Algebra 2 Flashcards: Graph Polynomial Functions And End Behavior

What this deck covers

This deck focuses on Graph Polynomial Functions And End Behavior, giving you a quick way to review the definitions, rules, and examples that matter most for Algebra 2.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Answer: 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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Answer: 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.