Questions: Choose the end behavior of the graph of each polynomial function. (a) f(x)=-4x^3+7x^2+3x-9 Falls to the left and rises to the right Rises to the left and falls to the right Rises to the left and rises to the right Falls to the left and falls to the right (b) f(x)=-4x^6+x^4+8x^3+9x Falls to the left and rises to the right Rises to the left and falls to the right Rises to the left and rises to the right Falls to the left and falls to the right (c) f(x)=2x(x-1)^2(x+3) Falls to the left and rises to the right Rises to the left and falls to the right Rises to the left and rises to the right Falls to the left and falls to the right

Solution Steps

To determine the end behavior of a polynomial function, we need to look at the leading term of the polynomial when it is expanded. The leading term is the term with the highest power of \( x \). The coefficient and the degree of this term will determine the end behavior.

1. For \( f(x) = -4x^3 + 7x^2 + 3x - 9 \):

   • The leading term is \( -4x^3 \).
   • Since the degree is odd and the leading coefficient is negative, the end behavior is: Falls to the left and rises to the right.

2. For \( f(x) = -4x^6 + x^4 + 8x^3 + 9x \):

   • The leading term is \( -4x^6 \).
   • Since the degree is even and the leading coefficient is negative, the end behavior is: Falls to the left and falls to the right.

3. For \( f(x) = 2x(x-1)^2(x+3) \):

   • Expand the polynomial to find the leading term.
   • The leading term is \( 2x^4 \).
   • Since the degree is even and the leading coefficient is positive, the end behavior is: Rises to the left and rises to the right.

For the polynomial \( f(x) = -4x^3 + 7x^2 + 3x - 9 \), the leading term is \( -4x^3 \). Since the degree \( 3 \) is odd and the leading coefficient \( -4 \) is negative, the end behavior is:

• Falls to the left (\( x \to -\infty \)) and rises to the right (\( x \to +\infty \)).

For the polynomial \( f(x) = -4x^6 + x^4 + 8x^3 + 9x \), the leading term is \( -4x^6 \). The degree \( 6 \) is even and the leading coefficient \( -4 \) is negative, so the end behavior is:

• Falls to the left (\( x \to -\infty \)) and falls to the right (\( x \to +\infty \)).

For the polynomial \( f(x) = 2x(x-1)^2(x+3) \), we expand it to find the leading term. The leading term is \( 2x^4 \). The degree \( 4 \) is even and the leading coefficient \( 2 \) is positive, thus the end behavior is:

• Rises to the left (\( x \to -\infty \)) and rises to the right (\( x \to +\infty \)).

Final Answer