Question

find the correct end behavior diagram for the given polynom
p(x) = 7x3 + 2x2 - 2x + 6
a

b

c

d

Explanation:

Step1: Determine Degree and Leading Coefficient

The polynomial is \( P(x) = 7x^3 + 2x^2 - 2x + 6 \). The degree (highest power of \( x \)) is 3 (odd), and the leading coefficient (coefficient of \( x^3 \)) is 7 (positive).

Step2: Analyze End Behavior

For polynomials:

• If degree is odd and leading coefficient is positive:
• As \( x \to +\infty \), \( P(x) \to +\infty \) (right end goes up).
• As \( x \to -\infty \), \( P(x) \to -\infty \) (left end goes down).

Looking at the options, the diagram with left end down and right end up matches this behavior. From the options, this corresponds to the diagram for option D (assuming the symbols: left end arrow down, right end arrow up). Wait, wait, let's recheck. Wait, maybe the symbols: let's parse the arrows. If D has left arrow up? No, wait, let's re-express. Wait, the standard end behavior for odd degree with positive leading coefficient: when \( x \to \infty \), \( y \to \infty \); when \( x \to -\infty \), \( y \to -\infty \). So the left end (as \( x \) approaches \( -\infty \)) should go down, and right end (as \( x \) approaches \( +\infty \)) should go up. So which option has that? Let's see the options:

• A: Maybe left up, right up? No.
• B: Left down, right down? No (even degree or negative leading coefficient).
• C: Left up, right up? No (even degree).
• D: Left up? No, wait, maybe the arrows: if D is left arrow up? No, wait, maybe the original symbols: let's assume the arrows are:

For odd degree, positive leading coefficient: \(
earrow \) on the right, \( \searrow \) on the left. So the diagram with left end down (arrow down) and right end up (arrow up). So among the options, D should be the one with left down and right up? Wait, maybe the user's diagram for D is left arrow up? No, maybe I misread. Wait, the polynomial is cubic, leading coefficient positive. So end behavior: as \( x \to \infty \), \( P(x) \to \infty \); as \( x \to -\infty \), \( P(x) \to -\infty \). So the graph should go down on the left (as \( x \) decreases) and up on the right (as \( x \) increases). So the correct diagram is the one with left end down and right end up. So if D has that, then D is correct.

Answer:

D (assuming the diagram for D has left end down and right end up, matching the end behavior of \( P(x) = 7x^3 + 2x^2 - 2x + 6 \))