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:

$\bullet$

Intercepts. Intercepts with the $x$-axis satisfy $y=0$. Intercepts with the $y$-axis satisfy $x=0$.

$\bullet$

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$\DG9/$ line $y=x$ is one if swapping $x$ and $y$ does not make a difference.

$\bullet$

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 $y(-x)=-y(x)$, i.e. if function $y(x)$ is antisymmetric.

$\bullet$

Singular points:

$\bullet$

corners where the direction of the curve changes by an angle less than 180$\DG9/$,

$\bullet$

cusps where it changes 180$\DG9/$,

$\bullet$

crossings where the curve crosses itself,

$\bullet$

positions of infinite curvature,

$\bullet$

...

If $y$ or any of its derivatives is infinite or not uniquely defined, the curve has a singularity at that point.

$\bullet$

A vertical asymptote $x_{\rm va}=A$ exists if $y\to\pm\infty$ for $x\to A$.

$\bullet$

A horizontal asymptote $y=A$ exists if $A=\lim_{x\to\pm\infty}y$ exists.

$\bullet$

Behavior for $x\to\pm\infty$ (e.g. $y\sim \vert x\vert^p$ for some $p$).

$\bullet$

An oblique asymptote $y_{\rm oa}=Ax+B$ exists if $A=\lim_{x\to\pm\infty} y'$ and $b=\lim_{x\to\pm\infty} y - Ax$ exist. (Or more simply if $\lim_{x\to\pm\infty} y-Ax-B = 0$.)

$\bullet$

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.

$\bullet$

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.

$\bullet$

Concavity is upward if $y''>0$, downward if $y''<0$.

$\bullet$

Inflection points are points where the concavity changes sign.

See [1, Chapters 13-15]