Latex graph of equations using Tikz

A graph is a pictorial representation of data connected by links. The graphs can be created through the tikz environment and without the tikz environment as well. But the graphs with equations are better by using the tikz environment.

In this topic, some examples are with equations, and some are the graphs with the use of vertices only.

Let-s consider an example to draw a graph for Temperature and Resistance.

Each step would include the explanation of the corresponding command. The description of that command will begin with the %.

The code to draw the graph is given below:

\documentclass[border=4.6pt]{standalone}% the standalone package is used to overcome the other preambles used in the sub-files.
% the length mentioned above is the length of the border of the whole graph page from the axis.
\usepackage{pgfplots} % the package used to implement the graphs
\pgfplotsset{width = 6.4cm, compat = newest} % it is the width of the graph. The lesser the width, the higher the graph will shrink.
\usepackage{sansmath} % it is the package used in math. It provides the math font.
\pgfplotsset{
% pgfplotsset is used to define the parameters used further to draw a graphs
  tick label style = { font=\tiny\sansmath\sffamily}, ,% it defines the font size of the label
  label style = { font=\small\sansmath\sffamily}
}
\begin{document}   
\begin{tikzpicture}  % tikz environment commonly used
  \begin{axis} [
      domain = 0:100,
      xlabel     = Temperature,  % You can label the axis according to your graph.
      ylabel     = Resistance,
      axis lines = left, % if you  use the right axis line, then the axis will be inverted
      clip       = true, 
      xmin = 0,  xmax = 100,
      ymin = 0, ymax = 100,
      xtick={38,80},  % xtick and ytick are used to determine the range of the drawn graph. The labels will be adjusted according to the xtick and ytick only.
      xticklabels={$40$,$70$},
      ytick={30,80},
      yticklabels={$30$,$70$},
    ]
   \addplot [smooth, dashed] % dashed option draws the dashed curve. You can also use dotted, dashdotted, dashdotdotted.
      coordinates { (10,5) (25,10) (35,20) (38,22) (45,26) (52,29)
                    (61,33) (72,40) (79,50) (80,59)};
    \addplot [smooth]  % the smooth option is used to illustrate the smooth curve
      coordinates { (20,22) (24,25) (30,29) (36,31) (45,36) (50,40)
                    (60,50) (65,59) (69,69) (70,77)};
    % you can include as many curves in your graph according to the requirements.

\end{axis}
\end{tikzpicture}
\end{document}

Output:

$Latex graph of equations using Tikz$

Determination of Axis of Graph

There are two types of axis commonly used. The first one is the simple graph, which contains four quadrants, and the other is the axis, which is used to draw bar graphs, etc.

To draw x and the y-axis, it is necessary to declare some packages, sets, and libraries. These will be listed in the code given below.

Let-s consider the first example to draw the axis of four quadrants.

\documentclass[12pt]{article} 
\usepackage{tikz} % package used for the tikz
\usepackage{pgfplots} % package used to draw plots
\usepgfplotslibrary{polar} % the polar library is used to implement the negative and positive axis
\usepgflibrary{shapes.geometric}
\usetikzlibrary{calc}
\pgfplotsset{my style/.append style={axis x line=middle, axis y line=
middle, xlabel={$x$}, ylabel={$y$}, axis equal }} % used to declare my style command.
% you can also align the axis to the left, right, etc. according to the requirements. But the middle would produce the axis with four quadrants, and the left or right would result in only one quadrant.
\begin{document}
\begin{tikzpicture} % tikz environment 

\begin{axis}
[my style, xtick={-4,-3,-2,...,4}, ytick={-4,-3,-2,...,4}, % you can also expand the points or range of the x-axis and the y-axis.
xmin=-4, xmax=4, ymin=-4, ymax=4]
\end{axis}
\end{tikzpicture}

\end{document}

Output:

$Latex graph of equations using Tikz$

There is only one difference between the two examples, i.e., to determine the type of style. The code to draw the axis suitable for bar graphs etc. that produces only one quadrant is given below:

\documentclass[10pt]{article} 
\usepackage{tikz} 
\usepackage{pgfplots}
\usepgfplotslibrary{polar} 
\usepgflibrary{shapes.geometric}
\usetikzlibrary{calc}
\pgfplotsset{my style/.append style={axis x line=left, axis y line=
left, xlabel={$x$}, ylabel={$y$}, axis equal }} %  the middle would produce the axis with four quadrants, and the left or right would result in only one quadrant.
\begin{document}
\begin{tikzpicture} % tikz environment 

\begin{axis}
[my style, xtick={-2,-1,...,2}, ytick={-2,-1,...,2}, % you can also expand the points or the range of the x-axis and the y-axis.
xmin=-2, xmax=2, ymin=-2, ymax=2]
\end{axis}
\end{tikzpicture}

\end{document}

You need to use left or right (for inverted axis) instead of the middle (for four quadrants) option in my style category. You can notice the change.

