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

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Answer:The table shows an exponential functionStep-by-step explanation:Linear vs Exponential FunctionsA 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

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