What's the function?
x= 2, y=14
x=10, y=54
x=8, y=44
x=?, y=34
x=?, y=104
x=?, y=31
I thought it was multiply by 5, add 4 but that won't work for the last one where y=31. Help!
I agree with you on your choice of the rule.
Functions can work with decimal numbers as well. So I would say x=5.4 solves the mystery of the last one.
To determine the function that relates the values of x and y in the provided examples, let's examine the patterns in the given data points.
First, we'll look at the relationship between the x-values and y-values. Here are the pairs of x and y values given:
1. (2, 14)
2. (10, 54)
3. (8, 44)
Upon examining the data, we can observe that the y-values do not seem to be directly related to the x-values through the same operation. The pattern we see is not apparent from the x-values alone. Therefore, it appears that there may be another factor involved.
Let's now consider the difference between y-values in adjacent examples:
1. (2, 14)
2. (10, 54)
- Difference: 54 - 14 = 40
3. (8, 44)
- Difference: 44 - 54 = -10
We can see that the difference between the y-values is not constant. Therefore, we can conclude that there is no linear relationship between x and y where the difference between y-values is the same for every pair.
To find the function that relates the x and y values, let's examine the differences between the y-values again:
1. (2, 14)
2. (10, 54)
- Difference: 40
3. (8, 44)
- Difference: -10
Comparing the differences, we can see that by taking the difference of the differences, we find a constant value:
40 - (-10) = 50
This suggests that the relationship between x and y may involve a quadratic or nonlinear function. To verify this, let's look at the difference between differences for the given data points:
1. (2, 14)
2. (10, 54)
- Difference: 50
3. (8, 44)
- Difference: 50
The difference of differences is the same for each pair of differences, indicating a constant value. This suggests that y-values are increasing at a consistent rate.
To determine the exact function, we can use the given data point (x=?, y=31) as well as the difference between the differences, which is 50.
Using the known data, we can calculate the x-value by finding the first difference:
x = 2 + (31 - 14) = 19
To calculate the y-value when x is unknown, we can use the second difference:
y = 31 + 50 = 81
Thus, the function that appears to fit the given data points is a quadratic function, where the difference between the differences is constant.