Question

use the leading - coefficient test to determine the end behavior of the graph of the given polynomial function. f(x)=7x^5 - 2x^3+3x^2 + 1
a. rises left & rises right.
b. falls left & rises right.
c. rises left & falls right.
d. falls left & falls right.
e. none of the above.

Explanation:

Step1: Identify degree and leading coefficient

The polynomial $f(x)=7x^{5}-2x^{3}+3x^{2}+1$ has degree $n = 5$ (the highest - power of $x$) and leading coefficient $a = 7$.

Step2: Apply leading - coefficient test for odd degree

For a polynomial with odd degree $n$ and positive leading coefficient $a>0$, as $x\to-\infty$, $y\to-\infty$ (falls left) and as $x\to+\infty$, $y\to+\infty$ (rises right).

Answer:

B. Falls left & rises right.