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Sagot :
Sure, let's address each part of the question step by step.
### Part A: End Behavior of the Function
End Behavior describes how the function behaves as [tex]\( x \)[/tex] approaches [tex]\(\infty\)[/tex] (positive infinity) and [tex]\( -\infty \)[/tex] (negative infinity).
As [tex]\( x \to \infty \)[/tex]:
- From the given values, we can see that [tex]\( y \)[/tex] approaches closer to -1, but never goes below -1 (e.g., [tex]\( y = -\frac{8}{9} \)[/tex] is very close to -1).
- Therefore, as [tex]\( x \)[/tex] approaches infinity, [tex]\( y \)[/tex] approaches the horizontal asymptote at [tex]\( y = -1 \)[/tex].
As [tex]\( x \to -\infty \)[/tex]:
- Looking at the table, as [tex]\( x \)[/tex] becomes more negative (e.g., -3), [tex]\( y \)[/tex] increases significantly (e.g., [tex]\( y = 80 \)[/tex]).
- Hence, as [tex]\( x \)[/tex] approaches negative infinity, [tex]\( y \)[/tex] grows larger and approaches infinity.
So, the end behavior is:
- As [tex]\( x \to \infty \)[/tex], [tex]\( y \to -1 \)[/tex].
- As [tex]\( x \to -\infty \)[/tex], [tex]\( y \to \infty \)[/tex].
### Part B: Positive and Negative Intervals
To determine the positive and negative intervals, we need to identify for which values of [tex]\( x \)[/tex] the [tex]\( y \)[/tex] values are positive or negative.
Positive:
- The positive [tex]\( y \)[/tex] values are for [tex]\( x = -3 \)[/tex] (80), [tex]\( x = -2 \)[/tex] (26), [tex]\( x = -1 \)[/tex] (8), and [tex]\( x = 0 \)[/tex] (2).
- Thus, the positive intervals are:
- From [tex]\( x = -3 \)[/tex] to [tex]\( x = -2 \)[/tex]
- From [tex]\( x = -2 \)[/tex] to [tex]\( x = -1 \)[/tex]
- From [tex]\( x = -1 \)[/tex] to [tex]\( x = 0 \)[/tex]
Negative:
- The negative [tex]\( y \)[/tex] values are for [tex]\( x = 1 \)[/tex] (0), [tex]\( x = 2 \)[/tex] (-[tex]\(\frac{2}{3}\)[/tex]), and [tex]\( x = 3 \)[/tex] (-[tex]\(\frac{8}{9}\)[/tex]).
- Thus, the negative intervals are:
- From [tex]\( x = 1 \)[/tex] to [tex]\( x = 2 \)[/tex]
- From [tex]\( x = 2 \)[/tex] to [tex]\( x = 3 \)[/tex]
Therefore,
- Positive intervals: [tex]\((-3, -2)\)[/tex], [tex]\((-2, -1)\)[/tex], [tex]\((-1, 0)\)[/tex]
- Negative intervals: [tex]\((1, 2)\)[/tex], [tex]\((2, 3)\)[/tex]
### Part C: Percent Rate of Change
The percent rate of change between consecutive [tex]\(y\)[/tex] values is calculated using the formula:
[tex]\[ \text{Percent Change} = \frac{\Delta y}{|y_{\text{initial}}|} \times 100\% \][/tex]
Let's calculate these step-by-step:
1. [tex]\( \text{From } x = -3 \text{ to } x = -2 \)[/tex]:
[tex]\[ \frac{26 - 80}{80} \times 100 = \frac{-54}{80} \times 100 = -67.5\% \][/tex]
2. [tex]\( \text{From } x = -2 \text{ to } x = -1 \)[/tex]:
[tex]\[ \frac{8 - 26}{26} \times 100 = \frac{-18}{26} \times 100 \approx -69.23\% \][/tex]
3. [tex]\( \text{From } x = -1 \text{ to } x = 0 \)[/tex]:
[tex]\[ \frac{2 - 8}{8} \times 100 = \frac{-6}{8} \times 100 = -75\% \][/tex]
4. [tex]\( \text{From } x = 0 \text{ to } x = 1 \)[/tex]:
[tex]\[ \frac{0 - 2}{2} \times 100 = \frac{-2}{2} \times 100 = -100\% \][/tex]
5. [tex]\( \text{From } x = 1 \text{ to } x = 2 \)[/tex]:
[tex]\[ \frac{-\frac{2}{3} - 0}{|0|} \text{ is undefined as the initial value is 0}\][/tex]
6. [tex]\( \text{From } x = 2 \text{ to } x = 3 \)[/tex]:
[tex]\[ \frac{-\frac{8}{9} - (-\frac{2}{3})}{|-\frac{2}{3}|} \times 100 = \frac{-\frac{8}{9} + \frac{2}{3}}{\frac{2}{3}} \times 100 = \frac{-\frac{8}{9} + \frac{6}{9}}{\frac{2}{3}} \times 100 = \frac{-\frac{2}{9}}{\frac{2}{3}} \times 100 = \frac{-\frac{2}{9}}{\frac{2}{9}} \times 100 = -22.2\% \][/tex]
Average Percent Change (where applicable):
[tex]\[ \text{Average} = \frac{-67.5 + (-69.23) + (-75) + (-100) + (-22.2)}{5} \approx -68.386\%\][/tex]
### Part D: Domain and Range
Domain:
- The domain is all [tex]\( x \)[/tex]-values for which the function is defined.
