Look at this table:
x y
0 2
1 6
2 18
3 54
4 162
Write a linear function (y=mx+b) or an exponential function (y=a(b)x) that models the data.
y=
To determine whether the function that models the data is linear or exponential, we need to observe the relationship between the x-values and the y-values.
Looking at the table, we can see that as the x-values increase, the corresponding y-values increase at an exponential rate. The y-values seem to be tripling each time we increase x by 1.
To represent this exponential growth, the function that models the data is:
y = a(b)^x
To find the values of a and b, we can look at the x and y values in the table.
Starting with the first row:
When x = 0, y = 2.
This gives us the equation y = a(b)^0. Since any number raised to the power of 0 equals 1, we get y = a(1), which simplifies to y = a.
So, a = 2.
Using this value of a, we can find the value of b. Observing the pattern in the table, we can see that as x increases by 1, the y-value triples. Therefore, we can say that:
y(1) = 3y(0)
6 = 3(2)
6 = 6
y(2) = 3y(1)
18 = 3(6)
18 = 18
y(3) = 3y(2)
54 = 3(18)
54 = 54
y(4) = 3y(3)
162 = 3(54)
162 = 162
This confirms that the common ratio (b) is 3.
Now, the complete function that models the data is:
y = 2(3)^x