曲线 的方程为 ,其中 。草绘 的图像,并在适用时用含有 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:
Since it is given that , dividing the inequality by reverses the inequality sign:
Hence, the domain of the curve is .
As approaches from the right or from the left, the denominator approaches , causing .
Thus, the equations of the vertical asymptotes are and .
2. Find the coordinates of the turning point.
Let the expression in the square root be .
Completing the square gives:
Because , the squared term . Hence, attains a maximum value of when .
The maximum of results in the minimum positive value for .
Substituting :
The coordinates of the turning point are .
3. Identify the line of symmetry.
The quadratic expression is symmetric about its vertex at , and the domain is also symmetric about .
Consequently, the curve is symmetric about the vertical line passing through its turning point.
The equation of the line of symmetry is .
4. Describe the sketch of the curve $C$.
To sketch the curve :
- Draw and label vertical dashed lines at (the -axis) and to represent the asymptotes.
- Plot and label the minimum turning point at .
- Draw a U-shaped curve that passes through the turning point, is visually symmetric about the line , and curves upwards to approach the asymptotes and smoothly as .
Asymptotes: and
Line of symmetry:
Turning point:
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