Look at this table:
x y
1 5.9
2 11.9
3 17.9
4 23.9
5 29.9
Write a linear function (y=mx+b) or an exponential function (y=a(b)x) that models the data.
y=
In order to determine if the given data can be modeled by a linear or exponential function, let's examine the relationship between the x-values and y-values.
By observing the x-values, we can notice that they increase by a constant difference of 1: 1, 2, 3, 4, 5. Therefore, this suggests a linear relationship.
By observing the y-values, they increase by a constant difference of 5.9: 5.9, 11.9, 17.9, 23.9, 29.9. However, this difference does not remain constant as x increases. Thus, it is not an exponential relationship.
To find the linear function that models the data, we need to find the slope (m) and y-intercept (b).
Let's calculate the difference in y-values for each step:
Δy1 = 11.9 - 5.9 = 6
Δy2 = 17.9 - 11.9 = 6
Δy3 = 23.9 - 17.9 = 6
Δy4 = 29.9 - 23.9 = 6
We can observe that the difference in y-values is constant at 6.
Now, let's calculate the difference in x-values for each step:
Δx1 = 2 - 1 = 1
Δx2 = 3 - 2 = 1
Δx3 = 4 - 3 = 1
Δx4 = 5 - 4 = 1
We can observe that the difference in x-values is also constant at 1.
Therefore, the slope (m) of the linear function can be calculated as the ratio of the change in y-values to the change in x-values: m = Δy / Δx = 6 / 1 = 6.
To find the y-intercept (b), we can choose any point from the table. Let's use the first point (1, 5.9):
y = mx + b
5.9 = 6(1) + b
5.9 = 6 + b
b = 5.9 - 6
b = -0.1
Finally, the linear function that models the given data is:
y = 6x - 0.1