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Sagot :
Certainly! Let's address each part of the question step by step.
### Creating the Graph of the Function
To understand the behavior of the function, we first need to plot the graph of the given data points. Here is a table of the given [tex]\( x \)[/tex] and [tex]\( y \)[/tex] values:
[tex]\[ \begin{tabular}{|c|c|c|c|c|c|c|c|} \hline $x$ & -3 & -2 & -1 & 0 & 1 & 2 & 3 \\ \hline $y$ & 120 & 24 & 0 & -6 & -7.5 & -7.875 & -7.96875 \\ \hline \end{tabular} \][/tex]
We plot these points on a graph and connect them to visualize the function.
### Part A: End Behavior of the Function
End behavior describes what happens to the values of [tex]\( y \)[/tex] (output) as [tex]\( x \)[/tex] (input) approaches positive or negative infinity.
- As [tex]\( x \to -\infty \)[/tex]: Observing the leftmost point where [tex]\( x = -3 \)[/tex], [tex]\( y = 120 \)[/tex]. The trend appears to be increasing sharply. Hence, as [tex]\( x \)[/tex] continues to decrease towards negative infinity, [tex]\( y \)[/tex] seems to approach positive infinity.
- As [tex]\( x \to +\infty \)[/tex]: Observing the rightmost point where [tex]\( x = 3 \)[/tex], [tex]\( y = -7.96875 \)[/tex]. The trend shows that [tex]\( y \)[/tex] decreases and the rate of change diminishes. It suggests that as [tex]\( x \)[/tex] increases to positive infinity, [tex]\( y \)[/tex] continues to decrease to negative infinity.
End Behavior:
- As [tex]\( x \to -\infty \)[/tex], [tex]\( y \to +\infty \)[/tex]
- As [tex]\( x \to +\infty \)[/tex], [tex]\( y \to -\infty \)[/tex]
### Part B: Positive and Negative Intervals
Positive intervals are where the function [tex]\( y \)[/tex] is greater than zero, and negative intervals are where [tex]\( y \)[/tex] is less than zero.
From the table:
- [tex]\( y \)[/tex] is positive for [tex]\( x = -3 \)[/tex] to [tex]\( -1 \)[/tex].
- [tex]\( y \)[/tex] is zero at [tex]\( x = -1 \)[/tex].
- [tex]\( y \)[/tex] is negative for [tex]\( x = -1 \)[/tex] to [tex]\( 3 \)[/tex].
Positive Interval:
- The function is positive for [tex]\( -3 < x < -1 \)[/tex].
Negative Interval:
- The function is negative for [tex]\( -1 < x < 3 \)[/tex].
### Part C: Percent Rate of Change
The percent rate of change between successive points is calculated using the formula:
[tex]\[ \text{Percent Rate of Change} = \left( \frac{y_{i+1} - y_i}{|y_i|} \right) \times 100\% \][/tex]
Let's compute this between each pair of points:
1. Between [tex]\( x = -3 \)[/tex] and [tex]\( x = -2 \)[/tex]:
- [tex]\( y \)[/tex]: from 120 to 24
- [tex]\( \text{Percent Rate of Change} = \left( \frac{24 - 120}{|120|} \right) \times 100\% = -80\% \)[/tex]
2. Between [tex]\( x = -2 \)[/tex] and [tex]\( x = -1 \)[/tex]:
- [tex]\( y \)[/tex]: from 24 to 0
- [tex]\( \text{Percent Rate of Change} = \left( \frac{0 - 24}{|24|} \right) \times 100\% = -100\% \)[/tex]
3. Between [tex]\( x = -1 \)[/tex] and [tex]\( x = 0 \)[/tex]:
- [tex]\( y \)[/tex]: from 0 to -6
- [tex]\( \text{Percent Rate of Change} = \left( \frac{-6 - 0}{|0|} \right) \times 100\% = \text{Undefined} \)[/tex] (cannot divide by zero)
4. Between [tex]\( x = 0 \)[/tex] and [tex]\( x = 1 \)[/tex]:
- [tex]\( y \)[/tex]: from -6 to -7.5
- [tex]\( \text{Percent Rate of Change} = \left( \frac{-7.5 + 6}{|6|} \right) \times 100\% = -25\% \)[/tex]
5. Between [tex]\( x = 1 \)[/tex] and [tex]\( x = 2 \)[/tex]:
- [tex]\( y \)[/tex]: from -7.5 to -7.875
- [tex]\( \text{Percent Rate of Change} = \left( \frac{-7.875 + 7.5}{|7.5|} \right) \times 100\% \approx -5\% \)[/tex]
6. Between [tex]\( x = 2 \)[/tex] and [tex]\( x = 3 \)[/tex]:
- [tex]\( y \)[/tex]: from -7.875 to -7.96875
- [tex]\( \text{Percent Rate of Change} = \left( \frac{-7.96875 + 7.875}{|7.875|} \right) \times 100\% \approx -1.19\% \)[/tex]
Percent Rate of Changes:
- From [tex]\( x = -3 \)[/tex] to [tex]\( x = -2 \)[/tex]: [tex]\(-80\%\)[/tex].
