# How do you sketch the general shape of f(x)=x^3-2x^2+1 using end behavior?

Sep 30, 2016

${x}^{3} - 2 {x}^{2} + 1 \setminus \to - \setminus \infty$ as $x \setminus \to - \setminus \infty$

${x}^{3} - 2 {x}^{2} + 1 \setminus \to + \setminus \infty$ as $x \setminus \to + \setminus \infty$

#### Explanation:

You need to know that the end behaviour of a polynomial depends on its degree:

• if the degree is even, both limits at $\setminus \pm \setminus \infty$ will be $+ \setminus \infty$
• if the degree is odd, you'll have the limit according to the direction: if $p \left(x\right)$ is your polynomial, then ${\lim}_{x \setminus \to - \setminus \infty} p \left(x\right) = - \setminus \infty$ and ${\lim}_{x \setminus \to + \setminus \infty} p \left(x\right) = + \setminus \infty$

This is easy to explain: an even degree means that you surely are the square of something: ${x}^{2}$ is the square of $x$, ${x}^{4}$ is the square of ${x}^{2}$, and so on. If $n$ is even, ${x}^{n}$ is the square of ${x}^{\frac{n}{2}}$. And since squares are always positive, the limits can only be $+ \setminus \infty$.

On the other hand, you can see an odd power as an even power of $x$ multiplied one more time by $x$. For example, see ${x}^{7}$ as ${x}^{6} \cdot x$.

We already observed that ${x}^{6}$ tends to positive infinity in both direction, so at $- \setminus \infty$ you'll have ${x}^{6} \cdot x \setminus \to \left(+ \setminus \infty\right) \left(- \setminus \infty\right) = - \setminus \infty$, while at $+ \setminus \infty$ you'll have ${x}^{6} \cdot x \setminus \to \left(+ \setminus \infty\right) \left(+ \setminus \infty\right) = + \setminus \infty$.

The reason for which the leading term is the only relevant one is simple, too: let's analyze your case: we have

${x}^{3} - 2 {x}^{2} + 1 = {x}^{3} \left(1 - \frac{2}{x} + \frac{1}{x} ^ 3\right)$

So, if we factor the greatest power of $x$, all the remaining terms will tend to zero as $x$ approaches infinity (in both direction), showing that (in this case) ${x}^{3}$ is the only relevant term to investigate the end behaviour.