Look at this table:
x y
1 15
2 45
3 135
4 405
5 1,215
Write a linear function (y=mx+b) or an exponential function (y=a(b)x) that models the data.
y=
To determine whether the data can be modeled with a linear function (y=mx+b) or an exponential function (y=a(b)^x), we need to observe the relationship between the x and y values.
If we divide each y-value by the previous x-value, we can see if there is a consistent ratio between them. Let's calculate the ratios:
y2/y1 = 45/15 = 3
y3/y2 = 135/45 = 3
y4/y3 = 405/135 = 3
y5/y4 = 1,215/405 = 3
As the ratio is the same (3) for each x-value, we can conclude that the data follows an exponential function.
To determine the specific equation for the exponential function, we can use the formula: y = a(b)^x.
Using the first data point (x=1, y=15), we can substitute these values into the equation:
15 = a(b)^1
Simplifying, we have:
15 = ab
Next, we can use another data point (x=2, y=45) to create a second equation:
45 = a(b)^2
Since we already know from the first equation that ab is equal to 15, we can substitute 15 for ab in the second equation:
45 = 15b^2
Dividing both sides of the equation by 15, we get:
3 = b^2
Taking the square root of both sides, we find:
b = ±√3
Since the values of b can be positive or negative, we need to consider both possibilities. Let's use the negative value for b to calculate a:
Substituting the values into the first equation:
15 = a(-√3)
To isolate a, we can divide both sides of the equation by -√3:
a = -15/√3
Therefore, the function that models the given data is:
y = (-15/√3)(-√3)^x
Simplifying, we get:
y = (-15/√3)(-3)^x
So, the exponential function that models the data is:
y = (-15/√3)(-3)^x