Question

What is the end behavior of the graph of the polynomial function y = 10x9 – 4x?

Answer 1

Answer:

The end behavior is to grow

Step-by-step explanation:

The first step is to identify the zeros of the function, it means, the values of x at which the function becomes zero. For achieving that, it necessary to factorize.

$f(x)=10x^9-4x$

$f(x)=x(10x^8-4)$

According to the previous, the zeros are:

• $x=0$
• $x=(4/10)^{1/8}$

If we replace those values in $f(x)$, we obtain:

• $f(x=0)=0$
• $f(x=(4/10)^{1/8})=0$

Now, imagine the following two situations:

• When x is extremely large with negative sign, or when x tends to $-\infty$: In that case, the equation would be:

$f(-\infty )=-\infty (10\cdot (-\infty )^8-4)$

The term $(-\infty )^8/[tex] equals [tex]\infty$ because the sign (-) is also raised to the power 8. The equation would be:

$f(-\infty )=-\infty (10\cdot (\infty )-4)$

If you multiply $\infty$ by 10 and subtract 4, the result is still $\infty$. The equation would be:

$f(-\infty )=-\infty \cdot (\infty )$

The only important thing in the previous expression is the multiplication of the signs, it means, a plus and a minus make a minus. So, $f(-\infty )=-\infty$, it means, THE GRAPH TENDS TO DECREASE WHEN X TENDS TO NEGATIVE INFINITIVE

• The second situation occurs when x is extremely large with positive sign, or when x tends to $\infty$: In that case, the equation would be:

$f(\infty )=\infty (10\cdot (\infty )^8-4)$

If you multiply $\infty$ by 10 and subtract 4, the result is still $\infty$. The equation would be:

$f(\infty )=\infty \cdot (\infty )$

The only important thing in the previous expression is the multiplication of the signs, it means, two pluses make a plus. So, $f(\infty )=\infty$, it means, THE GRAPH TENDS TO GROW WHEN X TENDS TO POSITIVE INFINITIVE

Thus, if you start giving arbitrary values to x, greater than $(4/10)^{1/8}=0.8917$, the value of $f(x)$ becomes greater. It means that the end behavior of the graph is to grow.

Please find attached the graph of the equation

Answer 2

Answer:

See below.

Step-by-step explanation:

The value of the highest degree ( x^9)  is 9 - odd.

So this well rise from negative infinity on the left and rise to positive infinity on the right - or, putting it in a different way, fall to the left and rise to the right.