Chapter 1.2, Problem 19Q

To determine

To describe: The way to transform the graph of $f$ to model the total cost for a senior citizen and then to determine the total cost for senior citizen to camp for $3$ days.

Answer to Problem 19Q

There should be a vertical shrink by a factor of $\frac{1}{2}$ and then translation of $30$ units down. Total cost for $3$ day camping for a senior citizen is $\$40$ .

Explanation of Solution

Given information: Function is $f(x)=20x+80$ . Senior citizens pay half of it and get an extra discount of $\$30$ .

Calculation:

As the total cost calculated by putting an arbitrary value of $x$ is reduced to half for senior citizens, there should be a vertical shrink by a factor of $\frac{1}{2}$ , thereafter discount of $\$30$ is dealt with by translation of $30$ units down,

  $\begin{array}{l}g(x)=\frac{1}{2}f(x)-30\\ \\ =\frac{1}{2}(20x+80)-30\\ \\ =10x+10\end{array}$

Putting $x=3$ in $g$ to calculate the cost of camping for $3$ days for a senior citizen,

  $\begin{array}{l}g(3)=10(3)+10\\ \\ =40\end{array}$

Thus, there should be a vertical shrink by a factor of $\frac{1}{2}$ and then translation of $30$ units down. Total cost for $3$ day camping for a senior citizen is $\$40$ .