Exercise 1.2.8

Question

What do all members of the family of linear functions $f(x)= 1 + m(x + 3)$ have in common? Sketch several members of the family.

Khagan Gulusoy · Accepted Answer

The graphs of all elements $f_m$ of the given family of linear functions $\{f_m\}_{m \in \mathbb{R}}$ are lines that pass through the point $(-3, 1)$ since
\begin{equation*}
    \forall m \in \mathbb{R}: \quad f_{m}(-3) = 1. 
\end{equation*}
Now we should sketch the graph of several members of the family described in the exercise. Let's take $m=1,$ $m=2$, $m=3.$ 
%If $m=1,$ we have $f(x)=1+x+3=x+4.$ If $m=2,$ $f(x)=1+2(x+3)=2x+7.$ If $m=3,$ $f(x)=1+3(x+3)=1+3x+9=3x+10.$
\begin{figure}[H] \centering
    \begin{tikzpicture}
    \begin{axis}[axis lines=middle, xmin=-6, xmax=1.5, ymax=4, ymin=-4, xlabel={$x$}, ylabel={$y$}, x label style = {at={(ticklabel cs:0.9)},anchor=south,above=4.8mm}, y label style={anchor=north, above=3mm}]
         \addplot[domain=-6:4, samples=2, mark=none, color=blue, ultra thick] {x+4};
         \addplot[domain=-6:4, samples=2, mark=none, color=red, ultra thick] {3*x+10};
         \addplot[domain=-6:4, samples=2, mark=none, color=green!60!black, ultra thick] {2*x+7};
    \end{axis}    
    \end{tikzpicture}
\end{figure}

Exercise 1.2.8

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Exercise 1.2.8

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