Polynomial Functions and End Behavior - AP Precalculus

1. What is the end behavior of $f(x) = -4x^3 + 2x - 5$ as $x \to -\text{infinity}$?

   $f(x) \to \text{infinity}$. Odd degree with negative leading coefficient: $x \to -\infty$ gives $f(x) \to +\infty$.

2. What is the end behavior of $f(x) = x^7 - x^5 + x$ as $x \to \text{infinity}$?

   $f(x) \to \text{infinity}$. Odd degree with positive leading coefficient goes to $+\infty$.

3. State the polynomial type for $f(x) = x^3 - 4x + 2$.

   Cubic polynomial. Degree 3 polynomials are called cubic.

4. What is the end behavior of $f(x) = 3x^2 - 7x + 4$ as $x \to \text{infinity}$?

   $f(x) \to \text{infinity}$. Even degree with positive leading coefficient goes to $+\infty$.

5. Determine the multiplicity of root $x = -1$ in $f(x) = (x+1)^2(x-2)$.

   2. The exponent on factor $(x+1)$.

6. What is the end behavior of $f(x) = -3x^4 + 6x^2 - 9$ as $x \to \text{infinity}$?

   $f(x) \to -\text{infinity}$. Even degree with negative leading coefficient goes to $-\infty$.

7. What is the y-intercept of $f(x) = 2x^4 - 3x^2 + 7$?

   7. Evaluate $f(0)$ to find y-intercept.

8. What is the end behavior of $f(x) = 5x^2 - 3x + 1$ as $x \to \text{infinity}$?

   $f(x) \to \text{infinity}$. Even degree with positive leading coefficient goes to $+\infty$.

9. State the end behavior of $f(x) = -x^6 + 2x^4 - 5$ as $x \to \text{infinity}$.

   $f(x) \to -\text{infinity}$. Even degree with negative leading coefficient goes to $-\infty$.

10. Identify the y-intercept in $f(x) = 6x^4 - x^2 + 2$.

    2. Evaluate $f(0)$ to find y-intercept.

11. What is the leading coefficient of $f(x) = 10x^3 - 5x^2 + 7$?

    10. The coefficient of the highest degree term.

12. What is the axis of symmetry for $f(x) = x^2 - 4x + 3$?

    $x = 2$. For quadratic $ax^2 + bx + c$, axis is $x = -\frac{b}{2a}$.

13. What is the leading term of $f(x) = x^8 - 3x^5 + 2x - 9$?

    $x^8$. The term with highest degree and its coefficient.

14. Identify the constant term in $f(x) = 4x^3 - x^2 + 6x - 9$.

    -9. The term without any variable.

15. State the end behavior of $f(x) = x^6 - 4x^3 + 5$ as $x \to -\text{infinity}$.

    $f(x) \to \text{infinity}$. Even degree with positive leading coefficient goes to $+\infty$.

16. State the y-intercept of $f(x) = -3x^3 + 4x^2 - x + 6$.

    6. Evaluate $f(0)$ to find y-intercept.

17. What is the degree of $f(x) = 4x^5 - 9$?

    5. The highest power of x.

18. Determine the degree of $f(x) = 9x^6 - 7x^4 + 3x^2$.

    6. The highest power of x.

19. Find the leading coefficient of $f(x) = 7x^5 - 4x^3 + x^2 - 8$.

    7. The coefficient of the highest degree term.

20. What is the end behavior of $f(x) = -x^5 + 3x^2 - 7$ as $x \to -\text{infinity}$?

    $f(x) \to \text{infinity}$. Odd degree with negative leading coefficient: $x \to -\infty$ gives $f(x) \to +\infty$.

21. What is the degree of $f(x) = -x^2 + 4x - 1$?

    2. The highest power of x.

22. What is the degree of the polynomial $f(x) = 3x^4 - 5x^3 + 2x^2 - x + 7$?

    4. The highest power of x determines the degree.

23. State the number of real roots for $f(x) = x^4 - 1$.

    2. Factor as $(x^2-1) = (x-1)(x+1)$ to find roots $\pm 1$.

24. What is the degree of $f(x) = x^7 - 2x^4 + x^2$?

    7. The highest power of x.

25. State the end behavior of $f(x) = x^9 - x^5$ as $x \to -\text{infinity}$.

    $f(x) \to -\text{infinity}$. Odd degree with positive leading coefficient: $x \to -\infty$ gives $f(x) \to -\infty$.

26. Identify the constant term in $f(x) = 5x^3 - 2x + 7$.

    7. The term without any variable.

27. State the polynomial function given roots $x = 1$, $x = -2$, and $x = 3$.

    $(x-1)(x+2)(x-3)$. Use $(x-r)$ for each root r.

28. Identify the degree of polynomial $f(x) = -5x^7 + 2x^4 - x$.

    $7$. The highest power of x.

29. State the leading coefficient of $f(x) = -8x^4 + 2x^3 - x + 1$.

    -8. The coefficient of the highest degree term.

30. State the constant term in $f(x) = 3x^5 - 5x^3 + 2$.

    2. The term without any variable.

31. Find the end behavior of $f(x) = 7x^2 - 4x + 1$ as $x \to -\text{infinity}$.

    $f(x) \to \text{infinity}$. Even degree with positive leading coefficient goes to $+\infty$.

32. Identify the leading term in $f(x) = 6x^7 - 3x^5 + 2x - 8$.

    $6x^7$. The term with highest degree and its coefficient.

33. Determine the multiplicity of root $x = 2$ in $f(x) = (x-2)^3(x+1)$.

    3. The exponent on factor $(x-2)$.

34. State the leading coefficient of $f(x) = -2x^3 + 4x^2 - x + 5$.

    -2. The coefficient of the highest degree term.

35. What is the end behavior of $f(x) = 2x^3 - 4x + 1$ as $x \to \text{infinity}$?

    $f(x) \to \text{infinity}$. Odd degree with positive leading coefficient goes to $+\infty$.

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