2−bf(x) = b(x^2 - 2x) = b[(x - 1)^2 - 1] = b(x - 1)^2 - b
Because b<0b < 0, the squared term b(x−1)2≤0b(x - 1)^2 \le 0. Hence, f(x)f(x) attains a maximum value of −b-b when x=1x = 1.
The maximum of f(x)f(x) results in the minimum positive value for y=1f(x)y = \frac{1}{\sqrt{f(x)}}.
Substituting x=1x = 1:
y=1b(1)2−2b(1)=1−by = \frac{1}{\sqrt{b(1)^2 - 2b(1)}} = \frac{1}{\sqrt{-b}}
The coordinates of the turning point are (1,1−b)\left(1, \frac{1}{\sqrt{-b}}\right).

3. Identify the line of symmetry.

The quadratic expression bx2−2bxbx^2 - 2bx is symmetric about its vertex at x=1x = 1, and the domain 0<x<20 < x < 2 is also symmetric about x=1x = 1.
Consequently, the curve CC is symmetric about the vertical line passing through its turning point.
The equation of the line of symmetry is x=1x = 1.

4. Describe the sketch of the curve $C$.

To sketch the curve CC:

  1. Draw and label vertical dashed lines at x=0x = 0 (the yy-axis) and x=2x = 2 to represent the asymptotes.
  2. Plot and label the minimum turning point at (1,1−b)\left(1, \frac{1}{\sqrt{-b}}\right).
  3. Draw a U-shaped curve that passes through the turning point, is visually symmetric about the line x=1x = 1, and curves upwards to approach the asymptotes x=0x = 0 and x=2x = 2 smoothly as y→∞y \to \infty.
答案

Asymptotes: x=0x = 0 and x=2x = 2
Line of symmetry: x=1x = 1
Turning point: (1,1−b)\left(1, \frac{1}{\sqrt{-b}}\right)

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