Answer:
1)
f(x)→ ∞ when x→∞ or x→ -∞.
2)
when x→ ∞ then f(x)→ -∞
and when x→ -∞ then f(x)→ ∞
Step-by-step explanation:
" The end behavior of a polynomial function is the behavior of the graph of as approaches positive infinity or negative infinity. The degree and the leading coefficient of a polynomial function determine the end behavior of the graph "
1)
a 14th degree polynomial with a positive leading coefficient.
Let f(x) be the polynomial function.
Since the degree is an even number and also the leading coefficient is positive so when we put negative or positive infinity to the function i.e. we put x→∞ or x→ -∞ ; it will always lead the function to positive infinity
i.e. f(x)→ ∞ when x→∞ or x→ -∞.
2)
a 9th degree polynomial with a negative leading coefficient.
As the degree of the polynomial is odd and also the leading coefficient is negative.
Hence when x→ ∞ then f(x)→ -∞ since the odd power of x will take it to positive infinity but the negative sign of the leading coefficient will take it to negative infinity.
When x→ -∞ then f(x)→ ∞; since the odd power of x will take it to negative infinity but the negative sign of the leading coefficient will take it to positive infinity.
Hence, when x→ ∞ then f(x)→ -∞
and when x→ -∞ then f(x)→ ∞
Even degree polynomials (like the 14 degree one) usually have the same end behaviour for the two ends (negative and positive). This his because if N is a positive whole number, we have that:
[tex](-A)^N = A^N[/tex]
And because the leading coefficient is positive, and a number with an even exponent is also positive, we can expect to see that the end behaviour of the 14th degree polynomial is that it increases to infinity.
For the 9th degree polynomial this does not happen, as here we have a negative leading coefficient and a odd exponent.
Then as x tends to infinity, the polynomial will tend to negative infinity.
as x tends to negative infinity, the polynomial will tend to infinity.
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