In most businesses, increasing prices of products can negatively impact the number of customers. 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.

a) Let the number of riders be a function of the fare charged. Graph the function, identify the graph of the function (line, parabola, hyperbola, or exponential), find the slope of the graph, find the price at which there will be no more riders, and the maximum number of riders possible.

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To begin, let's first identify the variables involved in this problem:

Let:
- x represent the fare charged in dollars
- y represent the number of riders

We are given that the bus company currently charges $2.25 for a ride and has an average of 800 riders per day. Based on the information provided in the survey, for every $0.25 increase in fare, the bus company will lose approximately 40 customers per day.

Therefore, we can formulate the relationship between the fare charged and the number of riders as follows:

y = 800 - 40(x - 2.25)

Now, let's analyze the equation and answer the questions:

a) Graphing the function:
To graph the function, we plot the points with the fare charged (x) on the x-axis and the number of riders (y) on the y-axis. Since the equation of the function is linear, the graph will be a straight line.

b) Finding the slope of the graph:
The slope of the graph represents the rate at which the number of riders changes concerning the fare charged. In this case, the slope is -40. This means that for every $0.25 increase in fare, the number of riders decreases by 40.

c) Finding the price at which there will be no more riders:
To find the price at which there will be no more riders, we need to set the number of riders (y) to zero and solve for x:

0 = 800 - 40(x - 2.25)

Now, we can solve for x:

800 = 40(x - 2.25)
800 = 40x - 90
890 = 40x
x = 22.25

Therefore, there will be no more riders if the fare is set to $22.25.

d) Finding the maximum number of riders possible:
To find the maximum number of riders possible, we need to identify the point on the graph where the number of riders (y) is at its highest value. Since the slope of the graph is negative, the maximum number of riders occurs at the lowest fare.

By observing the equation, we can see that the fare charged cannot go below $2.25 (the initial fare). Therefore, the maximum number of riders possible is 800.

In summary, the relationship between the fare charged and the number of riders is described by the linear function y = 800 - 40(x - 2.25), where x represents the fare charged, and y represents the number of riders. The graph of this function is a straight line. The slope of the graph is -40, indicating that for every $0.25 increase in fare, there is a decrease of 40 riders. The fare at which there will be no more riders is $22.25, and the maximum number of riders possible is 800.