Question

use the leading coefficient test to determine the end - behavior of the graph of the given polynomial function f(x)=2x^5 + 7x^2 - 3x + 5. choose the correct answer below. a. the graph of f(x) falls to the left and rises to the right. b. the graph of f(x) falls to the left and falls to the right. c. the graph of f(x) rises to the left and falls to the right. d. the graph of f(x) rises to the left and rises to the right.

Explanation:

Step1: Identify degree and leading coefficient

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

Step2: Apply leading - coefficient test

For a polynomial $y = a_nx^n+\cdots+a_0$, if $n$ is odd and $a_n>0$, as $x\to-\infty$, $y\to-\infty$ (graph falls to the left) and as $x\to+\infty$, $y\to+\infty$ (graph rises to the right). Here, $n = 5$ (odd) and $a=2>0$.

Answer:

A. The graph of $f(x)$ falls to the left and rises to the right