Chapter 1.2, Problem 41E

a.

To determine

To write: An expression for the vertical translation of the graph of $f$ to the graph of $g$ .

Answer to Problem 41E

The required expression is $f(x)+(c-b)=g(x)$.

Explanation of Solution

Given Information: Given functions representing two parallel lines are:

  $f(x)=mx+b$ and $g(x)=mx+c$ .

Calculation:

Let us translate $f(x)$ vertically by a factor $k$ . Then

  $f(x)+k=g(x)$

  $\Rightarrow mx+b+k=mx+c$

  $\Rightarrow b+k=c$

  $\Rightarrow k=c-b$

So, $f(x)$ must be translated vertically by $(c-b)$

  $units$ to obtain $g(x).$

Thus, the required expression is $f(x)+(c-b)=g(x)$ .

b.

To determine

To write: An expression for the horizontal translation of the graph of $f$ to the graph of $g$ .

Answer to Problem 41E

The required expression is $f(x+\frac{c-b}{m})=g(x)$ .

Explanation of Solution

Given Information: Given functions representing two parallel lines are:

  $f(x)=mx+b$ and $g(x)=mx+c$ .

Calculation:

Let us translate $f$ horizontally by a factor $h$ .

Then, $f(x+h)=g(x)$

  $\begin{array}{l}\Rightarrow m(x+h)+b=mx+c\\ \Rightarrow mx+mh+b=mx+c\\ \Rightarrow mh+b=c\\ \Rightarrow mh=c-b\\ \Rightarrow h=\frac{c-b}{m}\end{array}$

So, $f(x)$ must be translated by $\frac{c-b}{m}units$ horizontally to obtain $g(x)$ .

Thus, the required expression is $f(x+\frac{c-b}{m})=g(x)$ .