Determine if the table shows a linear or an exponential function

The table shows an exponential function

Step-by-step explanation:

Linear vs Exponential Functions

A linear function is written as:

where m and b are constants.

If a table contains a linear function, then for each pair of ordered pairs (x1,y1) and (x2,y2), the value of m must be constant.

The slope can be calculated as:

An exponential function is written as:

Where r is the ratio and yo is a constant.

If a table contains an exponential function, for two ordered pairs (x1,y1) and (x2,y2), the value of r must be constant.

The ratio can be calculated as:

Calculate the slope for (0,4) and (1,2):

Calculate the slope for (1,2) and (2,1):

Since the slope is not the same, the function is not linear.

Now calculate the ratio for (0,4) and (1,2)

The radical of index 1 is simply equal to its argument:

Now calculate the ratio for (0,4) and (2,1)

Testing other points we'll find the same ratio, thus the table is an exponential function