Graph Polynomial Functions and End Behavior - Algebra 2

1. Which end behavior matches $f(x)=-2x^6+5x$?

   As $x\to\pm\infty$, $f(x)\to-\infty$. Even degree $6$ with negative leading coefficient $-2$.

2. Identify the multiplicity of the zero $x=2$ in $f(x)=(x-2)^3(x+1)$.

   Multiplicity $3$. The factor $(x-2)^3$ has exponent $3$.

3. Identify the zeros of $f(x)=(x+5)^2(x-1)$.

   $x=-5$ (mult. $2$) and $x=1$ (mult. $1$). The squared factor gives multiplicity $2$, the linear factor gives multiplicity $1$.

4. Identify whether the graph crosses or bounces at $x=0$ for $f(x)=x^5(x-2)$.

   Crosses at $x=0$. Odd multiplicity $5$ causes the graph to cross the axis.

5. Which end behavior matches degree $7$ with leading coefficient $2$?

   As $x\to-\infty$, $f(x)\to-\infty$; $x\to\infty$, $f(x)\to\infty$. Odd degree with positive leading coefficient.

6. What is the multiplicity of a zero $r$ if $(x-r)^m$ is a factor of $f(x)$?

   Multiplicity is $m$. The exponent on the factor $(x-r)$ equals the multiplicity.

7. What does it indicate if $f(x)$ is written as $a\prod (x-r_i)^{m_i}$?

   Zeros are $r_i$ with multiplicities $m_i$. Factored form reveals each zero $r_i$ and its multiplicity $m_i$.

8. What are the real zeros of $f(x)=x(x-7)(x+2)$?

   $x=0$, $x=7$, and $x=-2$. Set each factor equal to zero and solve.

9. Identify whether the graph crosses or bounces at $x=-4$ for $f(x)=(x+4)^2(x-1)$.

   Bounces (touches and turns) at $x=-4$. Even multiplicity $2$ causes the graph to touch and turn around.

10. What is the zero of the factor $(x-\frac{1}{3})$?

    $x=\frac{1}{3}$. Set $x-\frac{1}{3}=0$ and solve for $x$.

11. What is the zero of the factor $(x+7)$?

    $x=-7$. Set $x+7=0$ and solve for $x$.

12. Identify the zeros of $f(x)=x^3-4x^2=x^2(x-4)$.

    $x=0$ (mult. $2$) and $x=4$ (mult. $1$). Factor out $x^2$ to reveal the zeros and their multiplicities.

13. Identify the zeros of $f(x)=(2x-6)(x+4)$.

    $x=3$ and $x=-4$. From $2x-6=0$, solve $x=3$; from $x+4=0$, solve $x=-4$.

14. Which end behavior matches $f(x)=-x^3+10x$?

    As $x\to-\infty$, $f(x)\to\infty$; $x\to\infty$, $f(x)\to-\infty$. Odd degree $3$ with negative leading coefficient $-1$.

15. What is the end behavior of $f(x)=a_nx^n+\dots$ determined by?

    The leading term $a_nx^n$. The highest degree term dominates behavior as $x$ approaches infinity.

16. Identify the end behavior of $f(x)=2(x-1)^2(x+3)$.

    As $x\to-\infty$, $f(x)\to-\infty$; $x\to\infty$, $f(x)\to\infty$. Odd degree $3$ with positive leading coefficient $2$.

17. Which factor gives the zero $x=-\frac{3}{2}$ in factored form?

    $(2x+3)$. Set $2x+3=0$ to find the zero $x=-\frac{3}{2}$.

18. What is the $x$-intercept coordinate of a zero $r$ of $f(x)$?

    $(r,0)$. Zeros occur where the graph crosses or touches the $x$-axis.

19. Identify the zeros of $f(x)=(x-3)(x+2)$.

    $x=3$ and $x=-2$. Set each factor equal to zero: $x-3=0$ and $x+2=0$.

20. What is the zero of $f(x)=(5-x)(x+1)$ coming from the factor $(5-x)$?

    $x=5$. Set $5-x=0$ to get $x=5$.

21. What is the $y$-intercept of $f(x)=-3(x+1)^2(x-2)$?

    $(0,6)$. Evaluate $f(0)=-3(0+1)^2(0-2)=-3(1)(-2)=6$.

22. What is the sign of $f(x)=-(x-2)^2(x+1)$ for very large positive $x$?

    $f(x)<0$ for large positive $x$. The negative leading coefficient dominates for large positive $x$.

23. Which end behavior matches $f(x)=3x^5-x^2+4$?

    As $x\to-\infty$, $f(x)\to-\infty$; $x\to\infty$, $f(x)\to\infty$. Odd degree $5$ with positive leading coefficient $3$.

24. Identify the $x$-intercepts of $f(x)=2(x-4)(x+1)$.

    $(4,0)$ and $(-1,0)$. The zeros $x=4$ and $x=-1$ correspond to $x$-intercepts.

25. Which end behavior matches degree $9$ with leading coefficient $-1$?

    As $x\to-\infty$, $f(x)\to\infty$; $x\to\infty$, $f(x)\to-\infty$. Odd degree with negative leading coefficient.

26. Identify the end behavior of $f(x)=-2(x-1)^2(x+2)^2$.

    As $x\to\pm\infty$, $f(x)\to-\infty$. Even degree $4$ with negative leading coefficient $-2$.

