Learn with Amar
Teaching video
A teaching recording for this chapter has not been published yet. Start with the worked example below and explore the related animations where available.
Grade 11 chapters and video availability01 · Read and understand
What you will learn
- Match one-sided values at a piecewise boundary.
- Justify the conclusion "3k−2=7 gives k=3" using the stated assumptions.
Before you start
Linear functions and limits from a table or graph.
Keep paper nearby. Read the question once for the context, then again to identify what is known and what you need to find.
Start with a question
Let f(x)=2x+1 for x<3 and f(x)=kx−2 for x≥3. Choose k so the graph has no jump.
Why this math matters
Match one-sided values at a piecewise boundary. This worked micro-lesson connects a precise mathematical condition to a conclusion you can check. The transfer task asks you to change the setting and decide which parts of the reasoning still apply.
Set up the model
A useful answer starts with clear assumptions:
- Each branch is linear and continuous on its own interval.
- The boundary value uses the second branch.
02 · Work through the example
Follow the reasoning, one step at a time.
Try to predict the next step before reading it. After each calculation, explain why the operation makes sense and how it helps answer the original question.
Work through it with Amar
See this example unfold.
The complete worked example, one idea at a time.
Make two formula pieces meet continuously
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Let f(x)=2x+1 for x<3 and f(x)=kx−2 for x≥3. Choose k so the graph has no jump.
Before you calculate
Read what is known and what you need to find. Make a prediction before moving to the first calculation.
Starts paused. Play advances through the full text at a reading pace; pause whenever you need more time. Previous, Next, and the phase buttons let you set your own pace. Playback pauses when this walkthrough leaves the screen or you switch tabs.
Your device’s reduced-motion setting keeps each phase still. Manual controls remain available. The full written solution stays below.
Build the model
The left-hand value approaches 2·3+1=7
The first rule determines the approaching height.
Work through the mathematics
The second rule gives f(3)=3k−2
The boundary belongs to the second branch.
Check the conclusion
3k−2=7 gives k=3
Matching these values makes the function continuous at the join.
The result
3k−2=7 gives k=3
Matching these values makes the function continuous at the join.
Common mistakes to catch
- Matching values is different from matching derivatives.
- The inequality signs decide which branch defines the endpoint.
03 · Practice independently
Try it before revealing the answer.
Use paper or a calculator as needed. Write your units and reasoning, then open the hint or explanation to check your approach.
Practice 1
Do the two slopes also match?
Show a hint
Compare the x coefficients.
Reveal answer and explanation
No; they are two and three
Continuity does not require equal slopes.
Practice 2
If k=2, what is the jump size?
Show a hint
Compare limiting heights seven and four.
Reveal answer and explanation
Three
The right value is 2·3−2=4.
Take the idea with you
Design a tiered rule that changes its rate without producing an abrupt jump.
04 · Reflect and continue
Can you explain it in your own words?
Before moving on, explain the main idea without looking at the worked example. Try both practice questions, check your reasoning, and name one mistake you now know how to avoid. Return to a step if you still need support.
Up next: Use a known root to determine a coefficient
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