Reciprocal graphs and asymptotes
Đã giới thiệu| English | Ý nghĩa |
|---|---|
| asymptote/ˈæsɪmptəʊt/ | tiếp tuyến |
Zero is a missing input, not a point to join across.
- An inverse model grows very large as its input approaches zero.
- This lesson studies asymptote 渐近线: A line that a curve approaches in a limiting direction.
The method
- For nonzero a, y=a/x has domain x≠0 and range y≠0, with asymptotes x=0 and y=0.
- Positive a gives branches in quadrants I and III; negative a gives II and IV.
- For y=a/x² the domain is still x≠0, but the graph is symmetric about the y-axis; both branches are positive if a>0 and negative if a<0.
How do the negative-x branches of y=4/x and y=4/x² differ?
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
Squaring removes the sign: y=4/x² has both branches above the axis, symmetric about the y-axis, while y=4/x keeps quadrant III negative. The reciprocal-square graph is not a pruned reciprocal graph.
A worked case
For y=4/x, checked points are (1,4), (2,2), (−1,−4), (−2,−2). As x approaches zero from the right, y grows positively; from the left it grows negatively. As |x| grows, y approaches zero. For y=4/x², (1,4) and (−1,4) have equal outputs, and both sides rise positively near zero. Intersecting y=4/x with y=x gives x²=4, so (2,2) and (−2,−2). Intersecting y=4/x² with y=1 gives x²=4, so (2,1) and (−2,1).
Reciprocal graphs and asymptotes
For nonzero a, y=a/x has domain x≠0 and range y≠0, with asymptotes x=0 and y=0
Connect graph shape to algebraic branches and allowed inputs.
Evaluate 4/x at x=−2.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
4/(−2)=−2.
Where it goes wrong
- Never plot x=0 or draw a stroke joining the two branches through it.
- The reciprocal-square graph is not the reciprocal graph with the negative branch removed; the entire left branch is reflected above the x-axis for positive a.
- These particular curves do not meet y=0, but “no curve can ever cross any asymptote” is not a valid general rule.
Not true: The graphs y=4/x and y=4/x² have the same outputs for every negative x. For negative x, 4/x is negative but 4/x² is positive.
Evaluate 4/x² at x=−2.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
Square the whole denominator input: 4/(−2)²=4/4=1.
For negative x, 4/x is negative but 4/x² is positive.
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
True. For negative x, 4/x is negative but 4/x² is positive.
Applying it
- For positive physical inputs, y=a/x can model inverse proportion and y=a/x² inverse-square proportion.
- If y=4/x represents a time and speed relation, the negative branch is mathematically valid but outside that model’s physical domain.
- Doubling a positive x halves 4/x and quarters 4/x².
- Check graph intersections in the original relation after clearing a nonzero denominator.
Find the positive x-coordinate where y=4/x meets y=x.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
With x≠0, 4/x=x gives x²=4; the positive root is 2.
Match each asymptote behaviour of y=4/x to its direction.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
Asymptotes describe limiting behaviour; never draw a stroke joining the branches through the excluded x=0.
In your course
- AQA A-level Mathematics (7357), specification section B.
- Every paper allows a calculator, but method marks still need the expression or equation written before its value.
- Asymptote: A line that a curve approaches in a limiting direction.
- For nonzero a, y=a/x has domain x≠0 and range y≠0, with asymptotes x=0 and y=0.
- Watch for: never plot x=0 or draw a stroke joining the two branches through it.