Graph Polynomial Functions and End Behavior - Algebra 2
Card 1 of 30
Identify the zeros of $f(x)=x^3-4x^2=x^2(x-4)$.
Identify the zeros of $f(x)=x^3-4x^2=x^2(x-4)$.
Tap to reveal answer
$x=0$ (mult. $2$) and $x=4$ (mult. $1$). Factor out $x^2$ to reveal the zeros and their multiplicities.
$x=0$ (mult. $2$) and $x=4$ (mult. $1$). Factor out $x^2$ to reveal the zeros and their multiplicities.
← Didn't Know|Knew It →
What is the zero of the factor $(x+7)$?
What is the zero of the factor $(x+7)$?
Tap to reveal answer
$x=-7$. Set $x+7=0$ and solve for $x$.
$x=-7$. Set $x+7=0$ and solve for $x$.
← Didn't Know|Knew It →
Identify the zeros of $f(x)=(x^2-9)(x-1)$ using factoring.
Identify the zeros of $f(x)=(x^2-9)(x-1)$ using factoring.
Tap to reveal answer
$x=-3$, $x=3$, and $x=1$. Factor $x^2-9=(x-3)(x+3)$ then set each factor to zero.
$x=-3$, $x=3$, and $x=1$. Factor $x^2-9=(x-3)(x+3)$ then set each factor to zero.
← Didn't Know|Knew It →
Which end behavior matches $f(x)=x^4-7x^2+1$?
Which end behavior matches $f(x)=x^4-7x^2+1$?
Tap to reveal answer
As $x\to\pm\infty$, $f(x)\to\infty$. Even degree $4$ with positive leading coefficient $1$.
As $x\to\pm\infty$, $f(x)\to\infty$. Even degree $4$ with positive leading coefficient $1$.
← Didn't Know|Knew It →
What is the $y$-intercept of $f(x)=-3(x+1)^2(x-2)$?
What is the $y$-intercept of $f(x)=-3(x+1)^2(x-2)$?
Tap to reveal answer
$(0,6)$. Evaluate $f(0)=-3(0+1)^2(0-2)=-3(1)(-2)=6$.
$(0,6)$. Evaluate $f(0)=-3(0+1)^2(0-2)=-3(1)(-2)=6$.
← Didn't Know|Knew It →
What is the $y$-intercept of $f(x)=(x-2)(x+3)$?
What is the $y$-intercept of $f(x)=(x-2)(x+3)$?
Tap to reveal answer
$(0,-6)$. Evaluate $f(0)=(0-2)(0+3)=(-2)(3)=-6$.
$(0,-6)$. Evaluate $f(0)=(0-2)(0+3)=(-2)(3)=-6$.
← Didn't Know|Knew It →
Identify the $x$-intercepts of $f(x)=2(x-4)(x+1)$.
Identify the $x$-intercepts of $f(x)=2(x-4)(x+1)$.
Tap to reveal answer
$(4,0)$ and $(-1,0)$. The zeros $x=4$ and $x=-1$ correspond to $x$-intercepts.
$(4,0)$ and $(-1,0)$. The zeros $x=4$ and $x=-1$ correspond to $x$-intercepts.
← Didn't Know|Knew It →
Identify the zeros of $f(x)=(x+5)^2(x-1)$.
Identify the zeros of $f(x)=(x+5)^2(x-1)$.
Tap to reveal answer
$x=-5$ (mult. $2$) and $x=1$ (mult. $1$). The squared factor gives multiplicity $2$, the linear factor gives multiplicity $1$.
$x=-5$ (mult. $2$) and $x=1$ (mult. $1$). The squared factor gives multiplicity $2$, the linear factor gives multiplicity $1$.
← Didn't Know|Knew It →
What is the multiplicity of a zero $r$ if $(x-r)^m$ is a factor of $f(x)$?
What is the multiplicity of a zero $r$ if $(x-r)^m$ is a factor of $f(x)$?
Tap to reveal answer
Multiplicity is $m$. The exponent on the factor $(x-r)$ equals the multiplicity.
Multiplicity is $m$. The exponent on the factor $(x-r)$ equals the multiplicity.
← Didn't Know|Knew It →
What is the end behavior of $f(x)=a_nx^n+\dots$ determined by?
What is the end behavior of $f(x)=a_nx^n+\dots$ determined by?
Tap to reveal answer
The leading term $a_nx^n$. The highest degree term dominates behavior as $x$ approaches infinity.
