Coordinate quadrants and basic graph families · Foundation
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| asymptote/ˈæsɪmptəʊt/ | 渐近线 | jiàn jìn xiàn |
What happens near the forbidden input?
- A table of a cube and a reciprocal gives very different values near zero. Can their graphs have the same shape?
- This lesson studies asymptote 渐近线: A line approached by a graph without a finite crossing in the stated example.
Choose the mathematical structure
- Coordinates are ordered (x,y). Signs identify the four quadrants. Use a value table, intercepts and symmetry to sketch a line, quadratic, cubic or reciprocal. For y=1/x, zero is excluded and the axes are asymptotes. Approximate intersections give graphical solutions.
- 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 approached by a graph without a finite crossing in the stated example.
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=x³, inputs -2,-1,0,1,2 give -8,-1,0,1,8. For y=1/x, inputs -2,-1,1,2 give -1/2,-1,1,1/2; never substitute zero. The point (-2,3) is in quadrant II. The graphs y=x² and y=4 meet at x=-2 and x=2. A quadratic y=(x-3)(x-7) has roots 3,7 and symmetry line x=5.
Coordinate quadrants and basic graph families
Coordinates are ordered (x,y)
Compare the model with the worked case and explain one change.
Find x³ at x=-2.
(-2)³=-8.
Test a tempting shortcut
- Join a reciprocal branch smoothly on its own side of zero; never draw a segment through the excluded input. A graph table needs enough points to reveal a turning point or shape.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
The graph of y=1/x has a point at x=0. This claim is false. Explain which definition or assumption it violates.
Find 1/x at x=2.
1/2=0.5.
The graph of y=1/x has a point at x=0.
Join a reciprocal branch smoothly on its own side of zero; never draw a segment through the excluded input. A graph table needs enough points to reveal a turning point or shape.
Interpret a new situation
- AQA A8/A11/A12 Foundation includes all quadrants, lines, quadratics, simple cubics and reciprocals. Higher exponential and degree-based trigonometric families are treated separately.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
Find the positive intersection input of y=x² and y=4.
x²=4 gives ±2; choose the positive value.
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
- 8300 · Foundation · 3.2. 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 approached by a graph without a finite crossing in the stated example. Choose the relationship, show the method, check its assumptions and interpret the result.