Question

Waiting in Line Suppose two employees at a fast-food restaurant can serve customers at the rate of 6 customers per minute. Further suppose that customers are arriving at the restaurant at the rate of \(x\) customers per minute. The average time \(T,\) in minutes, spent waiting in line and having your order taken and filled is given by the function \(T(x)=-\frac{1}{x-6},\) where \(0

Short answer

Graph \( T(x) = -\frac{1}{x-6} \) by shifting \( \frac{1}{x} \) 6 units right and reflecting vertically.

Step-by-step solution

Understand the Base Function

The given function is \[ T(x) = -\frac{1}{x-6} \]. This is a transformation of the base function \( f(x) = \frac{1}{x} \), which is a hyperbola.

Horizontal Shift

The function \( T(x) = -\frac{1}{x-6} \) represents a horizontal shift of the base function \( \frac{1}{x} \) by 6 units to the right. This is because \( x \) is replaced with \( x - 6 \).

Vertical Reflection

The negative sign in front of the transformed function, \( -\frac{1}{x-6} \), indicates a reflection in the x-axis. This means that the hyperbola is flipped vertically.

Identify Asymptotes

The vertical asymptote is shifted to \( x = 6 \). The horizontal asymptote remains at \( y = 0 \) because multiplying by -1 does not affect it.

Plotting the Function

To graph \( T(x) \), start by drawing the vertical asymptote at \( x = 6 \) and the horizontal asymptote at \( y = 0 \). Then, plot points on either side of \( x = 6 \) to get the shape of the hyperbola, reflecting it vertically as necessary.

Key concepts

Hyperbola

A hyperbola is a type of curve on a graph that is shaped like two opposite-facing, open curves. It is represented by an equation of the form \(\frac{1}{x}\). In the given exercise, the base function \(\frac{1}{x}\) is used. Hyperbolas have unique characteristics:

• Two branches that mirror each other.
• Asymptotes that the curves approach but never touch.
• A center point where the asymptotes intersect.

The hyperbola helps us understand how the average waiting time changes rapidly as the number of customers per minute approaches 6.

Horizontal Shift

A horizontal shift involves moving the entire graph of a function left or right. In our function \(T(x) = -\frac{1}{x-6}\), the graph is shifted 6 units to the right. This happens because we replace \(x\) with \(x - 6\). As a result,

• The graph starts at \(x = 6\) instead of \(x = 0\).
• It affects where the vertical asymptote is placed.

This shift allows us to model the wait times more accurately for our specific arrival rate.

Vertical Reflection

In a vertical reflection, the graph is flipped over the x-axis. In the function \(T(x) = -\frac{1}{x-6}\), the negative sign before the fraction causes this reflection. What this means is:

• All y-values of the function are multiplied by -1.
• If a point was above the x-axis, it moves to the same distance below and vice versa.
• In our case, the hyperbola that opened upwards and downwards is now flipped.

This reflection changes the direction of the wait time growth as customer arrival rates change.

Asymptotes

Asymptotes are lines that a graph approaches but never touches. They are essential in understanding hyperbolas. For the function \(T(x) = -\frac{1}{x-6}\):

• The vertical asymptote is at \(x = 6\); this is where the function goes to infinity.
• The horizontal asymptote remains at \(y = 0\), meaning the graph gets closer to 0 but never quite reaches it.

Identifying these asymptotes helps in knowing how the wait times behave as customer arrival approaches critical rates.