Question

look at this table:

x -1 0 1 2 3
y 1.62 4.86 14.58 43.74 131.22
Write a linear function (y = mx + b) or an exponential function (y=a(b)^x) that models the data

Answer 1

[AI-generated answer — Bot GPT 3.5]

Looking at the given table, we can observe that as the values of x increase, the values of y seem to be increasing at a much faster rate. This indicates that the data is likely to be modeled by an exponential function rather than a linear function.

To determine the exponential function that models the data, we can look for a pattern in the values. Let's take a look at the ratios between consecutive values of y:

4.86/1.62 ≈ 3
14.58/4.86 ≈ 3
43.74/14.58 ≈ 3
131.22/43.74 ≈ 3

The ratios are approximately equal to 3, indicating that each value of y is roughly 3 times the previous value.

By substituting the first point (x = -1, y = 1.62) into the exponential function form y = a(b)^x, we can find the base (b) of the exponential function:

1.62 = a(b)^-1

Rearranging the equation, we get:

(b)^-1 = 1.62/a

Taking the reciprocal of both sides, we have:

b = a/1.62

Now, we can substitute the second point (x = 0, y = 4.86) into the exponential function form:

4.86 = a(b)^0 = a

Thus, we know that a = 4.86.

Finally, substituting this value of a into the equation for b that we found earlier, we can determine the value of b:

b = 4.86/1.62 ≈ 3

Therefore, the exponential function that models the given data is:

y = 4.86(3)^x