Modulus graphs and piecewise solutions
Đã giới thiệu| English | Ý nghĩa |
|---|---|
| modulus/ˈmɒdjʊləs/ | mô-đun |
Values on either side give the same distance, but their algebraic signs differ.
- A sensor measures distance from a target value.
- This lesson studies modulus 绝对值: The nonnegative distance of a real number from zero.
The method
- For a real expression u, |u|=u when u≥0 and |u|=−u when u<0.
- Find where a linear expression is zero to locate the corner of its V-shaped modulus graph.
- Solve equations by checking both branches and their domains; the right-hand side must be nonnegative.
Why does |2x-4|=x+2 require checking x+2≥0?
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
|2x-4|≥0 always, so solutions need x≥-2. Each branch solution - x=6 from the right branch and x=2/3 from the left - must also satisfy its own branch condition; both do.
A worked case
For y=|2x−4|, the corner is (2,0). The left branch is y=4−2x for x<2; the right is y=2x−4 for x≥2. To solve |2x−4|=x+2, require x≥−2. The right branch gives 2x−4=x+2, hence x=6. The left gives 4−2x=x+2, hence x=2/3. Both satisfy their branch restrictions and the original equation. The graph meets the line at (6,8) and (2/3,8/3).
Modulus graphs and piecewise solutions
For a real expression u, |u|=u when u≥0 and |u|=−u when u<0
Connect graph shape to algebraic branches and allowed inputs.
Find the x-coordinate of the corner of y=|2x−4|.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
Set 2x−4=0, giving x=2.
Where it goes wrong
- Do not replace |u| by u on a negative branch.
- Squaring an equation can create candidates where its right-hand side is negative.
- The equation |2x−4|=−1 has no real solutions.
- The graph of |2x−4| has one corner and its slopes change there.
Not true: Every modulus equation has exactly two real solutions. A modulus equation can have no solution, one or two; |x|=−1 has none.
Find y=|2x−4| when x=−1.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
The inner value is 2(−1)−4=−6, whose modulus is 6.
A modulus equation can have no solution, one or two.
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
True. A modulus equation can have no solution, one or two; |x|=−1 has none.
Applying it
- For |2x−4|<6, use −6<2x−4<6 to obtain −1<x<5.
- For |2x−4|>6, use 2x−4<−6 OR 2x−4>6: x<−1 or x>5.
- A bounded distance gives an inside interval; an excessive distance gives two outside intervals.
- Test both boundary and interior values.
Find the larger solution of |2x−4|=x+2.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
On the x≥2 branch, 2x−4=x+2 gives x=6; the other solution is 2/3.
Match each modulus feature of y=|2x-4| to its location.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
The graph is V-shaped with the corner at the zero of the expression; each branch is an ordinary line on its own domain.
In your course
- AQA A-level Mathematics (7357), specification section B.
- Every paper allows a calculator, but method marks still need the expression or equation written before its value.
- Modulus: The nonnegative distance of a real number from zero.
- For a real expression u, |u|=u when u≥0 and |u|=−u when u<0.
- Watch for: do not replace |u| by u on a negative branch.