Question

How do you find the end behavior and state the possible number of $x$ intercepts and the value of the $y$ intercept given $-{{x}^{3}}-4x$ ?

Answer 1

Hint: In this question we have been given with a polynomial for which we have to find the end behavior and we have to state the possible number of $x$ intercepts and the $y$ intercepts. Now to find the end behavior of the polynomial, we will look at the degree of the leading coefficient of the polynomial and then deduce its value when $x$ approaches infinity. We will find the $y$ intercept by substituting $x=0$ and the $x$ intercept by substituting $y=0$.

Complete step-by-step solution:
We have the given polynomial as:
$\Rightarrow -{{x}^{3}}-4x$
We will consider the polynomial to be $y$ therefore, it can be written as:
$\Rightarrow y=-{{x}^{3}}-4x\to \left( 1 \right)$
Now to find the end behavior, we need to look at the degree of the leading coefficient of the polynomial. The degree is the value of the exponent on a term. In this case the degree will be the highest power of $x$ which is $-{{x}^{3}}$.
Now consider the term:
$\Rightarrow -{{x}^{3}}$
We will see that as $x\to \pm \infty$
Now as $x\to \infty$ then $-{{x}^{3}}\to -\infty$
And $x\to -\infty$ then $-{{x}^{3}}\to \infty$
Now the $y$ axis intercept occurs where $x=0$:
On substituting $x=0$ in equation $\left( 1 \right)$, we get:
$\Rightarrow y=-{{\left( 0 \right)}^{3}}-4\left( 0 \right)$
On simplifying, we get:
$\Rightarrow y=0$
Now the $x$ axis intercept occurs where $y=0$:
On substituting $y=0$ in equation $\left( 1 \right)$, we get:
$\Rightarrow 0=-{{x}^{3}}-4x$
On transferring the terms from the right-hand side to the left-hand side, we get:
$\Rightarrow {{x}^{3}}+4x=0$
On taking $x$ common, we get:
$\Rightarrow x\left( {{x}^{2}}+4 \right)=0$
We get:
$\Rightarrow x=0$ and ${{x}^{2}}+4=0$, now ${{x}^{2}}+4=0$ has no real solutions. The solution will be only $x=0$.
Therefore, we get the coordinates $\left( x,y \right)$ as $\left( 0,0 \right)$.

We can see that the graph concludes the finding.

Note: It is to be remembered that intercept is a point on the $y$ axis through which the slope of the given line passes. The general method of finding the intercept is substituting the value of $x=0$ and solving for $y$ and then substituting $y=0$ and solving for $x$. The coordinate $\left( x,y \right)$ generated is the intercept.