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

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

Flashcard 3: 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 4: 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 5: 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 6: 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 7: 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 8: 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 9: 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 10: 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 11: 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 12: 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 13: 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 14: 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 15: 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 16: 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 17: 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 18: 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 19: 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 20: 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 21: 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 22: 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 23: 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 24: 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 25: 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 26: 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.

Flashcard 27: 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 28: 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 29: 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 30: 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: $(x-4)(x+4)(x-1)(x+1)$.