5. Principal Farley has $200 to pay for some teachers to attend a technology conference. The company hosting the conference is allowing 2 teachers to attend for free. The number of teachers y that can be sent to the conference is given by the function y = 200/x + 2, where x is the cost per teacher.
a. Describe the reasonable domain and range values for this function.
b. Identify the vertical and horizontal asympototes.
c. Graph the function.
d. Give two points on the graph whose coordinates are whole numbers and describe what they mean in the context of this situation.
A:
a. ?
b. ?
c. ?
d. ?
a. In this context, the reasonable domain values for the function would be values of x that are greater than 0, since the cost per teacher cannot be negative or zero. However, it should also be noted that the cost per teacher should be less than or equal to the remaining budget after accounting for the 2 free teachers. So, the domain can be expressed as 0 < x ≤ (200 - 2) = 198.
The reasonable range values for the function would be values of y that are greater than or equal to 2, since the company is allowing 2 teachers to attend for free. However, the range should also consider the maximum number of teachers that can be sent to the conference based on the budget. So, the range can be expressed as 2 ≤ y ≤ (200/x + 2).
b. Vertical asymptotes occur when the function approaches infinity or negative infinity. In this case, if x approaches 0, the function y would approach positive infinity. So, x = 0 is a vertical asymptote.
There are no horizontal asymptotes in this function since the range values (y) can vary based on the cost per teacher (x). The function does not approach any specific value as x becomes large or small.
c. To graph the function, you can plot points by selecting different values of x within the domain and then calculating the corresponding y values using the given function. Connect the plotted points to visualize the graph. However, as an AI text-based bot, I am unable to provide a visual representation of the graph. You may use graphing tools or software to plot the graph of the function.
d. To find two points on the graph with whole numbers, you can choose values of x that are factors of 200 as it will ensure the resulting y values are whole numbers as well. For example, if x = 10, substituting it into the function y = (200/10) + 2 will give y = 22. So, one point on the graph would be (10, 22). This means that with a cost of $10 per teacher, 22 teachers can be sent to the conference.
Similarly, if x = 25, substituting it into the function y = (200/25) + 2 will give y = 10. So, another point on the graph would be (25, 10). This means that with a cost of $25 per teacher, 10 teachers can be sent to the conference.
These points provide examples of the cost per teacher (x) and the corresponding number of teachers that can be sent to the conference (y) within the context of the given situation.