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 12 chapters and video availability01 · Read and understand
What you will learn
- Find all closest points using symmetry and a nonnegative remainder.
- Justify the conclusion "Closest points are (±1/√2,1/2), at distance √3/2" using the stated assumptions.
Before you start
Parabolas, derivatives, and completing the square.
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
Find the points on y=x² closest to (0,1).
Why this math matters
Find all closest points using symmetry and a nonnegative remainder. 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:
- Every real point of the parabola is allowed.
- Distance is Euclidean.
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.
Minimize squared distance to avoid a square root
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Find the points on y=x² closest to (0,1).
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
Squared distance D²=x²+(x²−1)²=x⁴−x²+1
Minimizing distance is equivalent to minimizing its nonnegative square.
Work through the mathematics
D²=(x²−1/2)²+3/4
Completing the square reveals the global lower bound.
Check the conclusion
Closest points are (±1/√2,1/2), at distance √3/2
The squared term vanishes at both symmetric x values.
The result
Closest points are (±1/√2,1/2), at distance √3/2
The squared term vanishes at both symmetric x values.
Common mistakes to catch
- A visually nearest-looking vertex need not minimize distance.
- A stationary candidate must still be classified or compared.
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
What is the distance from the parabola's vertex to (0,1)?
Show a hint
Use the two vertical coordinates.
Reveal answer and explanation
One
The vertex is farther away than the two minimizing points.
Practice 2
Why are there two minimizers?
Show a hint
The distance expression depends only on x².
Reveal answer and explanation
Symmetry about the y-axis
Opposite x values produce the same height and distance.
Take the idea with you
Replace a distance objective with its square when that preserves the ordering and simplifies the algebra.
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: Distinguish net accumulation from total geometric area
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