Answer:
The table shows an exponential function
Step-by-step explanation:
Linear vs Exponential Functions
A linear function is written as:
[tex]y=mx+b[/tex]
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:
[tex]\displaystyle m=\frac{y_2-y_1}{x_2-x_1}[/tex]
An exponential function is written as:
[tex]y=y_o.r^x[/tex]
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:
[tex]\displaystyle r=\sqrt[x2-x1]{\frac{y2}{y1}}[/tex]
Calculate the slope for (0,4) and (1,2):
[tex]\displaystyle m=\frac{2-4}{1-0}=-2[/tex]
Calculate the slope for (1,2) and (2,1):
[tex]\displaystyle m=\frac{1-2}{2-1}=-1[/tex]
Since the slope is not the same, the function is not linear.
Now calculate the ratio for (0,4) and (1,2)
[tex]\displaystyle r=\sqrt[1-0]{\frac{1}{2}}[/tex]
The radical of index 1 is simply equal to its argument:
[tex]\displaystyle r=\frac{1}{2}[/tex]
Now calculate the ratio for (0,4) and (2,1)
[tex]\displaystyle r=\sqrt[2-0]{\frac{1}{4}}[/tex]
[tex]\displaystyle r=\sqrt{\frac{1}{4}}[/tex]
[tex]\displaystyle r=\frac{1}{2}[/tex]
Testing other points we'll find the same ratio, thus the table is an exponential function
