Look at this table:
x y
–
6
–
3.5
–
5
–
2.8
–
4
–
2.1
–
3
–
1.4
–
2
–
0.7
Write a linear function (y=mx+b) or an exponential function (y=a(b)x) that models the data.
y=
In order to determine whether the given data can be modeled by a linear or exponential function, we can examine how the y-values change with respect to the x-values.
If the y-values change at a constant rate as the x-values change, then the data can be modeled by a linear function. If the y-values change at a constant ratio as the x-values change, then the data can be modeled by an exponential function.
Examining the given data, we can see that as the x-values increase by 1, the y-values decrease by a constant amount. Therefore, the data can be modeled by a linear function.
To find the slope (m) and y-intercept (b) of the linear function, we can choose two points from the table and use them to calculate the slope.
Let's choose the points (-6, 0) and (-2, 2) from the table.
Slope (m) = (change in y) / (change in x)
= (0 - 2) / (-6 - (-2))
= -2 / -4
= 0.5
Using the slope and one of the points, we can find the y-intercept (b) by substituting the values of x, y, and m into the linear equation y = mx + b and solving for b.
0 = 0.5(-6) + b
0 = -3 + b
b = 3
Therefore, the linear function that models the given data is:
y = 0.5x + 3