To determine the end behavior of a polynomial function, we need to look at two things:
1. The degree of the polynomial (the highest power of x).
2. The leading coefficient (the number in front of the term with the highest power).
Here’s how it works:
- If the degree is even and the leading coefficient is positive, both ends of the graph will go up.
- If the degree is even and the leading coefficient is negative, both ends of the graph will go down.
- If the degree is odd and the leading coefficient is positive, the left end will go down and the right end will go up.
- If the degree is odd and the leading coefficient is negative, the left end will go up and the right end will go down.
Let's apply this to each function:
1.
f(x) = x³ - 2x² + 3
- Degree: 3 (odd)
- Leading coefficient: 1 (positive)
- End behavior: Left end goes down, right end goes up.
2.
f(x) = 3x⁶ - 7x⁴ - 2x⁹
- Degree: 9 (odd)
- Leading coefficient: -2 (negative)
- End behavior: Left end goes up, right end goes down.
3.
f(x) = -x³ + 3x² - 4
- Degree: 3 (odd)
- Leading coefficient: -1 (negative)
- End behavior: Left end goes up, right end goes down.
4.
f(x) = x⁴ - 4x² + 2x + 4
- Degree: 4 (even)
- Leading coefficient: 1 (positive)
- End behavior: Both ends go up.
5.
f(x) = x⁵ + 3x³ + 3
- Degree: 5 (odd)
- Leading coefficient: 1 (positive)
- End behavior: Left end goes down, right end goes up.
6.
f(x) = -x⁵ + 3x³ + 3
- Degree: 5 (odd)
- Leading coefficient: -1 (negative)
- End behavior: Left end goes up, right end goes down.
7.
f(x) = -x² - 8x - 15
- Degree: 2 (even)
- Leading coefficient: -1 (negative)
- End behavior: Both ends go down.
8.
f(x) = x³ - 3x² + 1
- Degree: 3 (odd)
- Leading coefficient: 1 (positive)
- End behavior: Left end goes down, right end goes up.
9.
f(x) = -2x² + 16x - 29
- Degree: 2 (even)
- Leading coefficient: -2 (negative)
- End behavior: Both ends go down.
10.
f(x) = -x⁴ + x³ - x²
- Degree: 4 (even)
- Leading coefficient: -1 (negative)
- End behavior: Both ends go down.
11.
f(x) = x⁴ - 6x³ + 8x²
- Degree: 4 (even)
- Leading coefficient: 1 (positive)
- End behavior: Both ends go up.
12.
f(x) = -x⁴ + 4x³ - 4x²
- Degree: 4 (even)
- Leading coefficient: -1 (negative)
- End behavior: Both ends go down.
Final Answer:
1. Left end goes down, right end goes up.
2. Left end goes up, right end goes down.
3. Left end goes up, right end goes down.
4. Both ends go up.
5. Left end goes down, right end goes up.
6. Left end goes up, right end goes down.
7. Both ends go down.
8. Left end goes down, right end goes up.
9. Both ends go down.
10. Both ends go down.
11. Both ends go up.
12. Both ends go down.
Parent Tip: Review the logic above to help your child master the concept of worksheet on polynomials.
