Question

which end behavior diagram describes the function, $f(x) = 3x^5 - 4x^4 + 7x^3 - 7$?

Explanation:

Step1: Identify the leading term

The function is \( f(x) = 3x^5 - 4x^4 + 7x^3 - 7 \). The leading term is \( 3x^5 \), which has a degree of 5 (odd) and a leading coefficient of 3 (positive).

Step2: Determine end behavior for odd degree with positive leading coefficient

For a polynomial with an odd degree and a positive leading coefficient:

• As \( x \to +\infty \), \( f(x) \to +\infty \) (because the leading term \( 3x^5 \) dominates, and for large positive \( x \), \( 3x^5 \) is positive and large).
• As \( x \to -\infty \), \( f(x) \to -\infty \) (because for large negative \( x \), \( x^5 \) is negative, and multiplying by the positive coefficient 3 gives a large negative value).

This means the graph should go down to the left (as \( x \to -\infty \)) and up to the right (as \( x \to +\infty \)). Looking at the diagrams, the one with the left arrow down and the right arrow up matches this behavior.

Answer:

The diagram with the left arrow pointing down and the right arrow pointing up (the second option among the given diagrams, typically represented as the one with the curve going from bottom - left to top - right in end behavior).