Explain the transformation needed to convert the following data to a linear data set.
{(1, 0.98), (2, 1.39), (3, 1.71), (4, 1.98), (5, 2.22), (6, 2.43)} (6 points)
To convert the given data set to a linear data set, we need to perform a transformation known as "linearization." This involves finding a linear equation in the form of y = mx + b that best represents the given data points.
To begin, let's plot the given data points on a graph to visualize the relationship between the x and y values.
Using software or graph paper, plot the points (1, 0.98), (2, 1.39), (3, 1.71), (4, 1.98), (5, 2.22), and (6, 2.43) on a graph.
Once you have the points plotted, try to draw a line that best fits the overall trend of the points. The line should be as close as possible to each point.
Adjust the line until it appears to pass through or come closest to the majority of the points.
Once you have drawn this best-fit line, you can determine the equation of this line by using the slope-intercept form, y = mx + b.
To find the slope (m) of the line, choose two points on the line, such as (1, 0.98) and (6, 2.43). Use the formula:
m = (y2 - y1) / (x2 - x1)
m = (2.43 - 0.98) / (6 - 1)
m = 1.45 / 5
m = 0.29
Now, we need to find the y-intercept (b) of the line. We can use any point on the line and substitute the coordinates into the equation y = mx + b. Let's use the point (1, 0.98):
0.98 = 0.29 * 1 + b
0.98 = 0.29 + b
b = 0.98 - 0.29
b = 0.69
So, the best-fit line equation for the given data set is y = 0.29x + 0.69. This equation represents the linearized form of the original data set.