End Behavior
In short: End behavior describes where a function’s outputs head when the input becomes extremely large or extremely negative, such as rising toward infinity or sinking toward negative infinity.
Algebra 2Functions and GraphsF-IF.C.7c
What is End Behavior?
The end behavior of a function is the story told by the far left and far right of its graph: where do the outputs go as \(x\) becomes enormously negative or enormously positive?
For a polynomial, the leading term decides everything. An even degree sends both ends the same way: up if the leading coefficient is positive, down if it is negative. An odd degree sends the ends opposite ways: up on the right for a positive coefficient, down on the right for a negative one. Middle terms matter near the origin, not at the edges.
How to spot it
- A question asks what happens to \(f(x)\) as \(x\) gets very large or very negative.
- A graph-matching task offers several curves and the only clue is the direction of the arrows on each end.
- The phrase as x approaches infinity, or the same with negative infinity, appears in the statement.
- A polynomial is given in standard form and only its degree and leading coefficient are needed.
The method
- Write the polynomial in standard form and find the term with the highest power.
- Note whether that degree is even or odd.
- Note whether the leading coefficient is positive or negative.
- Combine the two facts to state both ends, then sketch arrows to confirm your answer.
End Behavior: solved example
Problem
Describe the end behavior of \(f(x) = -2x^5 + 3x^2 – 7\) and of \(g(x) = 3x^4 – x\).
Solution
For \(f\): the leading term is \(-2x^5\), degree 5 (odd) with a negative coefficient. As \(x\) grows toward \(\infty\), \(f(x)\) heads toward \(-\infty\); as \(x\) heads toward \(-\infty\), \(f(x)\) climbs toward \(\infty\).
For \(g\): the leading term is \(3x^4\), degree 4 (even) with a positive coefficient, so both ends rise toward \(\infty\).
Try it
Question 1
Describe the ends of \(h(x) = -x^6 + 5x^3\).
Show the answer
Even degree with a negative leading coefficient: both ends fall toward \(-\infty\).
Question 2
Describe the ends of \(p(x) = 4x^3 – 9x + 1\).
Show the answer
Odd degree with a positive leading coefficient: the left end falls and the right end rises.
End Behavior vs. Asymptote
An asymptote is a line the graph creeps toward without touching, such as a horizontal line at height 2. End behavior is only a statement about direction at the far edges. Polynomials have no asymptotes, because their ends fly off to infinity instead of settling near a line.
Keep going
Lessons
Worksheets and quizzes
- Zeros and Graphs of Polynomial Functions – Worksheet
- Zeros and Graphs of Polynomial Functions – Quiz
Related words
- Polynomial
- Degree of a Polynomial
- Asymptote
- Limit at Infinity
- Multiplicity of a Zero
- Zeros of a Function
Whole dictionary: A–Z · Algebra 2 · Functions and Graphs