Graph the function

[SHAKUURAH]

In most businesses, increasing prices of their product can have a negative effect on the number of customers of the business. A bus company in a small town has an average number of riders of 800 per day. The bus company charges $2.25 for a ride. They conducted a survey of their customers and found that they will lose approximately 40 customers per day for each $.25 increase in fare.

Given the description above, graph the function, identify the graph of the function, and give the domain and range as shown in the graph, and also the domain and range of the entire function excluding unrealistic values.

 

[tkhunny]

In most businesses, increasing prices of their product can have a negative effect on the number of customers of the business. A bus company in a small town has an average number of riders of 800 per day. The bus company charges $2.25 for a ride. They conducted a survey of their customers and found that they will lose approximately 40 customers per day for each $.25 increase in fare.

Given the description above, graph the function, identify the graph of the function, and give the domain and range as shown in the graph, and also the domain and range of the entire function excluding unrealistic values.

Graph what function? Riders as a function of Price? It should say so.

The Range should be rather obvious, at least in one direction. If the price increases, the riders decrease. This makes [0,800]. No one is talking about an increase in passengers.

The Domain is easily found from one criterion, how much can the Range take? 800/40 = 20. This makes [2.25,2.25+0.25*20] or [2.25,7.25]. Again, no one is talking about a decrease in proces.

I'll let you draw the graph.

 

[soroban]

Hello, SHAKUURAH!

tkhunny is correct . . . some information is missing.
But I've seen similar problems, so I'll make some guesses . . .

In most businesses, increasing prices of their product can have a negative effect
on the number of customers of the business.
A bus company in a small town has an average number of riders of 800 per day.
The bus company charges $2.25 for a ride.
They conducted a survey of their customers and found that they will lose
approximately 40 customers per day for each $.25 increase in fare.

Given the description above:
(a) Graph the function
(b) Identify the graph of the function
(c) Give the domain and range as shown in the graph
(d) Give the domain and range of the entire function excluding unrealistic values.

I assume they want us to find the Revenue (income) function.

Let \(\displaystyle x\) = number of $0.25 increases in the fare.
\(\displaystyle \;\;\)The new fare will be: \(\displaystyle \,\$2.25\,+\,0.25x\)

But they will lose \(\displaystyle 40x\) customers.
\(\displaystyle \;\;\)The number of customers will be: \(\displaystyle \,800\,-\,40x\)

Hence, the revenue is: \(\displaystyle \,R \;=\;(2.25\,+\,0.25x)(800\,-\,40x)\)
\(\displaystyle \;\;\)which simplifies to: \(\displaystyle \,R\;=\;1800\;+\;110x\,-\,10x^2\)

This is a quadratic function whose graph is a parabola. (b)

Since the \(\displaystyle x^2\)-term is negative, the parabola opens downward.

Let \(\displaystyle R\,=\,0\), and we have: \(\displaystyle \,-10x^2\,+\,110x\,+\,1800\:=\:0\)

Divide by -10: \(\displaystyle \,x^2\,-\,11x\,-\,180\:=\;0\;\;\Rightarrow\;\;(x\,+\,9)(x\,-\,20)\:=\:0\)

The \(\displaystyle x\)-intecepts are: \(\displaystyle \,(-9,0),\;(20,0)\)

(a)                   |   * *
                      *         *
                   *  |            *
                 *    |              *
                *     |               *
                      |
             --*------+----------------*--
              -9      |                20 
                      |
              *       |                 *

(c) As shown in the graph, the domain is all real \(\displaystyle x:\;(-\infty,\,\infty)\)

The vertex is: \(\displaystyle \,(5.5,\,2102.50)\)
Hence, the range is: \(\displaystyle (-\infty,\,2102.50]\)


(d) Excuding unrealistic values . . .
The domain is: \(\displaystyle \,(-9,\,20)\) . . . excluding -9 and 20
The range is: \(\displaystyle (0,\,2102.50]\)

If \(\displaystyle x\,=\,-9\), they would reduce the price by 25¢ nine times.
They would have 1160 customers, but the price is $0 . . . no revenue!

If \(\displaystyle x\,=\,20\), they would raise the price by 25¢ twenty times.
They would charge $7.25 per ride, but they would have 0 customers . . . no revenue!