Reciprocal graphs and asymptotes
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| asymptote/ˈæsɪmptəʊt/ | 渐近线 | jiàn jìn xiàn |
An inverse model grows very large as its input approaches zero. Zero is a missing input, not a point to join across.
- An inverse model grows very large as its input approaches zero. Zero is a missing input, not a point to join across.
- This lesson studies asymptote 渐近线: A line that a curve approaches in a limiting direction.
Choose the mathematical structure
- 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.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines asymptote?
A line that a curve approaches in a limiting direction.
Work through a checked case
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
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.
4/(−2)=−2.
Test a tempting shortcut
- 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.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
The graphs y=4/x and y=4/x² have the same outputs for every negative x. This claim is false. Explain which definition or assumption it violates.
Evaluate 4/x² at x=−2.
Square the whole denominator input: 4/(−2)²=4/4=1.
The graphs y=4/x and y=4/x² have the same outputs for every negative x.
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.
Interpret a new situation
- 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.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
Find the positive x-coordinate where y=4/x meets y=x.
With x≠0, 4/x=x gives x²=4; the positive root is 2.
Match each part of a complete solution to its purpose.
An assumption justifies the model; a check tests the result; interpretation connects it to the question.
Use this in your course
- 7357 · A-level · B. Match the target tier and specification before assigning extensions.
- Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.
A line that a curve approaches in a limiting direction. Choose the relationship, show the method, check its assumptions and interpret the result.