Look at this table:
x y
–
6 36
–
5 31
–
4 26
–
3 21
–
2 16
Write a linear function (y=mx+b) or an exponential function (y=a(b)x) that models the data.
y=
To determine if the relationship given in the table is a linear or exponential function, we need to examine the pattern of change in the y-values with respect to the x-values.
In the given table, as the x-values decrease by 1, the y-values decrease by 5. This shows a consistent rate of change, indicating that the relationship is linear.
To find the slope, m, we can take any two points from the table and use the formula:
m = (change in y) / (change in x).
Let's use the points (x = -6, y = 36) and (x = -5, y = 31):
m = (31 - 36) / (-5 - (-6))
m = (31 - 36) / (-5 + 6)
m = -5 / 1
m = -5
Now that we have the slope, we can use the point-slope form of a linear equation, y - y₁ = m(x - x₁), and substitute any point from the table for (x₁, y₁).
Let's use the point (x = -6, y = 36):
y - 36 = -5(x - (-6))
y - 36 = -5(x + 6)
y - 36 = -5x - 30
y = -5x + 6
Therefore, the linear function that models the data is y = -5x + 6.