1.4: End Behavior | AP Precalculus | Code Scholars Tutoring

How it works

The leading term controls polynomial end behavior because it eventually dominates all lower-degree terms. Even degree gives matching end directions; odd degree gives opposite directions. The sign of the leading coefficient determines the right-hand end.

f(x)=a_nx^n+\cdots\quad\text{has the end behavior of }a_nx^n

Worked examples

Example 1

Describe the ends of $\displaystyle f(x)=-2x^4+3x-1$ .

Even degree and negative leading coefficient mean both ends go down: $\displaystyle f(x)\to-\infty$ as $\displaystyle x\to\pm\infty$ .

Example 2

Describe the ends of $\displaystyle g(x)=x^3-4x$ .

The positive cubic leading term gives $\displaystyle g(x)\to-\infty$ to the left and $\displaystyle g(x)\to\infty$ to the right.

Problems

Problem 1.4.1

Describe the ends of $\displaystyle f(x)=5x^6-x^2+2$ .

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The degree is even and the leading coefficient is positive, so both ends rise toward $\displaystyle \infty$ .

Problem 1.4.2

Describe the ends of $\displaystyle g(x)=-x^5+2x^2$ .

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The negative odd-degree leading term makes the left end rise and the right end fall.

Problem 1.4.3

Use the graph of $\displaystyle y=-x^3+3x$ to identify its end behavior.

The graph y = −x cubed + 3x rises to the left and falls to the right. — Follow the leading term at large input magnitude.

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The negative cubic leading term means $\displaystyle y\to\infty$ as $\displaystyle x\to-\infty$ , and $\displaystyle y\to-\infty$ as $\displaystyle x\to\infty$ .

The graph y = −x cubed + 3x rises to the left and falls to the right. The left end tends to positive infinity; the right end tends to negative infinity. — The left end tends to positive infinity; the right end tends to negative infinity.

Lesson Problems PDF Solutions PDF Unit MCQ

How it works

Worked examples

Example 1

Example 2

Problems

Problem 1.4.1

Problem 1.4.2

Problem 1.4.3

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