Problem Set - Graphing Rational Functions

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au
and

Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA
ram@math.wisc.edu

Last updates: 7 December 2009

Graphing Rational Functions

	Let $f\left(x\right)=y$ where $x^2+y^2=1$. Graph $f\left(x\right)$. Determine where $f\left(x\right)$ is defined. Determine where $f\left(x\right)$ is continuous. Determine where $f\left(x\right)$ is differentiable. Determine where $f\left(x\right)$ is increasing and where it is decreasing. Determine where $f\left(x\right)$ is concave up and where it is concave down. Determine what the critical pionts of $f\left(x\right)$ are. Determine what the points of inflection of $f\left(x\right)$ are. Determine what the asymptotes to $f\left(x\right)$ are (if $f\left(x\right)$ has asymptotes).
	Let $f\left(x\right)=\sqrt{1-x^2}$. Graph $f\left(x\right)$. Determine where $f\left(x\right)$ is defined. Determine where $f\left(x\right)$ is continuous. Determine where $f\left(x\right)$ is differentiable. Determine where $f\left(x\right)$ is increasing and where it is decreasing. Determine where $f\left(x\right)$ is concave up and where it is concave down. Determine what the critical pionts of $f\left(x\right)$ are. Determine what the points of inflection of $f\left(x\right)$ are. Determine what the asymptotes to $f\left(x\right)$ are (if $f\left(x\right)$ has asymptotes).
	Let $f\left(x\right)=\sqrt{a^2-x^2}$, where $a$ is a constant. Graph $f\left(x\right)$. Determine where $f\left(x\right)$ is defined. Determine where $f\left(x\right)$ is continuous. Determine where $f\left(x\right)$ is differentiable. Determine where $f\left(x\right)$ is increasing and where it is decreasing. Determine where $f\left(x\right)$ is concave up and where it is concave down. Determine what the critical pionts of $f\left(x\right)$ are. Determine what the points of inflection of $f\left(x\right)$ are. Determine what the asymptotes to $f\left(x\right)$ are (if $f\left(x\right)$ has asymptotes).
	Let ${\displaystyle f\left(x\right)=y}$, where ${\left(x-h\right)}^2+{\left(y-k\right)}^2=r^2$, where $h,k$ and $r$ are constants. Graph $f\left(x\right)$. Determine where $f\left(x\right)$ is defined. Determine where $f\left(x\right)$ is continuous. Determine where $f\left(x\right)$ is differentiable. Determine where $f\left(x\right)$ is increasing and where it is decreasing. Determine where $f\left(x\right)$ is concave up and where it is concave down. Determine what the critical pionts of $f\left(x\right)$ are. Determine what the points of inflection of $f\left(x\right)$ are. Determine what the asymptotes to $f\left(x\right)$ are (if $f\left(x\right)$ has asymptotes).
	Let ${\displaystyle f\left(x\right)=y}$, where $x^2+y^2-2hx-2ky+h^2+k^2=r^2$, where $h,k$ and $r$ are constants. Graph $f\left(x\right)$. Determine where $f\left(x\right)$ is defined. Determine where $f\left(x\right)$ is continuous. Determine where $f\left(x\right)$ is differentiable. Determine where $f\left(x\right)$ is increasing and where it is decreasing. Determine where $f\left(x\right)$ is concave up and where it is concave down. Determine what the critical pionts of $f\left(x\right)$ are. Determine what the points of inflection of $f\left(x\right)$ are. Determine what the asymptotes to $f\left(x\right)$ are (if $f\left(x\right)$ has asymptotes).
	Let ${\displaystyle f\left(x\right)=y}$ where ${\displaystyle \frac{x^2}{a^2}+\frac{y^2}{b^2}=1}$, where $a$ and $b$ are constants. Graph $f\left(x\right)$. Determine where $f\left(x\right)$ is defined. Determine where $f\left(x\right)$ is continuous. Determine where $f\left(x\right)$ is differentiable. Determine where $f\left(x\right)$ is increasing and where it is decreasing. Determine where $f\left(x\right)$ is concave up and where it is concave down. Determine what the critical pionts of $f\left(x\right)$ are. Determine what the points of inflection of $f\left(x\right)$ are. Determine what the asymptotes to $f\left(x\right)$ are (if $f\left(x\right)$ has asymptotes).
	Let ${\displaystyle f\left(x\right)=y}$ where ${\displaystyle x=a\cos \theta }$ and $y=b\sin \theta$, where $a$ and $b$ are constants. Graph $f\left(x\right)$. Determine where $f\left(x\right)$ is defined. Determine where $f\left(x\right)$ is continuous. Determine where $f\left(x\right)$ is differentiable. Determine where $f\left(x\right)$ is increasing and where it is decreasing. Determine where $f\left(x\right)$ is concave up and where it is concave down. Determine what the critical pionts of $f\left(x\right)$ are. Determine what the points of inflection of $f\left(x\right)$ are. Determine what the asymptotes to $f\left(x\right)$ are (if $f\left(x\right)$ has asymptotes).
