Question

(a) Determine the end behavior of the graph of the function

Answer 1

Answer

$y=x^4$

EXPLANATION

Problem Statement

The question gives us a polynomial:

$f(x)=x^2(x-10)(x+10)$

We are asked to find the end behavior of the function for large values of |x|.

Solution

To solve this question, we simply need to expand the function.

$\begin{gathered} f(x)=x^2(x-10)(x+10) \\ f(x)=x^2(x^2-100) \\ f(x)=x^4-100x^2 \end{gathered}$

For large values of x,

$\begin{gathered} x^4>100x^2 \\ \text{Divide both sides by }x^2 \\ \frac{x^4}{x^2}>\frac{100x^2}{x^2} \\ \\ x^2>100 \\ x^2>10^2 \\ x>10 \\ \\ \text{Thus, for any value of x > 10,} \\ x^4>100x^2\text{ holds true.} \\ \\ \text{This means that we can choose a sufficiently large enough }x\text{ to make the influence of }100x^2 \\ \text{negligible.} \\ \\ f(x)\approx x^4,\text{ for very large values of x.} \end{gathered}$

Final Answer

$y=x^4$