Question

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

1. goes if the leading coefficient is
2. goes if the leading coefficient is

the left side:

1. goes in the direction as the right if the degree is
2. goes in the direction as the right if the degree is

example 3: determine the end behavior for the following polynomials. limit notation is not ne
a) $f(x)=4x^{5}$
l: r:
b) $g(x)=\frac{1}{2}x^{4}$
l: r:
c) $y = - 2$
l: r:
d) $h(x)=3 - x^{5}$
l: r:
e) $k(x)=8x^{2}+4 - x^{5}$
l: r:
f) $m(x)=$

Explanation:

Step1: Recall right - hand end - behavior rules

For a polynomial \(y = a_nx^n+\cdots+a_0\), when considering the right - hand side (\(x\to+\infty\)), if the leading coefficient \(a_n>0\), the polynomial goes to \(+\infty\). If \(a_n < 0\), the polynomial goes to \(-\infty\).

Step2: Recall left - hand end - behavior rules

For the left - hand side (\(x\to-\infty\)), if the degree \(n\) of the polynomial is even, the left - hand end - behavior is the same as the right - hand end - behavior. If the degree \(n\) is odd, the left - hand end - behavior is the opposite of the right - hand end - behavior.

Example 3 a) \(f(x)=4x^5\)

• Left (L): The degree \(n = 5\) (odd) and leading coefficient \(a = 4>0\). As \(x\to-\infty\), \(y\to-\infty\).
• Right (R): As \(x\to+\infty\), since \(a = 4>0\), \(y\to+\infty\).

Example 3 b) \(g(x)=\frac{1}{2}x^4\)

• Left (L): The degree \(n = 4\) (even) and leading coefficient \(a=\frac{1}{2}>0\). As \(x\to-\infty\), \(y\to+\infty\).
• Right (R): As \(x\to+\infty\), since \(a=\frac{1}{2}>0\), \(y\to+\infty\).

Example 3 d) \(h(x)=3 - x^5=-x^5+3\)

• Left (L): The degree \(n = 5\) (odd) and leading coefficient \(a=- 1<0\). As \(x\to-\infty\), \(y\to+\infty\).
• Right (R): As \(x\to+\infty\), since \(a=-1<0\), \(y\to-\infty\).

Example 3 e) \(k(x)=8x^2 + 4-x^5=-x^5+8x^2 + 4\)

• Left (L): The degree \(n = 5\) (odd) and leading coefficient \(a=-1<0\). As \(x\to-\infty\), \(y\to+\infty\).
• Right (R): As \(x\to+\infty\), since \(a=-1<0\), \(y\to-\infty\).

Answer:

Example 3 a) L: \(y\to-\infty\), R: \(y\to+\infty\)
Example 3 b) L: \(y\to+\infty\), R: \(y\to+\infty\)
Example 3 d) L: \(y\to+\infty\), R: \(y\to-\infty\)
Example 3 e) L: \(y\to+\infty\), R: \(y\to-\infty\)