Whether the graph of a polynomial rises or falls can be determined by the Leading Coefficient Tests.
P(x) = a_{n}x^{n} + a_{n-1}x^{n-1} +............. a_{1}x + a_{0}
In the above polynomial, n is the degree and a_{n} is the leading coefficient.
Case |
End Behavior of Graph |
When n is odd and a_{n} is positive
Graph falls to the left and rises to the right
When n is odd and a_{n} is negative
Graph rises to the left and falls to the right
When n is even and a_{n} is positive
Graph rises to the left and right
When n is even and a_{n} is negative
Graph falls to the left and right
Example 1 :
Find the right-hand and left-hand behaviors of the graph of
f(x) = x^{5} + 2x^{3} - 3x + 5
Solution :
Because the degree is odd and the leading coefficient is positive, the graph falls to the left and rises to the right as shown in the figure.
Example 2 :
Determine the end behavior of the graph of the polynomial function below using Leading Coefficient Test.
P(x) = -x^{3} + 5x
Solution :
Because the degree is odd and the leading coefficient is negative, the graph rises to the left and falls to the right as shown in the figure.
Example 3 :
Determine the end behavior of the graph of the polynomial function below using Leading Coefficient Test.
P(x) = 2x^{2} - 2
Solution :
Because the degree is even and the leading coefficient is positive, the graph rises to the left and right as shown in the figure.
Example 4 :
Determine the end behavior of the graph of the polynomial function below using Leading Coefficient Test.
P(x) = -x^{2} + 1
Solution :
Because the degree is even and the leading coefficient is negative, the graph falls to the left and right as shown in the figure.
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