LEMMA

Advanced functions

Polynomial function behavior

ALEKS placement

The leading term decides the ends; the factors decide the middle.

What this covers

  • Read the degree and the sign of the leading coefficient for end behavior.
  • Even degree means both ends agree; odd degree means they oppose.
  • Each factor gives an x-intercept.
  • Even multiplicity touches the axis; odd multiplicity crosses it.

Worked example

Worked example

Describe f(x) = -(x - 1)²(x + 3).

  1. degree 3,leading coefficient<0Two factors of x times one more, then negated.
  2. x→−∞: f(x)→+∞;x→+∞: f(x)→−∞An odd degree gives opposite ends; the negative lead makes the right end fall.
  3. x=1(touch),x=−3(cross)Multiplicity 2 touches, multiplicity 1 crosses.
Another worked example

Find the limit of 2x³ − 9x + 1 as x → ∞.

  1. x→∞lim​(2x3−9x+1)Read this as: as x grows without bound to the right, what happens to the output?
  2. x→∞lim​(2x3−9x+1)=+∞For large x the leading term 2x³ outgrows the other terms. Its coefficient is positive, so the right end rises forever.

A common mistake

A mistake Lemma catches

From

degf=2degf=4

“The degree counts every x that multiplies out, not the number of factors: (x + 2)³ contributes 3 and (x − 1) contributes 1, so the degree is 3 + 1 = 4. Counting the two factors gives 2.”

Try one

Sample problem

Determine the limit as x approaches positive infinity.

x→∞lim​(x5+7x+2)

Practice polynomial behavior free

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Where this fits

Before this