Coordinate geometry and tangents · Higher
도입됨| English | 의미 |
|---|---|
| gradient/ˈɡreɪdɪənt/ | 기울기 |
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.
Two non-vertical lines are perpendicular. What is true of their gradients?
난이도 1/5 · 100 코인 + 오늘 첫 완료 시 소액 보너스
Perpendicular gradients satisfy m₁m₂=-1: each is the negative reciprocal of the other. Equal gradients give parallel lines instead.
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).
난이도 2/5 · 140 코인 + 오늘 첫 완료 시 소액 보너스
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).
난이도 2/5 · 140 코인 + 오늘 첫 완료 시 소액 보너스
Expand y-5=3x-6, so y=3x-1; the intercept is -1.
Perpendicular nonvertical lines always have equal gradients.
난이도 1/5 · 100 코인 + 오늘 첫 완료 시 소액 보너스
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.
난이도 2/5 · 140 코인 + 오늘 첫 완료 시 소액 보너스
The radius is the positive square root of 25, namely 5.
Match each feature of y=mx+c to its meaning.
난이도 2/5 · 140 코인 + 오늘 첫 완료 시 소액 보너스
Gradient measures steepness and the intercept fixes position; equal gradients with different intercepts never meet.
Use this in your course
- 8300 · Higher · 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.
The change in y divided by the corresponding change in x. Choose the relationship, show the method, check its assumptions and interpret the result.