Coordinate geometry and tangents · Higher · Géométrie analytique et tangentes · Niveau supérieur
| English | Français |
|---|---|
| gradient/ˈɡreɪdɪənt/ | pente |
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? · Quelle description définit correctement la pente ?
The change in y divided by the corresponding change in x. · Le changement en y divisé par le changement correspondant en 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 · Géométrie analytique et tangentes
A line through (x₁,y₁) with gradient m has y-y₁=m(x-x₁) · Une droite passant par (x₁,y₁) avec une pente m a pour équation y-y₁=m(x-x₁)
Compare the model with the worked case and explain one change. · Compare le modèle avec l'exemple résolu et explique un changement.
Find the gradient through (1,2) and (4,8). · Trouvez la pente passant par (1,2) et (4,8).
Gradient=(8-2)/(4-1)=6/3=2. · Pente=(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). · Trouvez l'ordonnée à l'origine de y-5=3(x-2).
Expand y-5=3x-6, so y=3x-1; the intercept is -1. · Développez y-5=3x-6, donc y=3x-1 ; l'ordonnée à l'origine est -1.
Perpendicular nonvertical lines always have equal gradients. · Les droites perpendiculaires non verticales ont toujours des pentes égales.
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. · Une ligne verticale n'a pas de pente finie ; ne l'obligez pas à s'écrire sous la forme y=mx+c. Lisez attentivement les signes du centre d'un cercle. Le rayon menant à une tangente est perpendiculaire à cette tangente.
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. · Trouvez le rayon de (x-2)²+(y+1)²=25.
The radius is the positive square root of 25, namely 5. · Le rayon est la racine carrée positive de 25, soit 5.
Match each part of a complete solution to its purpose. · Associer chaque partie d'une solution complète à son but.
An assumption justifies the model; a check tests the result; interpretation connects it to the question. · Une hypothèse justifie le modèle ; une vérification teste le résultat ; l'interprétation le relie à la question.
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.