Straight lines and gradients · Core
הוצג| English | משמעות |
|---|---|
| gradient/ˈɡreɪdɪənt/ | שיפוע |
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.
The method
- Gradient is change in y divided by change in x.
- A straight line has y=mx+c, where c is its y-intercept.
- Parallel lines have equal gradients.
A line passes through (1,2) and (4,8). What is its gradient?
רמת קושי 1/5 · 100 מטבעות + בונוס קטן בהשלמה הראשונה היום
Gradient = change in y / change in x = (8-2)/(4-1)=6/3=2. Dividing x-change by y-change inverts it.
A worked case
Through (2,5) with gradient 3, substitute to get 5=3×2+c, so c=-1 and y=3x-1. Points (1,2) and (4,8) give gradient (8-2)/(4-1)=2.
Straight lines and gradients
Gradient is change in y divided by change in x
Compare the model with the worked case and explain one change.
Find the gradient through (1,2) and (4,8).
רמת קושי 2/5 · 140 מטבעות + בונוס קטן בהשלמה הראשונה היום
Gradient=(8-2)/(4-1)=6/3=2.
Where it goes wrong
- 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.
Not true: Perpendicular nonvertical lines have equal gradients. Perpendicular gradients multiply to −1: a gradient of 2 needs −1/2. Parallel lines have equal gradients.
Find the y-intercept of y-5=3(x-2).
רמת קושי 2/5 · 140 מטבעות + בונוס קטן בהשלמה הראשונה היום
Expand y-5=3x-6, so y=3x-1; the intercept is -1.
Perpendicular nonvertical lines have equal gradients.
רמת קושי 1/5 · 100 מטבעות + בונוס קטן בהשלמה הראשונה היום
Perpendicular gradients multiply to −1: a gradient of 2 needs −1/2. Parallel lines have equal gradients.
Applying it
- Plot a straight line using two checked points and label its intercept.
- Perpendicular-gradient formulae and circle equations are not part of this Foundation/Core lesson.
For y=3x-1, find y when x=5.
רמת קושי 2/5 · 140 מטבעות + בונוס קטן בהשלמה הראשונה היום
Substitute x=5 into y=3x-1: y=15-1=14.
Match each part of y=3x-1 to its meaning.
רמת קושי 2/5 · 140 מטבעות + בונוס קטן בהשלמה הראשונה היום
In y=mx+c the multiplier m is the gradient and the constant c is where the line crosses the y-axis.
In your course
- OxfordAQA International GCSE Mathematics (9260), Core tier, specification section 3.2.
- Every paper allows a calculator, but method marks still need the expression or equation written before its value.
- Gradient: The change in y divided by the corresponding change in x.
- Gradient is change in y divided by change in x.
- Watch for: A vertical line has no finite gradient; do not force it into y=mx+c.
Write from a gradient of 4
- The line passes through (2, 1) and (5, 13).
- Find the gradient and the equation.
A straight line passes through (2, 1) and (5, 13). What are its gradient and its equation?
12 divided by 3 is 4. At x = 2, 4 times 2 minus 7 is 1.
What is the gradient?
רמת קושי 1/5 · 100 מטבעות + בונוס קטן בהשלמה הראשונה היום
12 divided by 3 is 4. Other options — 2 (the lesson gradient between (1, 2) and (4, 8)); 3 (the lesson gradient of y = 3x - 1).