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
- Compare a polynomial with its highest-degree term as input magnitude grows.
- Justify the conclusion "As x→∞, p→−∞; as x→−∞, p→∞" using the stated assumptions.
Before you start
Polynomial degree and powers.
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
Describe the two ends of p(x)=−2x⁵+7x²−3.
Why this math matters
Compare a polynomial with its highest-degree term as input magnitude grows. 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:
- The polynomial coefficients are fixed.
- The statement concerns limits at large positive or negative inputs.
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.
Read distant graph behavior from the leading term
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Describe the two ends of p(x)=−2x⁵+7x²−3.
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
p(x)/(−2x⁵)=1−7/(2x³)+3/(2x⁵)
Dividing by the leading term measures the relative size of lower powers.
Work through the mathematics
The ratio tends to one as |x| grows
Lower powers become negligible relative to x⁵.
Check the conclusion
As x→∞, p→−∞; as x→−∞, p→∞
Odd degree gives opposite end directions, and the negative leading coefficient determines which end rises.
The result
As x→∞, p→−∞; as x→−∞, p→∞
Odd degree gives opposite end directions, and the negative leading coefficient determines which end rises.
Common mistakes to catch
- A large lower-term coefficient does not change the eventual leading-degree dominance.
- End behavior is not a complete graph sketch.
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 are the ends of 3x⁴−100x?
Show a hint
Use even degree and a positive leading coefficient.
Reveal answer and explanation
Both rise to positive infinity
The fourth power eventually dominates the linear term.
Practice 2
Does end behavior identify every turning point?
Show a hint
Distant behavior is only part of a graph.
Reveal answer and explanation
No
Lower terms can create several local changes before the eventual end directions.
Take the idea with you
Use a leading-term comparison to test whether a polynomial extrapolation has a plausible long-range shape.
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: Recognize a power law on logarithmic axes
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