Sign Tables

$\newenvironment {prompt}{}{} \newcommand {\todo }[0]{} \newcommand {\npnoround }[0]{\nprounddigits {-1}} \newcommand {\npnoroundexp }[0]{\nproundexpdigits {-1}} \newcommand {\npunitcommand }[1]{\ensuremath {\mathrm {#1}}} \newcommand {\CancelColor }[0]{} \newcommand {\cancelto }[0]{1} \newcommand {\RR }[0]{\mathbb R} \newcommand {\R }[0]{\mathbb R} \newcommand {\N }[0]{\mathbb N} \newcommand {\Z }[0]{\mathbb Z} \newcommand {\d }[0]{\mathop {}\!d} \newcommand {\l }[0]{\ell } \newcommand {\ddx }[0]{\frac {d}{\d x}} \newcommand {\zeroOverZero }[0]{\ensuremath {\boldsymbol {\tfrac {0}{0}}}} \newcommand {\inftyOverInfty }[0]{\ensuremath {\boldsymbol {\tfrac {\infty }{\infty }}}} \newcommand {\zeroOverInfty }[0]{\ensuremath {\boldsymbol {\tfrac {0}{\infty }}}} \newcommand {\zeroTimesInfty }[0]{\ensuremath {\small \boldsymbol {0\cdot \infty }}} \newcommand {\inftyMinusInfty }[0]{\ensuremath {\small \boldsymbol {\infty -\infty }}} \newcommand {\oneToInfty }[0]{\ensuremath {\boldsymbol {1^\infty }}} \newcommand {\zeroToZero }[0]{\ensuremath {\boldsymbol {0^0}}} \newcommand {\inftyToZero }[0]{\ensuremath {\boldsymbol {\infty ^0}}} \newcommand {\numOverZero }[0]{\ensuremath {\boldsymbol {\tfrac {\#}{0}}}} \newcommand {\dfn }[0]{\textbf } \newcommand {\unit }[0]{\mathop {}\!\mathrm } \newcommand {\eval }[1]{\bigg [ #1 \bigg ]} \newcommand {\seq }[1]{\left ( #1 \right )} \newcommand {\epsilon }[0]{\varepsilon } \newcommand {\phi }[0]{\varphi } \newcommand {\iff }[0]{\Leftrightarrow } \DeclareMathOperator {\arccot }{arccot} \DeclareMathOperator {\arcsec }{arcsec} \DeclareMathOperator {\arccsc }{arccsc} \DeclareMathOperator {\si }{Si} \DeclareMathOperator {\proj }{\vec {proj}} \DeclareMathOperator {\scal }{scal} \DeclareMathOperator {\sign }{sign} \newcommand {\arrowvec }[0]{\overrightarrow } \newcommand {\vec }[0]{\mathbf } \newcommand {\veci }[0]{{\boldsymbol {\hat {\imath }}}} \newcommand {\vecj }[0]{{\boldsymbol {\hat {\jmath }}}} \newcommand {\veck }[0]{{\boldsymbol {\hat {k}}}} \newcommand {\vecl }[0]{\boldsymbol {\l }} \newcommand {\uvec }[1]{\mathbf {\hat {#1}}} \newcommand {\utan }[0]{\mathbf {\hat {t}}} \newcommand {\unormal }[0]{\mathbf {\hat {n}}} \newcommand {\ubinormal }[0]{\mathbf {\hat {b}}} \newcommand {\dotp }[0]{\bullet } \newcommand {\cross }[0]{\boldsymbol \times } \newcommand {\grad }[0]{\boldsymbol \nabla } \newcommand {\divergence }[0]{\grad \dotp } \newcommand {\curl }[0]{\grad \cross } \newcommand {\lto }[0]{\mathop {\longrightarrow \,}\limits } \newcommand {\bar }[0]{\overline } \newcommand {\surfaceColor }[0]{violet} \newcommand {\surfaceColorTwo }[0]{redyellow} \newcommand {\sliceColor }[0]{greenyellow} \newcommand {\vector }[1]{\left \langle #1\right \rangle } \newcommand {\sectionOutcomes }[0]{} \newcommand {\HyperFirstAtBeginDocument }[0]{\AtBeginDocument } \newcommand {\epstopdfsetup }[0]{\setkeys {ETE}} changed the theorem header of amsthm failed\MessageBreak }}} \newcommand {\GetTitleStringSetup }[0]{\setkeys {gettitlestring}} \newcommand {\Sectionformat }[2]{#1}$ $% This file was *autogenerated* from digInSignGraphs.sagetex.sage with % sagetex.py version 2015/08/26 v3.0-92d9f7a %b680affe2d395064b7c544c6fc40ebfd% md5sum of corresponding .sage file (minus "goboom","current_tex_line",and pause/unpause lines)$

