look at this table:
x 2 3 4 5 6
y -26 -36 -46 -56 -66
Write a linear function (y = mx + b) or an exponential function (y=a(b)^x) that models the data
To determine whether the table follows a linear or exponential pattern, we can calculate the differences between consecutive y-values. If the differences are consistent, then it is a linear relationship. However, if the differences increase or decrease exponentially, then it is an exponential function.
Calculating the differences between consecutive y-values:
-36 - (-26) = -10
-46 - (-36) = -10
-56 - (-46) = -10
-66 - (-56) = -10
Since the differences are consistent (-10) throughout the table, it implies that the relationship is linear. Therefore, we can write a linear function (y = mx + b) to model the data.
Using any two points from the table, let's say (2, -26) and (6, -66), we can find the slope (m) and y-intercept (b) of the linear function.
Step 1: Calculate the slope (m):
m = (y2 - y1) / (x2 - x1)
= (-66 - (-26)) / (6 - 2)
= (-66 + 26) / 4
= -40 / 4
= -10
Step 2: Determine the y-intercept (b):
Using the point (2, -26) and the slope (m) calculated in step 1, we can substitute these values into the linear equation y = mx + b and solve for b:
-26 = -10(2) + b
-26 = -20 + b
b = -26 + 20
b = -6
Therefore, the linear function that models the data is:
y = -10x - 6