The data (1, 5), (3, 16), (5, 35), (7, 67), (9, 110) could be modeled as a linear relationship if (2 points)
x is transformed to x2
y is transformed to y2
x is transformed to x3
y is transformed to y3
To determine if the given data can be modeled as a linear relationship, we need to check if transforming the variables x or y produces a linear relationship.
Transformation 1: x is transformed to x^2
To check if transforming x to x^2 creates a linear relationship, we need to calculate the transformed values of y.
Original Data: (1, 5), (3, 16), (5, 35), (7, 67), (9, 110)
Transformed Data (x^2): (1, 1), (9, 3), (25, 5), (49, 7), (81, 9)
If we plot the transformed data, we can see that it does not form a linear relationship. The points are not aligned in a straight line, so this transformation does not result in a linear relationship.
Transformation 2: y is transformed to y^2
Similarly, we can check if transforming y to y^2 creates a linear relationship by calculating the transformed values of x.
Original Data: (1, 5), (3, 16), (5, 35), (7, 67), (9, 110)
Transformed Data (y^2): (5, 25), (16, 256), (35, 1225), (67, 4489), (110, 12100)
Again, if we plot the transformed data, we can see that it does not form a linear relationship. The points are not aligned in a straight line, so this transformation does not result in a linear relationship.
Transformation 3: x is transformed to x^3
Next, let's check if transforming x to x^3 creates a linear relationship by calculating the transformed values of y.
Original Data: (1, 5), (3, 16), (5, 35), (7, 67), (9, 110)
Transformed Data (x^3): (1, 1), (27, 3), (125, 5), (343, 7), (729, 9)
If we plot the transformed data, we can see that it forms a linear relationship. The points are aligned in a straight line, so this transformation results in a linear relationship.
Transformation 4: y is transformed to y^3
Finally, let's check if transforming y to y^3 creates a linear relationship by calculating the transformed values of x.
Original Data: (1, 5), (3, 16), (5, 35), (7, 67), (9, 110)
Transformed Data (y^3): (5,125), (16, 4096), (35, 42875), (67, 300763), (110, 1331000)
Once again, if we plot the transformed data, we can see that it does not form a linear relationship. The points are not aligned in a straight line, so this transformation does not result in a linear relationship.
To summarize, out of the given transformations, only transforming x to x^3 results in a linear relationship when considering the given data.