Question

sketch the graph of a polynomial func

1. \\(\lim\limits_{x\to -\infty} p(x) = -\infty\\) and\\(\lim\limits_{x\to \infty} p(x) = \infty\\)

Explanation:

Step1: Identify polynomial degree parity

The end behavior $\lim_{x\to-\infty} p(x) = -\infty$ and $\lim_{x\to\infty} p(x) = \infty$ matches polynomials with odd degree and positive leading coefficient. For example, $p(x)=x$ (degree 1) or $p(x)=x^3$ (degree 3).

Step2: Choose simple polynomial example

Select $p(x)=x$, a basic odd-degree polynomial with positive leading coefficient.

Step3: Plot key end behavior

• As $x\to\infty$, $p(x)\to\infty$ (graph rises right).
• As $x\to-\infty$, $p(x)\to-\infty$ (graph falls left).
• The line passes through the origin $(0,0)$ and has a constant positive slope.

Answer:

A valid graph is a straight line (or any odd-degree polynomial curve) that rises to the right, falls to the left, and is continuous with no breaks. For the simple case of $p(x)=x$, the graph is a diagonal line passing through the origin, extending upward infinitely as $x\to\infty$ and downward infinitely as $x\to-\infty$.