The leading term $a_nx^n$. The highest degree term dominates behavior as $x$ approaches infinity.
← Didn't Know|Knew It →
What are the real zeros of $f(x)=x(x-7)(x+2)$?
What are the real zeros of $f(x)=x(x-7)(x+2)$?
Tap to reveal answer
$x=0$, $x=7$, and $x=-2$. Set each factor equal to zero and solve.
$x=0$, $x=7$, and $x=-2$. Set each factor equal to zero and solve.
← Didn't Know|Knew It →
Identify the zeros of $f(x)=(2x-6)(x+4)$.
Identify the zeros of $f(x)=(2x-6)(x+4)$.
Tap to reveal answer
$x=3$ and $x=-4$. From $2x-6=0$, solve $x=3$; from $x+4=0$, solve $x=-4$.
$x=3$ and $x=-4$. From $2x-6=0$, solve $x=3$; from $x+4=0$, solve $x=-4$.
← Didn't Know|Knew It →
What is the zero of $f(x)=(5-x)(x+1)$ coming from the factor $(5-x)$?
What is the zero of $f(x)=(5-x)(x+1)$ coming from the factor $(5-x)$?
Tap to reveal answer
$x=5$. Set $5-x=0$ to get $x=5$.
$x=5$. Set $5-x=0$ to get $x=5$.
← Didn't Know|Knew It →
Identify the end behavior of $f(x)=-(x-1)(x+2)(x-3)(x+4)$.
Identify the end behavior of $f(x)=-(x-1)(x+2)(x-3)(x+4)$.
Tap to reveal answer
As $x\to\pm\infty$, $f(x)\to-\infty$. Even degree $4$ with negative leading coefficient $-1$.
As $x\to\pm\infty$, $f(x)\to-\infty$. Even degree $4$ with negative leading coefficient $-1$.
← Didn't Know|Knew It →
Identify the end behavior of $f(x)=2(x-1)^2(x+3)$.
Identify the end behavior of $f(x)=2(x-1)^2(x+3)$.
Tap to reveal answer
As $x\to-\infty$, $f(x)\to-\infty$; $x\to\infty$, $f(x)\to\infty$. Odd degree $3$ with positive leading coefficient $2$.
As $x\to-\infty$, $f(x)\to-\infty$; $x\to\infty$, $f(x)\to\infty$. Odd degree $3$ with positive leading coefficient $2$.
← Didn't Know|Knew It →
What happens at $x=r$ if $r$ is a zero of odd multiplicity?
What happens at $x=r$ if $r$ is a zero of odd multiplicity?
Tap to reveal answer
The graph crosses the $x$-axis at $x=r$. Odd multiplicities cause the graph to pass through the $x$-axis.
The graph crosses the $x$-axis at $x=r$. Odd multiplicities cause the graph to pass through the $x$-axis.
← Didn't Know|Knew It →
What is the sign of $f(x)=-(x-2)^2(x+1)$ for very large positive $x$?
What is the sign of $f(x)=-(x-2)^2(x+1)$ for very large positive $x$?
Tap to reveal answer
$f(x)<0$ for large positive $x$. The negative leading coefficient dominates for large positive $x$.
$f(x)<0$ for large positive $x$. The negative leading coefficient dominates for large positive $x$.
← Didn't Know|Knew It →
Which factor gives the zero $x=-\frac{3}{2}$ in factored form?
Which factor gives the zero $x=-\frac{3}{2}$ in factored form?
Tap to reveal answer
$(2x+3)$. Set $2x+3=0$ to find the zero $x=-\frac{3}{2}$.
$(2x+3)$. Set $2x+3=0$ to find the zero $x=-\frac{3}{2}$.
← Didn't Know|Knew It →
What is the zero of the factor $(x-\frac{1}{3})$?
What is the zero of the factor $(x-\frac{1}{3})$?
Tap to reveal answer
$x=\frac{1}{3}$. Set $x-\frac{1}{3}=0$ and solve for $x$.
$x=\frac{1}{3}$. Set $x-\frac{1}{3}=0$ and solve for $x$.
← Didn't Know|Knew It →
Identify the zeros of $f(x)=(x-3)(x+2)$.
Identify the zeros of $f(x)=(x-3)(x+2)$.
Tap to reveal answer
$x=3$ and $x=-2$. Set each factor equal to zero: $x-3=0$ and $x+2=0$.
