Determine the end behavior for $y=10 x^{9}\left(6 x^{4}+10 x\right)$. As $x \rightarrow-\infty, y \rightarrow$ help (numbers) As $x \rightarrow \infty, y \rightarrow$ help (numbers)
Solution
Step 1 ：The end behavior of a function can be determined by looking at the highest degree term in the function. In this case, the highest degree term is \(60x^{13}\), which is obtained by multiplying \(10x^9\) and \(6x^4\).
Step 2 ：As \(x\) approaches negative infinity, the value of \(y\) will depend on the sign of the highest degree term. Since the degree is odd, the sign of \(y\) will be negative if \(x\) is negative.
Step 3 ：As \(x\) approaches positive infinity, the value of \(y\) will also depend on the sign of the highest degree term. Since the degree is odd, the sign of \(y\) will be positive if \(x\) is positive.
Step 4 ：Therefore, as \(x \rightarrow-\infty, y \rightarrow -\infty\) and as \(x \rightarrow \infty, y \rightarrow \infty\).
Step 5 ：Final Answer: As \(x \rightarrow-\infty, y \rightarrow \boxed{-\infty}\) and as \(x \rightarrow \infty, y \rightarrow \boxed{\infty}\)
