1.1 Introduction

Graphs are important for engineers for a number of reasons:

• Understanding relationships between variables.
• Summarizing data.
• Representing data (like in a Moody diagram).
• Interpolating data.
• Understanding the overal nature of data. See warming.jpg for an example that you simply could not appreciate by looking at a list of numbers.
• ...

Look for:

• Intercepts. Intercepts with the $x$-axis satisfy $y$ $\vphantom0\raisebox{1.5pt}{$=$}$ 0. Intercepts with the $y$-axis satisfy $x$ $\vphantom0\raisebox{1.5pt}{$=$}$ 0.
• A symmetry line exists if the curve is the same at both sides of the line. More precisely, a symmetry line acts as a mirror that mirrors the curve into itself. The $y$-axis is a symmetry line if the sign of $x$ does not make a difference. The $x$-axis is one if the sign of $y$ does not make a difference. The 45$\POW9,{\circ}$ line $y$ $\vphantom0\raisebox{1.5pt}{$=$}$ $x$ is one if swapping $x$ and $y$ does not make a difference.
• Symmetry points. Every point on the curve must have match at the exact opposite side of a symmetry point. Mathematically, if $\vec{r}_1$ is on the curve, then so must be $\vec{r}_{\rm S}-(\vec{r}_1-\vec{r}_{\rm S})$. The origin is a symmetry point if $y(-x)$ $\vphantom0\raisebox{1.5pt}{$=$}$ $-y(x)$, i.e. if function $y(x)$ is antisymmetric.
• Singular points:
  • corners where the direction of the curve changes by an angle less than 180$\POW9,{\circ}$,
  • cusps where it changes 180$\POW9,{\circ}$,
  • crossings where the curve crosses itself,
  • positions of infinite curvature,
  • ...
  If $y$ or any of its derivatives is infinite or not uniquely defined, the curve has a singularity at that point.
• A vertical asymptote $x_{\rm va}$ $\vphantom0\raisebox{1.5pt}{$=$}$ $A$ exists if $y\to\pm\infty$ for $x\to A$.
• A horizontal asymptote $y$ $\vphantom0\raisebox{1.5pt}{$=$}$ $A$ exists if $A$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\lim_{x\to\pm\infty}y$ exists.
• Behavior for $x\to\pm\infty$ (e.g. $y\sim \vert x\vert^p$ for some $p$).
• An oblique asymptote $y_{\rm oa}$ $\vphantom0\raisebox{1.5pt}{$=$}$ $Ax+B$ exists if $A$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\lim_{x\to\pm\infty} y'$ and $b$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\lim_{x\to\pm\infty}y-Ax$ exist. (Or more simply if $\lim_{x\to\pm\infty}y-Ax-B$ $\vphantom0\raisebox{1.5pt}{$=$}$ 0.)
• Extent in $x$ (the range of $x$-values of the curve) and extent in $y$ (the range of $y$-values of the curve). If $y$ is a given function of $x$, then the $x$-extent is the $x$-values for which $y$ can be computed, but the $y$ extent may not be so simple.
• Minima and maxima. A global maximum/minimum is the highest/lowest value of $y$ that can be found anywhere. You should find both the value of the maximum/minimum and its location(s). A local maximum/minimum is the highest/lowest value that can be found in a small vicinity around the localtion of the local maximum/minimum. Normally, you first find the local maxima/minima, and then, based on consideration of the entire graph, decide whether they are also global ones. The derivative changes sign at a maximum/minimum if defined at both sides of the maximum/minimum. So look for both zero derrivatives and singular points.
• Concavity is upward if $y''$ $\raisebox{.3pt}{$>$}$ 0, downward if $y''$ $\raisebox{.3pt}{$<$}$ 0.
• Inflection points are points where the concavity changes sign.

See [1, Chapters 13-15]