Final answer:
An equation to represent the constant of proportionality between the number of cats and dogs is c = 3d, depicted on a graph as a straight line with points where the number of cats is three times the number of dogs.
Step-by-step explanation:
The student's question revolves around establishing a mathematical equation that represents the constant of proportionality between the number of cats (c) and the number of dogs (d) in a pet shelter, with the constant given as 3. This situation can be mathematically expressed with the equation c = 3d. To graph this relationship, one would plot the number of dogs on the x-axis and the number of cats on the y-axis. Every point on the line y = 3x (where y represents cats and x represents dogs) would reflect a possible distribution of cats and dogs that adheres to the given constant of proportionality of 3.
To create a graph, you would start by plotting the point (0,0) because if there are no dogs, there will be no cats according to this relationship. Next, if you have 1 dog, you would have 3 cats, and you would plot the point (1,3). Continue plotting points where the number of cats is always three times the number of dogs, and connect these points to form a straight line, as direct proportionality results in a linear graph.
Note that this straightforward relationship assumes that the proportionality constant remains fixed and does not consider real-world limitations, such as the shelter's capacity or the likelihood of fluctuations in the population of cats or dogs.