Modulus graphs and piecewise solutions
導入済み| English | 意味 |
|---|---|
| modulus/ˈmɒdjʊləs/ | 絶対値 |
A sensor measures distance from a target value. Values on either side give the same distance, but their algebraic signs differ.
- A sensor measures distance from a target value. Values on either side give the same distance, but their algebraic signs differ.
- This lesson studies modulus 绝对值: The nonnegative distance of a real number from zero.
Choose the mathematical structure
- 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.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Why does |2x-4|=x+2 require checking x+2≥0?
難易度 1/5 · 100 コイン + 今日初回完了時に小ボーナス付与
|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.
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.
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|.
難易度 2/5 · 140 コイン + 今日初回完了時に小ボーナス付与
Set 2x−4=0, giving x=2.
Test a tempting shortcut
- 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.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
Every modulus equation has exactly two real solutions. This claim is false. Explain which definition or assumption it violates.
Find y=|2x−4| when x=−1.
難易度 2/5 · 140 コイン + 今日初回完了時に小ボーナス付与
The inner value is 2(−1)−4=−6, whose modulus is 6.
Every modulus equation has exactly two real solutions.
難易度 1/5 · 100 コイン + 今日初回完了時に小ボーナス付与
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.
Interpret a new situation
- 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.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
Find the larger solution of |2x−4|=x+2.
難易度 2/5 · 140 コイン + 今日初回完了時に小ボーナス付与
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.
難易度 2/5 · 140 コイン + 今日初回完了時に小ボーナス付与
The graph is V-shaped with the corner at the zero of the expression; each branch is an ordinary line on its own domain.
Use this in your course
- 7357 · A-level · B. 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 nonnegative distance of a real number from zero. Choose the relationship, show the method, check its assumptions and interpret the result.