Did you remember the method used to solve $(x-2)(x+3)>0$ ? Did it involve finding roots? One method is to collect information in something called a sign table (or sign chart, sign diagram or sign graph). Why this method works is a consequence of the intermediate value theorem:

A function can change its sign only at a value where it is not continuous or at a zero.

With this in mind, suppose that $f$ is a function which is continuous on the interval $(a,b)$ and suppose also that $f$ has no zeros in this interval:

If $f(c)>0$ for some $c$ in the interval $(a,b)$ then $f(x)>0$ for all $x$ in $(a,b)$
If $f(c)<0$ for some $c$ in the interval $(a,b)$ then $f(x)<0$ for all $x$ in $(a,b)$

A sign table indicates the intervals where a given function is positive and where it is negative. Given $f(x)$ , a sign table is produced by finding all values in the domain of $f(x)$ where $f(x)$ is zero or discontinuous. The values are then put on a number line. Finally, the sign of $f(x)$ between each of these values is found, either by sampling or from the properties if the given function.

Let’s try some examples:

Consider the function $f(x) = (x-2)(x+3)$ Construct a sign table for $f(x)$ to indicate the intervals where $f(x)$ is positive and the intervals where $f(x)$ is negative

First, as f(x) is a polynomial, it is continuous at all real numbers. Therefore we only need to find the zeros of $f(x)$ . If $f(x) = 0$ then $(x-2)(x+3) = 0$ So $(x-2)= 0 \text { or } (x+3) = 0$ Which means that if $f(x)=0$ then $x=2$ or $x=-3$ . With this, we indicate these values on a number line:

Now we can check points between the zeros to find where $f(x)$ is positive and negative:

$\begin{align*} f(-4)&=\answer[given]{6},\\ f(0)&=\answer[given]{-6},\\ f(3)&=\answer[given]{6}. \end{align*}$

From this we can make a sign table:

Consider the function $f(x) = \frac {(x-4)(x+4)}{(x+2)}$ Construct a sign table for $f(x)$ to indicate the intervals where $f(x)$ is positive and the intervals where $f(x)$ is negative

First, as f(x) is a rational function, it is continuous at all values in its domain. In particular, the only values where $f(x)$ is not continuous is where $f(x)$ is undefined. This will happen when its denominator is zero: $\begin{align*} (x+2) &= 0 \\ x &=\answer[given]{-2} \end{align*}$

So we will have $x=2$ as one of our labels on our sign table. Next, we need to find where $f(x)=0$ . Here:

$(x-4)(x+4) = 0$ So $f(x)=0$ when $x=4$ or $x=-4$

Now we can label our sign table

Now we can check points between the zeros and where $f(x)$ is not continuous to find where $f(x)$ is positive and negative:

$\begin{align*} f(-5)&=\answer[given]{-3},\\ f(-3)&=\answer[given]{7},\\ f(0)&=\answer[given]{-8},\\ f(5)&=\answer[given]{\frac{9}{7}}. \end{align*}$

From this we can make a sign table: