SAT-43: Graphs of Higher-Degree Polynomials — End Behavior & Multiplicity

SAT-43: Graphs of Higher-Degree Polynomials — End Behavior & Multiplicity

Description: Cubics, quartics and beyond have wavy graphs. Two ideas explain them: end behavior (what the tails do) and multiplicity (how the curve behaves at each root).

End behavior

The tails depend on the degree (highest power) and the leading coefficient (sign of the front number):

• Even degree (like x², x⁴): both tails go the same way — both up (if leading coeff > 0) or both down (< 0).
• Odd degree (like x³, x⁵): tails go opposite ways.

(Oʻzbekcha: juft daraja → ikkala uchi bir tomonga; toq daraja → qarama-qarshi tomonlarga.)

Multiplicity at a root

The multiplicity is how many times a factor repeats. It controls what the graph does at that x-intercept:

• Odd multiplicity (1, 3, …) → the graph crosses the x-axis.
• Even multiplicity (2, 4, …) → the graph touches and turns back (bounces).

Memory hook: odd = cross, even = bounce. (Oʻzbekcha: toq → kesib oʻtadi, juft → tegib qaytadi.)

Worked Example

Describe y = (x − 1)²(x + 2).

• Degree = 2 + 1 = 3 (odd), leading coeff positive → left tail down, right tail up.
• Root x = 1 has multiplicity 2 (even) → graph touches at x = 1.
• Root x = −2 has multiplicity 1 (odd) → graph crosses at x = −2.

(Oʻzbekcha: x = 1 da tegib qaytadi, x = −2 da kesib oʻtadi.)

Practice

At x = 3, does y = (x − 3)⁴(x + 1) cross or touch the x-axis?

Key words — Kalit soʻzlar

• Degree — daraja
• Leading coefficient — bosh koeffitsiyent
• End behavior — uchlarining xatti-harakati
• Multiplicity — karralilik (takrorlanish soni)
• Root / Zero — ildiz / nol
• Cross — kesib oʻtish
• Touch / Bounce — tegib qaytish
• Even / Odd — juft / toq
• Tail — uch (dum)

Summary

• End behavior: even degree → same-direction tails; odd degree → opposite tails (sign of leading coeff flips them).
• Multiplicity: odd → cross, even → touch/bounce.
• Total degree = sum of all the multiplicities.