Let's analyze the end behavior of each polynomial function given:
1. Polynomial: [tex]\( p(x) = 3x^2 - 7 \)[/tex]
- Degree of the polynomial: 2 (The highest power of [tex]\( x \)[/tex] is 2)
- Leading coefficient: 3 (The coefficient of the highest power of [tex]\( x \)[/tex])
For polynomials with an even degree:
- If the leading coefficient is positive, both ends of the polynomial will rise.
- If the leading coefficient is negative, both ends of the polynomial will fall.
Since the degree is 2 (even) and the leading coefficient is 3 (positive), the end behavior is:
- Rises right, rises left
2. Polynomial: [tex]\( p(x) = 2x^5 + 8 \)[/tex]
- Degree of the polynomial: 5 (The highest power of [tex]\( x \)[/tex] is 5)
- Leading coefficient: 2 (The coefficient of the highest power of [tex]\( x \)[/tex])
For polynomials with an odd degree:
- If the leading coefficient is positive, the polynomial will rise to the right and fall to the left.
- If the leading coefficient is negative, the polynomial will fall to the right and rise to the left.
Since the degree is 5 (odd) and the leading coefficient is 2 (positive), the end behavior is:
- Rises right, falls left
3. Polynomial: [tex]\( p(x) = -3x^3 + 2x^2 - x - 1 \)[/tex]
- Degree of the polynomial: 3 (The highest power of [tex]\( x \)[/tex] is 3)
- Leading coefficient: -3 (The coefficient of the highest power of [tex]\( x \)[/tex])
For polynomials with an odd degree:
- If the leading coefficient is positive, the polynomial will rise to the right and fall to the left.
- If the leading coefficient is negative, the polynomial will fall to the right and rise to the left.
Since the degree is 3 (odd) and the leading coefficient is -3 (negative), the end behavior is:
- Falls right, rises left
4. Polynomial: [tex]\( p(x) = -4x^4 + 4x^2 - 3 \)[/tex]
- Degree of the polynomial: 4 (The highest power of [tex]\( x \)[/tex] is 4)
- Leading coefficient: -4 (The coefficient of the highest power of [tex]\( x \)[/tex])
For polynomials with an even degree:
- If the leading coefficient is positive, both ends of the polynomial will rise.
- If the leading coefficient is negative, both ends of the polynomial will fall.
Since the degree is 4 (even) and the leading coefficient is -4 (negative), the end behavior is:
- Falls right, falls left
So, the end behaviors for each polynomial are:
1. [tex]\( p(x) = 3x^2 - 7 \)[/tex] ⟹ Rises right, rises left
2. [tex]\( p(x) = 2x^5 + 8 \)[/tex] ⟹ Rises right, falls left
3. [tex]\( p(x) = -3x^3 + 2x^2 - x - 1 \)[/tex] ⟹ Falls right, rises left
4. [tex]\( p(x) = -4x^4 + 4x^2 - 3 \)[/tex] ⟹ Falls right, falls left