$x=3$ and $x=-2$. Set each factor equal to zero: $x-3=0$ and $x+2=0$.
← Didn't Know|Knew It →
What is the end behavior when $n$ is even and $a_n<0$?
What is the end behavior when $n$ is even and $a_n<0$?
Tap to reveal answer
As $x\to\pm\infty$, $f(x)\to-\infty$. Even degree with negative leading coefficient creates downward parabola-like ends.
As $x\to\pm\infty$, $f(x)\to-\infty$. Even degree with negative leading coefficient creates downward parabola-like ends.
← Didn't Know|Knew It →
What does it indicate if $f(x)$ is written as $a\prod (x-r_i)^{m_i}$?
What does it indicate if $f(x)$ is written as $a\prod (x-r_i)^{m_i}$?
Tap to reveal answer
Zeros are $r_i$ with multiplicities $m_i$. Factored form reveals each zero $r_i$ and its multiplicity $m_i$.
Zeros are $r_i$ with multiplicities $m_i$. Factored form reveals each zero $r_i$ and its multiplicity $m_i$.
← Didn't Know|Knew It →
What is the maximum number of turning points a degree $n$ polynomial can have?
What is the maximum number of turning points a degree $n$ polynomial can have?
Tap to reveal answer
At most $n-1$ turning points. Turning points occur where the derivative equals zero, limited by degree.
At most $n-1$ turning points. Turning points occur where the derivative equals zero, limited by degree.
← Didn't Know|Knew It →
What is the maximum number of real zeros a degree $n$ polynomial can have?
What is the maximum number of real zeros a degree $n$ polynomial can have?
Tap to reveal answer
At most $n$ real zeros. The Fundamental Theorem of Algebra limits real zeros to the degree.
At most $n$ real zeros. The Fundamental Theorem of Algebra limits real zeros to the degree.
← Didn't Know|Knew It →
What is the $y$-intercept of a polynomial function $f(x)$ in terms of $f$?
What is the $y$-intercept of a polynomial function $f(x)$ in terms of $f$?
Tap to reveal answer
$(0,f(0))$. Set $x=0$ and evaluate the function to find where it crosses the $y$-axis.
$(0,f(0))$. Set $x=0$ and evaluate the function to find where it crosses the $y$-axis.
← Didn't Know|Knew It →
What is the end behavior when $n$ is odd and $a_n<0$?
What is the end behavior when $n$ is odd and $a_n<0$?
Tap to reveal 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.
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.
← Didn't Know|Knew It →
What is the zero of the factor $(4x-1)$?
What is the zero of the factor $(4x-1)$?
Tap to reveal answer
$x=\frac{1}{4}$. Set $4x-1=0$ to get $4x=1$, so $x=\frac{1}{4}$.
$x=\frac{1}{4}$. Set $4x-1=0$ to get $4x=1$, so $x=\frac{1}{4}$.
← Didn't Know|Knew It →
Identify the zeros of $f(x)=(x^2+5x)(x-2)$ using factoring.
Identify the zeros of $f(x)=(x^2+5x)(x-2)$ using factoring.
Tap to reveal answer
$x=0$, $x=-5$, and $x=2$. Factor $x^2+5x=x(x+5)$ then set factors equal to zero.
$x=0$, $x=-5$, and $x=2$. Factor $x^2+5x=x(x+5)$ then set factors equal to zero.
← Didn't Know|Knew It →
Identify the zeros of $f(x)=(x^2-4x+4)(x+1)$.
Identify the zeros of $f(x)=(x^2-4x+4)(x+1)$.
Tap to reveal answer
$x=2$ (mult. $2$) and $x=-1$ (mult. $1$). Factor $x^2-4x+4=(x-2)^2$ to identify the repeated zero.
$x=2$ (mult. $2$) and $x=-1$ (mult. $1$). Factor $x^2-4x+4=(x-2)^2$ to identify the repeated zero.
← Didn't Know|Knew It →
Identify the zeros of $f(x)=(x^2-16)(x^2-1)$.
Identify the zeros of $f(x)=(x^2-16)(x^2-1)$.
Tap to reveal answer
$x=-4$, $x=4$, $x=-1$, and $x=1$. Factor each quadratic: $(x-4)(x+4)(x-1)(x+1)$.
$x=-4$, $x=4$, $x=-1$, and $x=1$. Factor each quadratic: $(x-4)(x+4)(x-1)(x+1)$.
← Didn't Know|Knew It →