Look at this table:x y1–22–43–84–165–32Write a linear (y=mx+b),

Look at this table:x y1–22–43–84–165–32Write a linear (y=mx+b),

Question
Functions
asked 2020-11-27

Look at this table: 

\(\begin{array}{|c|c|}\hline x & y \\ \hline 1 & 2 \\ \hline 2 & 4 \\ \hline 3 & 8 \\ \hline 4 & 16 \\ \hline 5 & 32 \\ \hline \end{array}\)

Write a linear \((y=mx+b),\) quadratic \((y=ax2)\), or exponential \((y=a(b)x)\) function that models the data.

\(y=?\)

Answers (1)

2020-11-28
The x-values are increasing by 1 each time and the y-values are doubling each time. Since the y-values are increasing by a constant factor of 2, the model must be exponential. The data would have to increase by a constant amount to be linear and the second difference of the y-values would have to be constant for the model to be quadratic.
Notice that each yy-coordinate is a power of 2. Since \(\displaystyle{2}={2}^{{1}},{4}={2}^{{2}},{8}={2}^{{3}},{16}={2}^{{4}}\), and \(\displaystyle{32}={2}^{{5}}\), then each y-coordinate is a power of 2 where the exponent is the x-coordinate. The equation is then \(\displaystyle{y}={2}^{{x}}\).
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