Question

1. which graph shows the proper end behavior for the equation $y = x^4 + 35x^3 - 23x^2 - 43$ as x becomes a large positive or negative number?

a. graph
b. graph
c. graph
d. graph

Explanation:

Step1: Determine the leading term

The given polynomial is \( y = x^4 + 35x^3 - 23x^2 - 43 \). The leading term is \( x^4 \) (the term with the highest degree).

Step2: Analyze the degree and leading coefficient

The degree of the polynomial (the exponent of the leading term) is 4, which is even. The leading coefficient (the coefficient of the leading term) is 1, which is positive.

Step3: Recall end - behavior rules for polynomials

For a polynomial \( y = a_nx^n+a_{n - 1}x^{n - 1}+\cdots+a_1x + a_0 \):

• If the degree \( n \) is even:
• If the leading coefficient \( a_n>0 \), as \( x

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

• If the leading coefficient \( a_n < 0 \), as \( x

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

• If the degree \( n \) is odd:
• If the leading coefficient \( a_n>0 \), as \( x

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

• If the leading coefficient \( a_n < 0 \), as \( x

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

Since our polynomial has an even degree (4) and a positive leading coefficient (1), as \( x
ightarrow+\infty \), \( y
ightarrow+\infty \) and as \( x
ightarrow-\infty \), \( y
ightarrow+\infty \).

Looking at the graphs:

• Graph A: As \( x

ightarrow+\infty \), \( y
ightarrow+\infty \) and as \( x
ightarrow-\infty \), \( y
ightarrow+\infty \) (matches our end - behavior).

• Graph B: As \( x

ightarrow-\infty \), \( y
ightarrow-\infty \) (does not match, since degree is even and leading coefficient positive).

• Graph C: As \( x

ightarrow-\infty \), \( y
ightarrow-\infty \) and as \( x
ightarrow+\infty \), \( y
ightarrow-\infty \) (does not match).

• Graph D: As \( x

ightarrow+\infty \), \( y
ightarrow-\infty \) and as \( x
ightarrow-\infty \), \( y
ightarrow-\infty \) (does not match, since leading coefficient is positive and degree is even).

Answer:

A