- From [tex]\( x = -2 \)[/tex] to [tex]\( x = -1 \)[/tex]: [tex]\(-100\%\)[/tex].
- From [tex]\( x = -1 \)[/tex] to [tex]\( x = 0 \)[/tex]: [tex]\(\text{Undefined}\)[/tex].
- From [tex]\( x = 0 \)[/tex] to [tex]\( x = 1 \)[/tex]: [tex]\(-25\%\)[/tex].
- From [tex]\( x = 1 \)[/tex] to [tex]\( x = 2 \)[/tex]: [tex]\(-5\%\)[/tex].
- From [tex]\( x = 2 \)[/tex] to [tex]\( x = 3 \)[/tex]: [tex]\(-1.19\%\)[/tex].
### Part D: Domain and Range
Domain: The set of all possible [tex]\( x \)[/tex]-values.
- For the provided data: The domain is [tex]\([-3, 3]\)[/tex].
Range: The set of all possible [tex]\( y \)[/tex]-values.
- For the provided data: The range is [tex]\([-7.96875, 120]\)[/tex].
Summary:
- Domain: [tex]\([-3, 3]\)[/tex]
- Range: [tex]\([-7.96875, 120]\)[/tex]
This detailed analysis addresses each part of the question using the provided data.
### Creating the Graph of the Function
To understand the behavior of the function, we first need to plot the graph of the given data points. Here is a table of the given [tex]\( x \)[/tex] and [tex]\( y \)[/tex] values:
[tex]\[ \begin{tabular}{|c|c|c|c|c|c|c|c|} \hline $x$ & -3 & -2 & -1 & 0 & 1 & 2 & 3 \\ \hline $y$ & 120 & 24 & 0 & -6 & -7.5 & -7.875 & -7.96875 \\ \hline \end{tabular} \][/tex]
We plot these points on a graph and connect them to visualize the function.
### Part A: End Behavior of the Function
End behavior describes what happens to the values of [tex]\( y \)[/tex] (output) as [tex]\( x \)[/tex] (input) approaches positive or negative infinity.
- As [tex]\( x \to -\infty \)[/tex]: Observing the leftmost point where [tex]\( x = -3 \)[/tex], [tex]\( y = 120 \)[/tex]. The trend appears to be increasing sharply. Hence, as [tex]\( x \)[/tex] continues to decrease towards negative infinity, [tex]\( y \)[/tex] seems to approach positive infinity.
- As [tex]\( x \to +\infty \)[/tex]: Observing the rightmost point where [tex]\( x = 3 \)[/tex], [tex]\( y = -7.96875 \)[/tex]. The trend shows that [tex]\( y \)[/tex] decreases and the rate of change diminishes. It suggests that as [tex]\( x \)[/tex] increases to positive infinity, [tex]\( y \)[/tex] continues to decrease to negative infinity.
End Behavior:
- As [tex]\( x \to -\infty \)[/tex], [tex]\( y \to +\infty \)[/tex]
- As [tex]\( x \to +\infty \)[/tex], [tex]\( y \to -\infty \)[/tex]
### Part B: Positive and Negative Intervals
Positive intervals are where the function [tex]\( y \)[/tex] is greater than zero, and negative intervals are where [tex]\( y \)[/tex] is less than zero.
From the table:
- [tex]\( y \)[/tex] is positive for [tex]\( x = -3 \)[/tex] to [tex]\( -1 \)[/tex].
