Question

quick question: if you were given the equation ( f(x)=-3x^{5}+6x ), what behavior would the ends of the graph have?
both ends will point up
both ends will point down
the left end will start down at a large negative and the right end will go upwards
the left end will start up at a large positive and the right end will go downwards

Explanation:

Step1: Determine the leading term

The leading term of the polynomial \(f(x)=-3x^{5}+6x\) is \(-3x^{5}\).

Step2: Analyze the end - behavior of a polynomial \(y = ax^{n}\)

For a polynomial \(y = ax^{n}\), when \(n\) is odd:

• If \(a>0\), as \(x

ightarrow+\infty\), \(y
ightarrow+\infty\) and as \(x
ightarrow-\infty\), \(y
ightarrow-\infty\).

• If \(a < 0\), as \(x

ightarrow+\infty\), \(y
ightarrow-\infty\) and as \(x
ightarrow-\infty\), \(y
ightarrow+\infty\).
Here \(n = 5\) (odd) and \(a=-3<0\).
As \(x
ightarrow+\infty\), \(y=-3x^{5}+6x\approx - 3x^{5}
ightarrow-\infty\) (the term \(6x\) is negligible compared to \(-3x^{5}\) for large \(|x|\)).
As \(x
ightarrow-\infty\), \(y=-3x^{5}+6x\approx - 3x^{5}\). Let \(t=-x\), then \(y=-3(-t)^{5}=3t^{5}\) and as \(t
ightarrow+\infty\) (when \(x
ightarrow-\infty\)), \(y = 3t^{5}
ightarrow+\infty\).

Answer:

The left end will start up at a large negative and the right end will go downwards.