	Let ${\displaystyle f\left(x\right)=\left(b/a\right)\sqrt{a^2-x^2}}$ where $a$ and $b$ are constants. Graph $f\left(x\right)$. Determine where $f\left(x\right)$ is defined. Determine where $f\left(x\right)$ is continuous. Determine where $f\left(x\right)$ is differentiable. Determine where $f\left(x\right)$ is increasing and where it is decreasing. Determine where $f\left(x\right)$ is concave up and where it is concave down. Determine what the critical pionts of $f\left(x\right)$ are. Determine what the points of inflection of $f\left(x\right)$ are. Determine what the asymptotes to $f\left(x\right)$ are (if $f\left(x\right)$ has asymptotes).
	Let ${\displaystyle f\left(x\right)=y}$, where $x^2-y^2=1$. Graph $f\left(x\right)$. Determine where $f\left(x\right)$ is defined. Determine where $f\left(x\right)$ is continuous. Determine where $f\left(x\right)$ is differentiable. Determine where $f\left(x\right)$ is increasing and where it is decreasing. Determine where $f\left(x\right)$ is concave up and where it is concave down. Determine what the critical pionts of $f\left(x\right)$ are. Determine what the points of inflection of $f\left(x\right)$ are. Determine what the asymptotes to $f\left(x\right)$ are (if $f\left(x\right)$ has asymptotes).
	Let ${\displaystyle f\left(x\right)=y}$ where ${\displaystyle \frac{x^2}{a^2}-\frac{y^2}{b^2}=1}$, where $a$ and $b$ are constants. Graph $f\left(x\right)$. Determine where $f\left(x\right)$ is defined. Determine where $f\left(x\right)$ is continuous. Determine where $f\left(x\right)$ is differentiable. Determine where $f\left(x\right)$ is increasing and where it is decreasing. Determine where $f\left(x\right)$ is concave up and where it is concave down. Determine what the critical pionts of $f\left(x\right)$ are. Determine what the points of inflection of $f\left(x\right)$ are. Determine what the asymptotes to $f\left(x\right)$ are (if $f\left(x\right)$ has asymptotes).
	Let ${\displaystyle f\left(x\right)=a{x}^2-b}$, where $a$ and $b$ are constants. Graph $f\left(x\right)$. Determine where $f\left(x\right)$ is defined. Determine where $f\left(x\right)$ is continuous. Determine where $f\left(x\right)$ is differentiable. Determine where $f\left(x\right)$ is increasing and where it is decreasing. Determine where $f\left(x\right)$ is concave up and where it is concave down. Determine what the critical pionts of $f\left(x\right)$ are. Determine what the points of inflection of $f\left(x\right)$ are. Determine what the asymptotes to $f\left(x\right)$ are (if $f\left(x\right)$ has asymptotes).
	Let ${\displaystyle f\left(x\right)=y}$, where $x=2y^2-1$. Graph $f\left(x\right)$. Determine where $f\left(x\right)$ is defined. Determine where $f\left(x\right)$ is continuous. Determine where $f\left(x\right)$ is differentiable. Determine where $f\left(x\right)$ is increasing and where it is decreasing. Determine where $f\left(x\right)$ is concave up and where it is concave down. Determine what the critical pionts of $f\left(x\right)$ are. Determine what the points of inflection of $f\left(x\right)$ are. Determine what the asymptotes to $f\left(x\right)$ are (if $f\left(x\right)$ has asymptotes).
	Let ${\displaystyle f\left(x\right)=y}$, where $x=\cos 2\theta$ and $y=\cos \theta$. Graph $f\left(x\right)$. Determine where $f\left(x\right)$ is defined. Determine where $f\left(x\right)$ is continuous. Determine where $f\left(x\right)$ is differentiable. Determine where $f\left(x\right)$ is increasing and where it is decreasing. Determine where $f\left(x\right)$ is concave up and where it is concave down. Determine what the critical pionts of $f\left(x\right)$ are. Determine what the points of inflection of $f\left(x\right)$ are. Determine what the asymptotes to $f\left(x\right)$ are (if $f\left(x\right)$ has asymptotes).
	Let ${\displaystyle f\left(x\right)=b\sqrt{x-a}}$ where $a$ and $b$ are constants. Graph $f\left(x\right)$. Determine where $f\left(x\right)$ is defined. Determine where $f\left(x\right)$ is continuous. Determine where $f\left(x\right)$ is differentiable. Determine where $f\left(x\right)$ is increasing and where it is decreasing. Determine where $f\left(x\right)$ is concave up and where it is concave down. Determine what the critical pionts of $f\left(x\right)$ are. Determine what the points of inflection of $f\left(x\right)$ are. Determine what the asymptotes to $f\left(x\right)$ are (if $f\left(x\right)$ has asymptotes).
	Let ${\displaystyle f\left(x\right)=\sqrt{x+2}}$. Graph $f\left(x\right)$. Determine where $f\left(x\right)$ is defined. Determine where $f\left(x\right)$ is continuous. Determine where $f\left(x\right)$ is differentiable. Determine where $f\left(x\right)$ is increasing and where it is decreasing. Determine where $f\left(x\right)$ is concave up and where it is concave down. Determine what the critical pionts of $f\left(x\right)$ are. Determine what the points of inflection of $f\left(x\right)$ are. Determine what the asymptotes to $f\left(x\right)$ are (if $f\left(x\right)$ has asymptotes).