Output:

$Latex graph of equations using Tikz$

Note: The number of dots (..) between the numbers specified in the -my style, xtick={-2,-1,...,2}, ytick={-2,-1,...,2} - command should be equal. Otherwise, Latex will consider an error.

Graph of equations

You can also draw graphs with the use of equations. The packages, sets, and libraries will be the same as used for other types of graphs.

First-order equations

The example is to draw a graph with the use of only one order equation.

The equation is given below:

$Latex graph of equations using Tikz$

The code to draw such type of equations is given below:

\documentclass[12pt]{article} 
\usepackage{tikz} 
\usepackage{pgfplots}
\usepgfplotslibrary{polar} % to determine the polar coordinates
\usepgflibrary{shapes.geometric}
\usetikzlibrary{calc}
\pgfplotsset{my style/.append style={axis x line=middle, axis y line=
middle, xlabel={$x$}, ylabel={$y$}, axis equal }} %  the middle would produce the axis with four quadrants, and the left or right would result in only one quadrant.
\begin{document}

\begin{tikzpicture}
\begin{axis}[my style]
\addplot[domain=-6:1] {-x+3}; % the domain signifies the range for the graph
\addplot[domain=1:6] {x};
\end{axis}
\end{tikzpicture}

\end{document}

Output:

$Latex graph of equations using Tikz$

Second-order equation

You can also draw the graph for the equations of second order. The code and the commands will be similar to the above type of graphs.

Let-s draw the graph for y=x²

The code for such type of graphs is given below:

\documentclass[12pt]{article} 
\usepackage{tikz} 
\usepackage{pgfplots}
\usepgfplotslibrary{polar} % to determine the polar coordinates
\usepgflibrary{shapes.geometric}
\usetikzlibrary{calc}
\pgfplotsset{my style/.append style={axis x line=middle, axis y line=
middle, xlabel={$x$}, ylabel={$y$}, axis equal }} %  the middle would produce the axis with four quadrants, and the left or right would result in only one quadrant.
\begin{document}

\begin{tikzpicture}
\begin{axis}[my style]
\addplot {x^2}; % here, the second-order equation is written. To draw a graph for any equation, you need to mention the equation in this section. If you want to specify the domain, then you can include the domain option.

\end{axis}
\end{tikzpicture}

\end{document}

Output:

$Latex graph of equations using Tikz$

Similarly, you can draw any shape and graph in Latex using the methods discussed so far.

Polar Curve

Latex also provides the feature to draw the area enclosed by the polar curve.

Here, the code will be similar as the above method.

Let-s draw the area enclosed by the curve, r²=2 sin?θ

In this case, a few changes are necessary. The first one is to include the samples. The samples are used to draw smooth graphs. It is used in complex graphs, such as polar graphs.

You need to include samples=100 in the code.

The other is to specify the domain. It depends on the user to determine the range for a domain according to the requirements for the graph.

To draw the polar curve, you need to use polaraxis environment instead of the axis environment, which is given below:

\begin{polaraxis}
............
\end{polaraxis}

The equation r²=2 sin?θ will now become,r=√2 sin?θ

The axis used to draw polar curves are different from the usual axis.

To implement polar curves, following preamble must be declared:

\pgfplotsset{my polar style/.append style={xticklabels={,, $\frac{\pi}{6}$, $\frac{\pi}{3}$, $\frac{\pi}{2}$, $\frac{2\pi}{3}$, $\frac{5\pi}{6}$, $\pi$, $\frac{7\pi}{6}$, $\frac{4\pi}{3}$, $\frac{3\pi}{2}$, $\frac{5\pi}{3}$,$\frac{11\pi}{6}$,}, thin }}

All the above parameters are the tick labels of the graph used to determine the polar curve.

The code to draw the polar graph is given below:

\documentclass[12pt]{article} 
\usepackage{tikz} 
\usepackage{pgfplots}
\usepgfplotslibrary{polar} % to determine the polar coordinates
\usepgflibrary{shapes.geometric}
\usetikzlibrary{calc}
\pgfplotsset{my polar style/.append style={xticklabels={,,
$\frac{\pi}{6}$, $\frac{\pi}{3}$, $\frac{\pi}{2}$, $\frac{2\pi}{3}$,
$\frac{5\pi}{6}$, $\pi$, $\frac{7\pi}{6}$, $\frac{4\pi}{3}$,
$\frac{3\pi}{2}$, $\frac{5\pi}{3}$,$\frac{11\pi}{6}$,}, thin }} % you can also mention thick instead of thin.
\begin{document}

\begin{tikzpicture}
\begin{polaraxis}[my polar style ] % the name of style depends on your choice. But make sure that the name used above in the append style should be same as the name used here.
\addplot[domain=0:90, samples=50]{sqrt(2*(sin(x))};
\addplot[domain=90:180, samples=50]{sqrt(2*(sin(x)))}; % you can modify the domain according to the requirements. The polar curve will be adjusted according to the specified domain or range. You can also change the sample size accordingly.
\end{polaraxis}
\end{tikzpicture}


\end{document}

Output: