Calculate the differences and ratios between consecutive terms. Are either of these the same throughout the sequence?
Type of Function: Exponential Function: y=3* 3^x
Practice makes perfect
Finding the Model
We want to tell whether the table of values represents a linear, exponential, or quadratic function. To do so, we will analyze how the consecutive terms are related to each other.
x
-1
0
1
2
3
y
1
3
9
27
81
Let's begin with 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 of the y-values 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 (0,3) and (1,9). We will start by substituting 0 and 3 for x and y, respectively.
We can write a partial equation of the function represented by the table.
y=3b^x
To find the value of b we will substitute 1 for x and 9 for y into our partial equation.
