Modulus graphs and piecewise solutions
| English | 中文 | Pinyin |
|---|---|---|
| modulus/ˈmɒdjʊləs/ | 绝对值 | jué duì zhí |
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.
Which description correctly defines modulus?
The nonnegative distance of a real number from zero.
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|.
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.
The inner value is 2(−1)−4=−6, whose modulus is 6.
Every modulus equation has exactly two real solutions.
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.
On the x≥2 branch, 2x−4=x+2 gives x=6; the other solution is 2/3.
Match each part of a complete solution to its purpose.
An assumption justifies the model; a check tests the result; interpretation connects it to the question.
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.