Question

example 2: sketch a polynomial function with the following limits: a) $lim_{x
ightarrow-infty}f(x)=-infty$, $lim_{x
ightarrowinfty}f(x)=-infty$. polynomial end behavior. for polynomial equations, it is easiest to find the

Explanation:

Step1: Analyze end - behavior rules

For a polynomial function \(y = a_nx^n+\cdots+a_0\), if \(n\) is even and \(a_n<0\), \(\lim_{x
ightarrow-\infty}f(x)=-\infty\) and \(\lim_{x
ightarrow\infty}f(x)=-\infty\). We can choose a simple even - degree polynomial, say \(y=-x^2\).

Step2: Sketch the function

The function \(y = -x^2\) is a parabola opening downwards. Its vertex is at the origin \((0,0)\). We plot a few points: when \(x = - 2\), \(y=-4\); when \(x=-1\), \(y = - 1\); when \(x = 1\), \(y=-1\); when \(x = 2\), \(y=-4\). Then we connect these points with a smooth curve.

Answer:

Sketch a parabola opening downwards with vertex at the origin. For example, the graph of \(y=-x^2\) which passes through points \((-2, - 4),(-1,-1),(0,0),(1,-1),(2,-4)\) and has the given end - behavior \(\lim_{x
ightarrow-\infty}f(x)=-\infty\) and \(\lim_{x
ightarrow\infty}f(x)=-\infty\).