	Let ${\displaystyle f\left(x\right)=-\sqrt{x+2}}$. Graph $f\left(x\right)$. Determine where $f\left(x\right)$ is defined. Determine where $f\left(x\right)$ is continuous. Determine where $f\left(x\right)$ is differentiable. Determine where $f\left(x\right)$ is increasing and where it is decreasing. Determine where $f\left(x\right)$ is concave up and where it is concave down. Determine what the critical pionts of $f\left(x\right)$ are. Determine what the points of inflection of $f\left(x\right)$ are. Determine what the asymptotes to $f\left(x\right)$ are (if $f\left(x\right)$ has asymptotes).
	Let ${\displaystyle f\left(x\right)=y}$ where $y^2\left(x^2-\mathrm{x}\right)=x^2-1$. Graph $f\left(x\right)$. Determine where $f\left(x\right)$ is defined. Determine where $f\left(x\right)$ is continuous. Determine where $f\left(x\right)$ is differentiable. Determine where $f\left(x\right)$ is increasing and where it is decreasing. Determine where $f\left(x\right)$ is concave up and where it is concave down. Determine what the critical pionts of $f\left(x\right)$ are. Determine what the points of inflection of $f\left(x\right)$ are. Determine what the asymptotes to $f\left(x\right)$ are (if $f\left(x\right)$ has asymptotes).
	Let ${\displaystyle f\left(x\right)=y}$ where ${\displaystyle x=\frac{y^2-1}{y^2+1}}$. Graph $f\left(x\right)$. Determine where $f\left(x\right)$ is defined. Determine where $f\left(x\right)$ is continuous. Determine where $f\left(x\right)$ is differentiable. Determine where $f\left(x\right)$ is increasing and where it is decreasing. Determine where $f\left(x\right)$ is concave up and where it is concave down. Determine what the critical pionts of $f\left(x\right)$ are. Determine what the points of inflection of $f\left(x\right)$ are. Determine what the asymptotes to $f\left(x\right)$ are (if $f\left(x\right)$ has asymptotes).
	Let ${\displaystyle f\left(x\right)=\frac{\sqrt{1+x}}{\sqrt{1-x}}}$. Graph $f\left(x\right)$. Determine where $f\left(x\right)$ is defined. Determine where $f\left(x\right)$ is continuous. Determine where $f\left(x\right)$ is differentiable. Determine where $f\left(x\right)$ is increasing and where it is decreasing. Determine where $f\left(x\right)$ is concave up and where it is concave down. Determine what the critical pionts of $f\left(x\right)$ are. Determine what the points of inflection of $f\left(x\right)$ are. Determine what the asymptotes to $f\left(x\right)$ are (if $f\left(x\right)$ has asymptotes).
	Let ${\displaystyle f\left(x\right)=\frac{x^2}{\sqrt{x+1}}}$. Graph $f\left(x\right)$. Determine where $f\left(x\right)$ is defined. Determine where $f\left(x\right)$ is continuous. Determine where $f\left(x\right)$ is differentiable. Determine where $f\left(x\right)$ is increasing and where it is decreasing. Determine where $f\left(x\right)$ is concave up and where it is concave down. Determine what the critical pionts of $f\left(x\right)$ are. Determine what the points of inflection of $f\left(x\right)$ are. Determine what the asymptotes to $f\left(x\right)$ are (if $f\left(x\right)$ has asymptotes).
	Let ${\displaystyle f\left(x\right)=x\sqrt{32-x^2}}$. Graph $f\left(x\right)$. Determine where $f\left(x\right)$ is defined. Determine where $f\left(x\right)$ is continuous. Determine where $f\left(x\right)$ is differentiable. Determine where $f\left(x\right)$ is increasing and where it is decreasing. Determine where $f\left(x\right)$ is concave up and where it is concave down. Determine what the critical pionts of $f\left(x\right)$ are. Determine what the points of inflection of $f\left(x\right)$ are. Determine what the asymptotes to $f\left(x\right)$ are (if $f\left(x\right)$ has asymptotes).
	Let ${\displaystyle f\left(x\right)=x\sqrt{1-x^2}}$. Graph $f\left(x\right)$. Determine where $f\left(x\right)$ is defined. Determine where $f\left(x\right)$ is continuous. Determine where $f\left(x\right)$ is differentiable. Determine where $f\left(x\right)$ is increasing and where it is decreasing. Determine where $f\left(x\right)$ is concave up and where it is concave down. Determine what the critical pionts of $f\left(x\right)$ are. Determine what the points of inflection of $f\left(x\right)$ are. Determine what the asymptotes to $f\left(x\right)$ are (if $f\left(x\right)$ has asymptotes).

References [PLACEHOLDER]

[BG] A. Braverman and D. Gaitsgory, Crystals via the affine Grassmanian, Duke Math. J. 107 no. 3, (2001), 561-575; arXiv:math/9909077v2, MR1828302 (2002e:20083)