Four friends went to a festival, where each had to pay an entrance fee and buy tickets for each ride they went on.
The table here shows how much money each of them spent at the festival and the number of rides each went on.
Individual
Number of Rides Amount of Money Spent (Dollars)
Friend 1
4
10.5
Friend 2
7
15.75
Friend 3
3
8.75
Friend 4
6
14
Determine if the relationship between the amount of money spent and the number of rides each person went on is a linear relationship. Explain your reasoning.
If the table of values represents a linear function, write the equation of the linear function. Interpret the value of the rate of change and the initial value in the context of the problem.
To determine if the relationship between the amount of money spent and the number of rides each person went on is a linear relationship, we can create a scatter plot of the data points and see if they form a straight line.
After plotting the data points, it seems that they do fall along a straight line, indicating a linear relationship between the amount of money spent and the number of rides.
To find the equation of the linear function, we can use the slope-intercept form y = mx + b, where y is the amount of money spent and x is the number of rides.
First, we need to calculate the slope (m) using two points on the line. Let's use Friend 1 and Friend 2:
m = (15.75 - 10.5) / (7 - 4) = 5.25 / 3 = 1.75
Now, we can use the slope and one data point to find the y-intercept (b). Let's use Friend 1:
10.5 = 1.75(4) + b
10.5 = 7 + b
b = 3.5
Therefore, the equation of the linear function is y = 1.75x + 3.5, where y is the amount of money spent and x is the number of rides.
In the context of the problem, the rate of change (1.75) represents the cost per ride at the festival. This means that for each additional ride, the amount spent increases by $1.75. The initial value (3.5) represents the fixed cost of attending the festival, regardless of the number of rides taken.