Cogito
Integrated Math 3 · Chapter 8 · Lesson 3
Piecewise Models and Systems
When one rule does not cover everything.
12 problems · about 21 minutes · F-IF.C.7, A-REI.C.6
What this lesson teaches
The student builds piecewise models and solves systems arising from them.
- A piecewise function applies different rules on different intervals.
- The pieces agreeing at a boundary makes the graph continuous there.
- Setting two models equal finds the break-even point.
Warm Up
Straightforward practice. Get the method working first.
5 problems4u + 50 = 6u + 30. What is u?
Answer 10
Why 10.
What does the intersection of two cost models represent?
Answer The usage at which both options cost the same.
Why The break-even point.
3u + 20 = 5u + 10. What is u?
Answer 5
Why 10 = 2u.
C = 4u + 10 at u = 10. What is C?
Answer 50
Why 40 + 10.
Below the break-even point, which plan is cheaper here? 1 the lower fixed fee, 2 the higher.
Answer 1
Why Few units means the fee dominates.
Build It Up
The same ideas with more to keep track of.
3 problemsA piecewise rule changes at x = 4. Is 4 a break point? 1 yes, 0 no.
Answer 1
Why That is where the rule changes.
Both pieces agree at the boundary. Is the graph continuous there? 1 yes, 0 no.
Answer 1
Why They meet at the same height.
C = 2u + 30 at u = 25. What is C?
Answer 80
Why 50 + 30.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsPut the comparison process in order.
Answer 1. Write a function for each option. 2. Set the two equal. 3. Solve for the break-even usage. 4. State which option wins on each side of it.
Why The answer is a recommendation, not just a number.
5u + 40 = 9u + 20. What is u?
Answer 5
Why 20 = 4u.
The Break-Even: 2u + 30 = 4u + 10. What is u?
Answer 10
Why 10.
The Cost: Plan A is C = 2u + 30. What is the cost at 10 units, in dollars?
Answer 50 dollars
Why $50.