Question

Determine if the table shows a linear or an exponential function​

Answer 1

Like ya cuttt g..........

Answer 2

The table shows an exponential function

Explanation:

Linear vs Exponential Functions

A linear function is written as:

`y=mx+b`

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:

`\displaystyle m=(y_2-y_1)/(x_2-x_1)`

An exponential function is written as:

`y=y_o.r^x`

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:

`\displaystyle r=\sqrt[x2-x1]{(y2)/(y1)}`

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

`\displaystyle m=(2-4)/(1-0)=-2`

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

`\displaystyle m=(1-2)/(2-1)=-1`

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

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

`\displaystyle r=\sqrt[1-0]{(1)/(2)}`

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

`\displaystyle r=(1)/(2)`

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

`\displaystyle r=\sqrt[2-0]{(1)/(4)}`

`\displaystyle r=\sqrt{(1)/(4)}`

`\displaystyle r=(1)/(2)`

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