Coordinate geometry and tangents · Higher
| English | 中文 | Pinyin |
|---|---|---|
| gradient/ˈɡreɪdɪənt/ | 斜率 | xié lǜ |
How does a path's slope become an equation?
- A path rises 6 metres over a horizontal distance of 3 metres. Its gradient connects a diagram to an equation.
- This lesson studies gradient 斜率: The change in y divided by the corresponding change in x.
Choose the mathematical structure
- A line through (x₁,y₁) with gradient m has y-y₁=m(x-x₁). Parallel lines have equal gradients. Finite perpendicular gradients multiply to -1. A circle has (x-a)²+(y-b)²=r².
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines gradient?
The change in y divided by the corresponding change in x.
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.
Through (2,5) with gradient 3, y-5=3(x-2), so y=3x-1. A perpendicular through the same point has y-5=-(x-2)/3. The circle (x-2)²+(y+1)²=25 has centre (2,-1) and radius 5.
Coordinate geometry and tangents
A line through (x₁,y₁) with gradient m has y-y₁=m(x-x₁)
Compare the model with the worked case and explain one change.
Find the gradient through (1,2) and (4,8).
Gradient=(8-2)/(4-1)=6/3=2.
Test a tempting shortcut
- A vertical line has no finite gradient; do not force it into y=mx+c. Read the signs of a circle's centre carefully. The radius to a tangent is perpendicular to the tangent.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
Perpendicular nonvertical lines always have equal gradients. This claim is false. Explain which definition or assumption it violates.
Find the y-intercept of y-5=3(x-2).
Expand y-5=3x-6, so y=3x-1; the intercept is -1.
Perpendicular nonvertical lines always have equal gradients.
A vertical line has no finite gradient; do not force it into y=mx+c. Read the signs of a circle's centre carefully. The radius to a tangent is perpendicular to the tangent.
Interpret a new situation
- Before solving a line-circle intersection, predict whether there are zero, one or two intersections. Substitution produces a quadratic whose discriminant checks the prediction.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
Find the radius of (x-2)²+(y+1)²=25.
The radius is the positive square root of 25, namely 5.
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
- 4MA1 · Higher · 3. 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.
The change in y divided by the corresponding change in x. Choose the relationship, show the method, check its assumptions and interpret the result.