- [tex]\( y \)[/tex] is zero at [tex]\( x = -1 \)[/tex].
- [tex]\( y \)[/tex] is negative for [tex]\( x = -1 \)[/tex] to [tex]\( 3 \)[/tex].
Positive Interval:
- The function is positive for [tex]\( -3 < x < -1 \)[/tex].
Negative Interval:
- The function is negative for [tex]\( -1 < x < 3 \)[/tex].
### Part C: Percent Rate of Change
The percent rate of change between successive points is calculated using the formula:
[tex]\[ \text{Percent Rate of Change} = \left( \frac{y_{i+1} - y_i}{|y_i|} \right) \times 100\% \][/tex]
Let's compute this between each pair of points:
1. Between [tex]\( x = -3 \)[/tex] and [tex]\( x = -2 \)[/tex]:
- [tex]\( y \)[/tex]: from 120 to 24
- [tex]\( \text{Percent Rate of Change} = \left( \frac{24 - 120}{|120|} \right) \times 100\% = -80\% \)[/tex]
2. Between [tex]\( x = -2 \)[/tex] and [tex]\( x = -1 \)[/tex]:
- [tex]\( y \)[/tex]: from 24 to 0
- [tex]\( \text{Percent Rate of Change} = \left( \frac{0 - 24}{|24|} \right) \times 100\% = -100\% \)[/tex]
3. Between [tex]\( x = -1 \)[/tex] and [tex]\( x = 0 \)[/tex]:
- [tex]\( y \)[/tex]: from 0 to -6
- [tex]\( \text{Percent Rate of Change} = \left( \frac{-6 - 0}{|0|} \right) \times 100\% = \text{Undefined} \)[/tex] (cannot divide by zero)
4. Between [tex]\( x = 0 \)[/tex] and [tex]\( x = 1 \)[/tex]:
- [tex]\( y \)[/tex]: from -6 to -7.5
- [tex]\( \text{Percent Rate of Change} = \left( \frac{-7.5 + 6}{|6|} \right) \times 100\% = -25\% \)[/tex]
5. Between [tex]\( x = 1 \)[/tex] and [tex]\( x = 2 \)[/tex]:
- [tex]\( y \)[/tex]: from -7.5 to -7.875
- [tex]\( \text{Percent Rate of Change} = \left( \frac{-7.875 + 7.5}{|7.5|} \right) \times 100\% \approx -5\% \)[/tex]
6. Between [tex]\( x = 2 \)[/tex] and [tex]\( x = 3 \)[/tex]:
- [tex]\( y \)[/tex]: from -7.875 to -7.96875
- [tex]\( \text{Percent Rate of Change} = \left( \frac{-7.96875 + 7.875}{|7.875|} \right) \times 100\% \approx -1.19\% \)[/tex]
Percent Rate of Changes:
- From [tex]\( x = -3 \)[/tex] to [tex]\( x = -2 \)[/tex]: [tex]\(-80\%\)[/tex].
- From [tex]\( x = -2 \)[/tex] to [tex]\( x = -1 \)[/tex]: [tex]\(-100\%\)[/tex].
- From [tex]\( x = -1 \)[/tex] to [tex]\( x = 0 \)[/tex]: [tex]\(\text{Undefined}\)[/tex].
- From [tex]\( x = 0 \)[/tex] to [tex]\( x = 1 \)[/tex]: [tex]\(-25\%\)[/tex].
- From [tex]\( x = 1 \)[/tex] to [tex]\( x = 2 \)[/tex]: [tex]\(-5\%\)[/tex].
- From [tex]\( x = 2 \)[/tex] to [tex]\( x = 3 \)[/tex]: [tex]\(-1.19\%\)[/tex].
### Part D: Domain and Range
Domain: The set of all possible [tex]\( x \)[/tex]-values.
- For the provided data: The domain is [tex]\([-3, 3]\)[/tex].
Range: The set of all possible [tex]\( y \)[/tex]-values.
- For the provided data: The range is [tex]\([-7.96875, 120]\)[/tex].
Summary:
- Domain: [tex]\([-3, 3]\)[/tex]
- Range: [tex]\([-7.96875, 120]\)[/tex]
This detailed analysis addresses each part of the question using the provided data.
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