Exercise 1.2.7

Question

\begin{enumerate}[label=(\alph*)]
    \item Find an equation for the family of linear functions with slope $2$ and sketch several members of the family.
    \item Find an equation for the family of linear functions such that $f(2) = 1.$ Sketch several members of the family.
    \item Which function belongs to both families?
\end{enumerate}

Khagan Gulusoy · Accepted Answer

\begin{enumerate}[label=(\alph*)]
    \item %Family of linear functions with slope $2$ can be described by $2x+b,$ where $b$ is the $y$-intercept of the function. To find the formulas of the examples of the family, we should take some random $b$. For example, $b=2,\ b=4, \ b=-1.$ Now we can sketch the graphs of the equations $y=2x+2,\ y=2x+4, \ y=2x-1.$
    Family of linear functions $\{f_b\}_{b \in \mathbb{R}}$ with slope $2$ can be described by the general formula $f_b(x)=2x+b,$ where $b \in \mathbb{R}$ is the $y$-intercept of the function. For example, $b=2,\ b=4$ and $b=-1$ give us formulas $f_2(x)=2x+2$, $f_4(x)=2x+4$, $f_{-1}(x)=2x-1.$
    \begin{figure}[H] \centering
    \begin{tikzpicture}
    \begin{axis}[axis lines=middle, xmin=-3.2, xmax=3.2, 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=-3.2:3.2, samples=2, mark=none, color=red, ultra thick] {2*x+2};
         \addplot[domain=-3.2:3.2, samples=2, mark=none, color=blue, ultra thick] {2*x+4};
         \addplot[domain=-3.2:3.2, samples=2, mark=none, color=green!60!black, ultra thick] {2*x-1};
    \end{axis}    
    \end{tikzpicture}
    \end{figure}

\item Our goal is to find an equation family $y$ such that for $x=2$, we would have $y=1.$ 
    \begin{align*}
        y-1 &= m(x-2)\
        y &= mx + (1-2m).
    \end{align*}
    These properties are true, for instance, for the equations $y=x-1, \ y=0.5x, \ y=2x-3:$
    \begin{figure}[H] \centering
    \begin{tikzpicture}
    \begin{axis}[axis lines=middle, xmin=-3.2, xmax=3.2, 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=-3.2:3.2, samples=2, mark=none, color=blue, ultra thick] {x-1};
         \addplot[domain=-3.2:3.2, samples=2, mark=none, color=red, ultra thick] {0.5*x};
         \addplot[domain=-3.2:3.2, samples=2, mark=none, color=green!60!black, ultra thick] {2*x-3};
         \addplot[mark=*, mark options={scale=1, fill=black}] coordinates {(2,1)};
    \end{axis}    
    \end{tikzpicture}
    \end{figure}

\item To find out which equation belongs to both families described in (a) and (b), we should substitute the coordinates of the point $(2,1) $ to the (a) family equation $y=2x+b.$ We have:
    $$1=2\cdot 2+b;$$
    $$1=4+b;$$
    $$b=-3.$$
    Thus, the formula that describes both families is $y=2x-3.$
\end{enumerate}

Exercise 1.2.7

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Exercise 1.2.7

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