27. Identify the end behavior of $f(x)=(x-1)^2(x+2)^2$.

    As $x\to\pm\infty$, $f(x)\to\infty$. Even degree $4$ with positive leading coefficient $1$.

28. What is the zero of the factor $(4x-1)$?

    $x=\frac{1}{4}$. Set $4x-1=0$ to get $4x=1$, so $x=\frac{1}{4}$.

29. Identify the end behavior of $f(x)=-(x-1)(x+2)(x-3)(x+4)$.

    As $x\to\pm\infty$, $f(x)\to-\infty$. Even degree $4$ with negative leading coefficient $-1$.

30. What is the end behavior when $n$ is even and $a_n<0$?

    As $x\to\pm\infty$, $f(x)\to-\infty$. Even degree with negative leading coefficient creates downward parabola-like ends.

31. Identify the zeros of $f(x)=(x^2-16)(x^2-1)$.

    $x=-4$, $x=4$, $x=-1$, and $x=1$. Factor each quadratic: $(x-4)(x+4)(x-1)(x+1)$.

32. Identify the zeros of $f(x)=(x^2-4x+4)(x+1)$.

    $x=2$ (mult. $2$) and $x=-1$ (mult. $1$). Factor $x^2-4x+4=(x-2)^2$ to identify the repeated zero.

33. What happens at $x=r$ if $r$ is a zero of odd multiplicity?

    The graph crosses the $x$-axis at $x=r$. Odd multiplicities cause the graph to pass through the $x$-axis.

34. How many times does $f(x)=(x-1)^2(x+3)$ cross the $x$-axis?

    Once (at $x=-3$). Only the zero with odd multiplicity causes a crossing.

35. Which end behavior matches degree $6$ with leading coefficient $-3$?

    As $x\to\pm\infty$, $f(x)\to-\infty$. Even degree with negative leading coefficient.

36. How many $x$-intercepts does $f(x)=(x-1)^2(x+3)^2$ have?

    $2$ $x$-intercepts: $x=1$ and $x=-3$. Each distinct zero corresponds to one $x$-intercept regardless of multiplicity.

37. What is the maximum number of real zeros a degree $n$ polynomial can have?

    At most $n$ real zeros. The Fundamental Theorem of Algebra limits real zeros to the degree.

38. What happens at $x=r$ if $r$ is a zero of even multiplicity?

    The graph touches and turns at $x=r$. Even multiplicities cause the graph to bounce off the $x$-axis.

39. Which end behavior matches $f(x)=x^4-7x^2+1$?

    As $x\to\pm\infty$, $f(x)\to\infty$. Even degree $4$ with positive leading coefficient $1$.

40. Identify the zeros of $f(x)=-x^2(x-4)^3$ and their multiplicities.

    $x=0$ (mult. $2$) and $x=4$ (mult. $3$). Count multiplicities from the exponents on each factor.

41. What is the end behavior when $n$ is odd and $a_n>0$?

    As $x\to-\infty$, $f(x)\to-\infty$; $x\to\infty$, $f(x)\to\infty$. Odd degree with positive leading coefficient rises from left to right.

42. Identify the zeros of $f(x)=(x^2-9)(x-1)$ using factoring.

    $x=-3$, $x=3$, and $x=1$. Factor $x^2-9=(x-3)(x+3)$ then set each factor to zero.

43. How many real zeros (counting multiplicity) does $f(x)=(x-2)^2(x+1)^3$ have?

    $5$ zeros counting multiplicity. Count each zero according to its multiplicity: $2+3=5$.

44. What is the end behavior when $n$ is even and $a_n>0$?

    As $x\to\pm\infty$, $f(x)\to\infty$. Even degree with positive leading coefficient creates upward parabola-like ends.

45. Identify the zeros of $f(x)=(x^2+5x)(x-2)$ using factoring.

    $x=0$, $x=-5$, and $x=2$. Factor $x^2+5x=x(x+5)$ then set factors equal to zero.

46. What is the maximum number of turning points a degree $n$ polynomial can have?

    At most $n-1$ turning points. Turning points occur where the derivative equals zero, limited by degree.

47. What is the leading coefficient of $f(x)=-4(x-1)^2(x+3)^3$?

    $-4$. The coefficient of the expanded leading term is $-4$.

48. What is the $y$-intercept of a polynomial function $f(x)$ in terms of $f$?

    $(0,f(0))$. Set $x=0$ and evaluate the function to find where it crosses the $y$-axis.

49. What is the degree and leading coefficient of $f(x)=-x(x-2)^2(x+1)$?

    Degree $4$, leading coefficient $-1$. Sum all exponents: $1+2+1=4$; coefficient is $-1$.

50. What does the Factor Theorem state for $f(x)$ and a number $r$?

    $f(r)=0$ iff $(x-r)$ is a factor of $f(x)$. A zero exists if and only if its corresponding linear factor divides $f(x)$.

51. What is the definition of a zero (root) of a polynomial function $f(x)$?

    A number $r$ such that $f(r)=0$. When $f(r)=0$, the function equals zero at that input.

52. What is the degree of $f(x)=2(x-1)^4(x+5)$?

    Degree $5$. Sum the exponents: $4+1=5$.

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

    $(0,-6)$. Evaluate $f(0)=(0-2)(0+3)=(-2)(3)=-6$.

54. What is the end behavior when $n$ is odd and $a_n<0$?

    As $x\to-\infty$, $f(x)\to\infty$; $x\to\infty$, $f(x)\to-\infty$. Odd degree with negative leading coefficient falls from left to right.

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