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The table represents an exponential function.
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline
[tex]$x$[/tex] & -3 & -2 & -1 & 0 & 1 & 2 & 3 \\
\hline
[tex]$y$[/tex] & 192 & 48 & 12 & 3 & [tex]$\frac{3}{4}$[/tex] & [tex]$\frac{3}{16}$[/tex] & [tex]$\frac{3}{64}$[/tex] \\
\hline
\end{tabular}

Does the function in the table represent growth or decay?

A. The function represents exponential decay because the base equals 4.

B. The function represents exponential growth because the base equals 4.

C. The function represents exponential decay because the base equals [tex]$\frac{1}{4}$[/tex].

D. The function represents exponential growth because the base equals [tex]$\frac{1}{4}$[/tex].


Sagot :

To determine whether the function in the table represents exponential growth or decay, we need to analyze the ratios between successive [tex]\( y \)[/tex] values. These ratios will help us identify the base of the exponential function.

Let's calculate these ratios step-by-step.

1. We have the data points:
[tex]\[ \begin{array}{|c|c|c|c|c|c|c|c|} \hline x & -3 & -2 & -1 & 0 & 1 & 2 & 3 \\ \hline y & 192 & 48 & 12 & 3 & \frac{3}{4} & \frac{3}{16} & \frac{3}{64} \\ \hline \end{array} \][/tex]

2. Calculate the ratios between successive [tex]\( y \)[/tex] values:
[tex]\[ \text{Ratio between } y(-2) \text{ and } y(-3): \frac{y(-2)}{y(-3)} = \frac{48}{192} = 0.25 \][/tex]
[tex]\[ \text{Ratio between } y(-1) \text{ and } y(-2): \frac{y(-1)}{y(-2)} = \frac{12}{48} = 0.25 \][/tex]
[tex]\[ \text{Ratio between } y(0) \text{ and } y(-1): \frac{y(0)}{y(-1)} = \frac{3}{12} = 0.25 \][/tex]
[tex]\[ \text{Ratio between } y(1) \text{ and } y(0): \frac{y(1)}{y(0)} = \frac{0.75}{3} = 0.25 \][/tex]
[tex]\[ \text{Ratio between } y(2) \text{ and } y(1): \frac{y(2)}{y(1)} = \frac{\frac{3}{16}}{\frac{3}{4}} = 0.25 \][/tex]
[tex]\[ \text{Ratio between } y(3) \text{ and } y(2): \frac{y(3)}{y(2)} = \frac{\frac{3}{64}}{\frac{3}{16}} = 0.25 \][/tex]

3. We see that the ratio is consistent across all pairs and equals 0.25. This means that the function can be defined with a base of 0.25 in the exponential function form.

Since the base of the exponential function is [tex]\( 0.25 \)[/tex], which is less than 1, this indicates that the function represents exponential decay.

Therefore, the correct statement is:
[tex]\[ \boxed{\text{The function represents exponential decay because the base equals } \frac{1}{4}.} \][/tex]
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