Consider the leading term of the polynomial function. What is the end behavior of the graph?
$$2x^7 – 8x^6 – 3x^5 – 3$$
What are the steps to do this?
I know with polynomial functions it goes from left to right, so would it be down then up?
My reasoning:
Since the term $2x^7$ is the largest term and the exponent is positive the right side of the graph would go up, but I have no clue on how to determine the left side of the graph.
$\endgroup$ 13 Answers
$\begingroup$$(-x)^7=-(x^7)$, so it will tend to $-\infty$ as $x\to -\infty$. This is true for any polynomial of odd degree, where the leading coefficient is positive.
$\endgroup$ $\begingroup$You can also factor the leading term and look at the function when it goes to infinity: $ \lim_{x \to \pm\infty}x^7(2 – \frac{8}{x} – \frac{3}{x^2} –\frac{3}{x^7}).$ I think this gives a better intuition of what happens at infinities.
$\endgroup$ $\begingroup$I don't know how to comment on the leading term and how it effects the graph, but if you want to get a complete picture of the graph you should consider carrying out the first derivative test to find the critical points and how the function behaves in the intervals separated by those critical points.
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