Question

sketch the graph of a polynomial function that could match each statement. 12. \\(\lim\limits_{x\to -\infty} p(x) = -\infty\\) and \\(\lim\limits_{x\to \infty} p(x) = \infty\\) 13. \\(\lim\limits_{x\to -\infty} p(x) = -\infty\\) and \\(\lim\limits_{x\to \infty} p(x) = -\infty\\) 14. \\(\lim\limits_{x\to -\infty} p(x) = \infty\\) and \\(\lim\limits_{x\to \infty} p(x) = -\infty\\)

Explanation:

Step1: Analyze degree/leading coefficient for Q12

For $\lim_{x \to -\infty} p(x) = -\infty$ and $\lim_{x \to \infty} p(x) = \infty$, the polynomial has odd degree and positive leading coefficient (e.g., $p(x)=x^3$).

Step2: Sketch Q12 graph

As $x\to-\infty$, graph falls to $-\infty$; as $x\to\infty$, graph rises to $\infty$. It crosses the x-axis at least once, with a cubic-like curve shape.

Step3: Analyze degree/leading coefficient for Q13

For $\lim_{x \to -\infty} p(x) = -\infty$ and $\lim_{x \to \infty} p(x) = -\infty$, the polynomial has even degree and negative leading coefficient (e.g., $p(x)=-x^2$).

Step4: Sketch Q13 graph

As $x\to\pm\infty$, graph falls to $-\infty$. It opens downward, with a parabola-like (or higher even degree) curve shape.

Step5: Analyze degree/leading coefficient for Q14

For $\lim_{x \to -\infty} p(x) = \infty$ and $\lim_{x \to \infty} p(x) = -\infty$, the polynomial has odd degree and negative leading coefficient (e.g., $p(x)=-x^3$).

Step6: Sketch Q14 graph

As $x\to-\infty$, graph rises to $\infty$; as $x\to\infty$, graph falls to $-\infty$. It crosses the x-axis at least once, with an inverted cubic-like curve shape.

Answer:

1. For $\boldsymbol{\lim_{x \to -\infty} p(x) = -\infty}$ and $\boldsymbol{\lim_{x \to \infty} p(x) = \infty}$:

Sketch an odd-degree polynomial with positive leading coefficient (e.g., cubic curve): starts at $-\infty$ when $x\to-\infty$, crosses the x-axis, and rises to $\infty$ as $x\to\infty$.

1. For $\boldsymbol{\lim_{x \to -\infty} p(x) = -\infty}$ and $\boldsymbol{\lim_{x \to \infty} p(x) = -\infty}$:

Sketch an even-degree polynomial with negative leading coefficient (e.g., downward-opening parabola): falls to $-\infty$ on both ends of the x-axis, opening downward.

1. For $\boldsymbol{\lim_{x \to -\infty} p(x) = \infty}$ and $\boldsymbol{\lim_{x \to \infty} p(x) = -\infty}$:

Sketch an odd-degree polynomial with negative leading coefficient (e.g., inverted cubic curve): starts at $\infty$ when $x\to-\infty$, crosses the x-axis, and falls to $-\infty$ as $x\to\infty$.