Look at this table:
x y
1 14
2 23
3 32
4 41
5 50
Write a linear function (y=mx+b) or an exponential function (y=a(b)x) that models the data.
y=
To determine whether the data can be represented by a linear or exponential function, we can examine the rate of change between the x and y values.
If we calculate the difference between consecutive y values, we can see that the difference is 9 for each pair of values:
23-14 = 9
32-23 = 9
41-32 = 9
50-41 = 9
This shows that the change in the y values is constant, indicating a linear relationship.
Additionally, we can calculate the difference between consecutive x values, which is always 1. This confirms that the relationship is linear.
Using the slope-intercept form of a linear equation, y = mx + b, we can determine the equation:
m = (y2 - y1) / (x2 - x1)
m = (23 - 14) / (2 - 1) = 9
Using the first point (x=1, y=14), we can substitute the values into the equation and solve for b:
14 = 9(1) + b
14 = 9 + b
b = 14 - 9
b = 5
Therefore, the linear function that models the data is:
y = 9x + 5