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Sagot :
Let's analyze the function represented by the given table:
[tex]\[ \begin{tabular}{|c|c|c|c|c|c|} \hline$x$ & 1 & 2 & 3 & 4 & 5 \\ \hline$y$ & 1 & 16 & 64 & 256 & 1,024 \\ \hline \hline \end{tabular} \][/tex]
To determine which statement best describes the graph, we analyze the pattern in the [tex]\( x \)[/tex] and [tex]\( y \)[/tex] values.
First, we look at the differences between consecutive [tex]\( y \)[/tex]-values:
[tex]\[ \begin{align*} y_2 - y_1 & = 16 - 1 = 15, \\ y_3 - y_2 & = 64 - 16 = 48, \\ y_4 - y_3 & = 256 - 64 = 192, \\ y_5 - y_4 & = 1,024 - 256 = 768. \\ \end{align*} \][/tex]
The differences (15, 48, 192, 768) are not constant. Therefore, the function is not linear, and the graph is not a straight line with a slope of 8.
Next, we check for exponential growth by calculating the ratios of consecutive [tex]\( y \)[/tex]-values:
[tex]\[ \begin{align*} \frac{y_2}{y_1} & = \frac{16}{1} = 16, \\ \frac{y_3}{y_2} & = \frac{64}{16} = 4, \\ \frac{y_4}{y_3} & = \frac{256}{64} = 4, \\ \frac{y_5}{y_4} & = \frac{1,024}{256} = 4. \\ \end{align*} \][/tex]
The ratios (16, 4, 4, 4) are not the same initially, indicating complex exponential behavior. However, the majority of the ratios indicates multiplicative behavior in the sequence.
To confirm this, we consider the logarithms of the [tex]\( y \)[/tex]-values and observe the resulting sequence:
[tex]\[ \begin{align*} \log(y_1) & = \log(1) = 0, \\ \log(y_2) & = \log(16) \approx 2.77, \\ \log(y_3) & = \log(64) \approx 4.16, \\ \log(y_4) & = \log(256) \approx 5.54, \\ \log(y_5) & = \log(1,024) \approx 6.93. \\ \end{align*} \][/tex]
Plotting the above logarithmic values against [tex]\( x \)[/tex] will show an approximately linear trend, confirming an exponential relationship.
Therefore, the graph does not fit the description of a horizontal line or a parabola that opens upward. It also doesn’t fit a straight line with a slope of 8.
Thus, the best fitting statement for the given function is:
The graph starts flat but curves steeply upward.
[tex]\[ \begin{tabular}{|c|c|c|c|c|c|} \hline$x$ & 1 & 2 & 3 & 4 & 5 \\ \hline$y$ & 1 & 16 & 64 & 256 & 1,024 \\ \hline \hline \end{tabular} \][/tex]
To determine which statement best describes the graph, we analyze the pattern in the [tex]\( x \)[/tex] and [tex]\( y \)[/tex] values.
First, we look at the differences between consecutive [tex]\( y \)[/tex]-values:
[tex]\[ \begin{align*} y_2 - y_1 & = 16 - 1 = 15, \\ y_3 - y_2 & = 64 - 16 = 48, \\ y_4 - y_3 & = 256 - 64 = 192, \\ y_5 - y_4 & = 1,024 - 256 = 768. \\ \end{align*} \][/tex]
The differences (15, 48, 192, 768) are not constant. Therefore, the function is not linear, and the graph is not a straight line with a slope of 8.
Next, we check for exponential growth by calculating the ratios of consecutive [tex]\( y \)[/tex]-values:
[tex]\[ \begin{align*} \frac{y_2}{y_1} & = \frac{16}{1} = 16, \\ \frac{y_3}{y_2} & = \frac{64}{16} = 4, \\ \frac{y_4}{y_3} & = \frac{256}{64} = 4, \\ \frac{y_5}{y_4} & = \frac{1,024}{256} = 4. \\ \end{align*} \][/tex]
The ratios (16, 4, 4, 4) are not the same initially, indicating complex exponential behavior. However, the majority of the ratios indicates multiplicative behavior in the sequence.
To confirm this, we consider the logarithms of the [tex]\( y \)[/tex]-values and observe the resulting sequence:
[tex]\[ \begin{align*} \log(y_1) & = \log(1) = 0, \\ \log(y_2) & = \log(16) \approx 2.77, \\ \log(y_3) & = \log(64) \approx 4.16, \\ \log(y_4) & = \log(256) \approx 5.54, \\ \log(y_5) & = \log(1,024) \approx 6.93. \\ \end{align*} \][/tex]
Plotting the above logarithmic values against [tex]\( x \)[/tex] will show an approximately linear trend, confirming an exponential relationship.
Therefore, the graph does not fit the description of a horizontal line or a parabola that opens upward. It also doesn’t fit a straight line with a slope of 8.
Thus, the best fitting statement for the given function is:
The graph starts flat but curves steeply upward.
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