THE COMPOUND // FIELD NOTES // WEEK 25
Boats, the discriminant, and functions
Field Book
Lessons 92 to 95
Due
Test 22 due · Quiz 5 · Supplement 25
Method
Paper first, every step
The discriminant reads a quadratic's future without solving it. The Examiner appreciates anything that audits before it acts.
What you have to be able to do
- Set up boat-in-river and plane-in-wind problems
- Read root behavior from the discriminant
- Use functional notation and test whether a relation is a function
The procedures
Boat and wind problemsField Book Lesson 92
- Downstream the rates add: B + W. Upstream they subtract: B - W.
- Write one distance equation per leg and solve the system.
The discriminantField Book Lesson 93
- Compute b2 - 4ac before solving.
- Positive means two real roots, zero means one repeated root, negative means a complex pair.
FunctionsField Book Lesson 94
- A function sends each input to exactly one output. Not every relation qualifies.
- f(x) names the output at x: substitute and evaluate.
Nonlinear systems, both optionsField Book Lesson 95
- Substitution when a variable is exposed, elimination when the squared terms align.
- Sketch first: the picture predicts how many answers to expect.
Worked: reading the discriminant
- For x2 + 4x + 1 = 0, compute b2 - 4ac.
- 16 - 4 = 12.
- Positive and not a perfect square: two real irrational roots.
Terms on the record
| Term | Meaning |
|---|
| current | the water's own rate, added or subtracted from the boat's |
| discriminant | b2 - 4ac, which tells the nature of the roots |
| function | a relation sending each input to exactly one output |