Question

which is the correct right end behavior of the function? (graph it)
y = 3x(x - 4)(x - 1)²
options (partially visible):
y → -∞ as x → -∞
y → -∞ as x → ∞
y → ∞ as x → ∞
y → ∞ as x → -∞

Explanation:

Step1: Determine the leading term

First, we expand the function \( y = 3x(x - 4)(x - 1)^2 \) to find the leading term. The leading term of a polynomial is the term with the highest degree. When we multiply out the factors, the highest - degree term comes from multiplying the highest - degree terms of each factor.

The factor \( x \) has degree 1, \( (x - 4) \) has degree 1, and \( (x - 1)^2=x^{2}-2x + 1\) has degree 2. So the degree of the polynomial is \( 1+1 + 2=4\)? Wait, no, wait: \( y=3x(x - 4)(x - 1)^2=3x(x - 4)(x^{2}-2x + 1)\).

Multiply \( x(x - 4)=x^{2}-4x\), then multiply \((x^{2}-4x)(x^{2}-2x + 1)=x^{2}(x^{2}-2x + 1)-4x(x^{2}-2x + 1)=x^{4}-2x^{3}+x^{2}-4x^{3}+8x^{2}-4x=x^{4}-6x^{3}+9x^{2}-4x\). Then multiply by \( 3x\): \( y = 3x(x^{4}-6x^{3}+9x^{2}-4x)=3x^{5}-18x^{4}+27x^{3}-12x^{2}\).

So the leading term is \( 3x^{5}\), and the degree of the polynomial \( n = 5\) (odd) and the leading coefficient \( a = 3\) (positive).

Step2: Analyze the end - behavior of the polynomial

For a polynomial function \( y=a_{n}x^{n}+a_{n - 1}x^{n - 1}+\cdots+a_{1}x + a_{0}\), the end - behavior is determined by the leading term \( a_{n}x^{n}\):

• If \( n \) is odd:
• If \( a_{n}>0\), as \( x

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

• If \( a_{n}<0\), as \( x

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

Since our leading term is \( 3x^{5}\), \( n = 5\) (odd) and \( a_{n}=3>0\). So as \( x
ightarrow\infty\) (right - end behavior), \( y
ightarrow\infty\).

Answer:

As \( x
ightarrow\infty\), \( y
ightarrow\infty\) (the option with \( y
ightarrow\infty \) as \( x
ightarrow\infty\))