Chapter 1.6, Problem 58E

Solution

The transformation that converts the graph showing temperature in degree Fahrenheit to degree Celsius.

The required transformation is a shift of $32$ units down then shrinks by factor $5/9$ .

Given:

The graph shows temperature in degree Fahrenheit.in place-M for $24$ hours.

  

Concept Used:

The graph of $y=f\left(x\right)$ is translated or shifted horizontally left or right according as:

1) If $y=f\left(x-c\right)$ , then graph shifted by c units right from $y=f\left(x\right)$ .

2) If $y=f\left(x+c\right)$ , then graph shifted by c units left from $y=f\left(x\right)$ .

And, the graph of $y=f\left(x\right)$ is translated or shifted vertically up or down according as:

1) If $y=f\left(x\right)+c$ , then graph shifted by c units up from $y=f\left(x\right)$ .

2) If $y=f\left(x\right)-c$ , then graph shifted by c units down from $y=f\left(x\right)$ .

The transformation of horizontal stretches or shrinks from $y=f\left(x\right)$ is as follows:

The graph of $y=f\left(\frac{x}{c}\right)$ transform to horizontal stretches by factor c , when $c>1$ and horizontal shrinks by factor c when $c<1$ .

The transformation of vertical stretches or shrinks from $y=f\left(x\right)$ is as follows:

The graph of $y=c\cdot f\left(x\right)$ transform to vertical stretches by factor c , when $c>1$ and shrinks by factor c when $c<1$ .

Calculation:

The graph in degree Fahrenheit, $F\left(t\right)$ is given need to convert in degree Celsius $C\left(t\right)$ .

Since, the equation to convert degree Fahrenheit, $F\left(t\right)$ to Celsius, $C\left(t\right)$ is, $C\left(t\right)=\left(5/9\right)\left[F\left(t\right)-32\right]$ .

The graph given in $y=F\left(t\right)$ can be converted to that of $C\left(t\right)$ by sequence of transformations,

1) The vertical shift of $32$ units down that convert $y=F\left(t\right)$ to $y=F\left(t\right)-32$ which is equal to $F\left(t\right)$ .

2) A vertical shrink by factor $5/9$ that transform $y=F\left(t\right)-32$ to $y=\left(5/9\right)\left[F\left(t\right)-32\right]$ .

Hence, the transformation that converts the graph in degree Fahrenheit to degree Celsius is a shift of $32$ units down then shrinks by factor $5/9$ .

Conclusion The required transformation is a shift of $32$ units down then shrinks by factor $5/9$ .