Chapter 3.6, Problem 51E

a.

To determine

To find: The dependent and independent variables in the function, $s\left(t\right)=-2\left|t-20\right|+40$ .

Answer to Problem 51E

Dependent variable is $s\left(t\right)$ (weekly sales) and independent variable is $t$ (time).

Explanation of Solution

Given: The function is $s\left(t\right)=-2\left|t-20\right|+40$ .

Calculation: Dependent variables are the variables whose value depends on other variable. Here, the value of $s\left(t\right)$ is dependent on the value of $t$ . So, $s\left(t\right)$ is a dependent variable.

Independent variables are the variables whose value does not depend on the other variable. Here, the value of $t$ is independent of the value of $s\left(t\right)$ . So, $t$ is an independent variable.

b.

To determine

To graph: The function, $s\left(t\right)=-2\left|t-20\right|+40$ using the transformations on the graph, $f\left(x\right)=\left|x\right|$ .

Explanation of Solution

Given information: The function is $s\left(t\right)=-2\left|t-20\right|+40$ .

Graph : Comparing the given function, $s\left(t\right)=-2\left|t-20\right|+40$ with the function, $f\left(x\right)=\left|x\right|$ .

The variable $x$ is replaced with the variable $t$ . The graph of $g\left(t\right)=\left|t-20\right|+40$ is plotted by shifting the function $f\left(t\right)=\left|t\right|$ to the right by $20$ units and upwards by $40$ units. The coefficient $-2$ will invert the graph upside down and stretches the function by the factor 2, $g\left(t\right)=\left|t-20\right|+40$ by a factor 2. So, the graph of the function, $s\left(t\right)=-2\left|t-20\right|+40$ will be:

  

Interpretation : The graph is obtained by shifting the function, $f\left(t\right)=\left|t\right|$ to the right by $20$ units and upwards by $40$ units and then the graph is inverted upside down and finally stretches the function by the factor of $2$ .