Math · College algebra · Worked example
Find the end behavior of a polynomial
Describe the end behavior of f(x) = 4x² − 3x⁵ + 7.
Find the leading term
Order the terms by degree: −3x⁵ + 4x² + 7. The leading term is −3x⁵; 4x² is only written first.
Read the degree and the sign
The degree, 5, is odd, and the leading coefficient, −3, is negative.
Test large inputs
At x = 10 the leading term swamps the others, and at x = −10 it has the opposite sign.
State the end behavior
As x → ∞, f(x) → −∞, and as x → −∞, f(x) → ∞: the graph rises on the left and falls on the right, like y = −x³.
Result
The graph rises to the left and falls to the right: f(x) → ∞ as x → −∞, and f(x) → −∞ as x → ∞.
Your turn
Describe the end behavior of g(x) = (2 − x)(x + 1)(x + 4)².
Show the answer and explanation
Both ends fall.
The leading term is (−x)(x)(x²) = −x⁴: even degree with a negative leading coefficient.
Keep exploring
Graph plots f near the origin; zoom out and the ends follow −3x⁵.
Return to the concept →Sources and scope
Authored study material. Tool results depend on the stated inputs and model assumptions.
Try in the workspace
Open the example inputs, change a value and keep a useful result on your board.
Graph the polynomial Check the values in Math Open worked example on a board Degree and leading term in Math ReferenceYour existing work stays on this device. Examples open as editable copies.