Calculate the difference and ratio between consecutive terms. Is either of these the same throughout the sequence?
Model That Best Describes the Data: Exponential Equation: y=2^x
Practice makes perfect
Finding the Model
We want to identify which kind of model best describes the data, linear, quadratic or exponential. To do so we will calculate the difference and ratio between consecutive terms.
x
0
1
2
3
4
y
1
2
4
8
16
Let's begin by calculating the first differences.
The first differences are not all equal. Therefore, the table of values does not represent a linear function. Let's find the second differences and compare them.
The second differences are not all equal. Therefore, the table of values does not represent a quadratic function. Let's find the ratios and compare them.
The ratios of successive y-values are equal. Therefore, the table of values can be modeled by an exponential function.
Finding the Equation
Let's recall the general form of this type of function.
y=ab^x
We will use two ordered pairs given in the table to find the values of a and b. For simplicity, let's use (0,1) and (1,2). We will start by substituting 0 and 1 for x and y, respectively.
We can write a partial equation of the function represented by the table.
y=(1)b^x ⇔ y=b^x
To find the value of b we will substitute 1 for x and 2 for y into our partial equation.