- Given the x-values in the table: [tex]\( x = -3, -2, -1, 0, 1, 2, 3 \)[/tex]
- The domain is: [tex]\([-3, 3]\)[/tex]
Range:
- The range consists of all [tex]\( y \)[/tex]-values the function takes.
- Given the y-values in the table: [tex]\( y = 80, 26, 8, 2, 0, -\frac{2}{3}, -\frac{8}{9} \)[/tex]
- The range is: [tex]\([-8/9, 80]\)[/tex]
Thus, summarizing:
- Part A:
- As [tex]\( x \to \infty \)[/tex], [tex]\( y \to -1 \)[/tex]
- As [tex]\( x \to -\infty \)[/tex], [tex]\( y \to \infty \)[/tex]
- Part B:
- Positive intervals: [tex]\((-3, -2)\)[/tex], [tex]\((-2, -1)\)[/tex], [tex]\((-1, 0)\)[/tex]
- Negative intervals: [tex]\((1, 2)\)[/tex], [tex]\((2, 3)\)[/tex]
- Part C:
- Percent changes: -67.5%, -69.23%, -75%, -100%, -22.2%
- Average percent change (where applicable): approximately -68.386%
- Part D:
- Domain: [tex]\([-3, 3]\)[/tex]
- Range: [tex]\([-8/9, 80]\)[/tex]
### Part A: End Behavior of the Function
End Behavior describes how the function behaves as [tex]\( x \)[/tex] approaches [tex]\(\infty\)[/tex] (positive infinity) and [tex]\( -\infty \)[/tex] (negative infinity).
As [tex]\( x \to \infty \)[/tex]:
- From the given values, we can see that [tex]\( y \)[/tex] approaches closer to -1, but never goes below -1 (e.g., [tex]\( y = -\frac{8}{9} \)[/tex] is very close to -1).
- Therefore, as [tex]\( x \)[/tex] approaches infinity, [tex]\( y \)[/tex] approaches the horizontal asymptote at [tex]\( y = -1 \)[/tex].
As [tex]\( x \to -\infty \)[/tex]:
- Looking at the table, as [tex]\( x \)[/tex] becomes more negative (e.g., -3), [tex]\( y \)[/tex] increases significantly (e.g., [tex]\( y = 80 \)[/tex]).
- Hence, as [tex]\( x \)[/tex] approaches negative infinity, [tex]\( y \)[/tex] grows larger and approaches infinity.
So, the end behavior is:
- As [tex]\( x \to \infty \)[/tex], [tex]\( y \to -1 \)[/tex].
- As [tex]\( x \to -\infty \)[/tex], [tex]\( y \to \infty \)[/tex].
### Part B: Positive and Negative Intervals
To determine the positive and negative intervals, we need to identify for which values of [tex]\( x \)[/tex] the [tex]\( y \)[/tex] values are positive or negative.
Positive:
- The positive [tex]\( y \)[/tex] values are for [tex]\( x = -3 \)[/tex] (80), [tex]\( x = -2 \)[/tex] (26), [tex]\( x = -1 \)[/tex] (8), and [tex]\( x = 0 \)[/tex] (2).
- Thus, the positive intervals are:
- From [tex]\( x = -3 \)[/tex] to [tex]\( x = -2 \)[/tex]
- From [tex]\( x = -2 \)[/tex] to [tex]\( x = -1 \)[/tex]
- From [tex]\( x = -1 \)[/tex] to [tex]\( x = 0 \)[/tex]
Negative:
- The negative [tex]\( y \)[/tex] values are for [tex]\( x = 1 \)[/tex] (0), [tex]\( x = 2 \)[/tex] (-[tex]\(\frac{2}{3}\)[/tex]), and [tex]\( x = 3 \)[/tex] (-[tex]\(\frac{8}{9}\)[/tex]).
- Thus, the negative intervals are:
- From [tex]\( x = 1 \)[/tex] to [tex]\( x = 2 \)[/tex]
- From [tex]\( x = 2 \)[/tex] to [tex]\( x = 3 \)[/tex]
Therefore,
- Positive intervals: [tex]\((-3, -2)\)[/tex], [tex]\((-2, -1)\)[/tex], [tex]\((-1, 0)\)[/tex]
- Negative intervals: [tex]\((1, 2)\)[/tex], [tex]\((2, 3)\)[/tex]
### Part C: Percent Rate of Change
The percent rate of change between consecutive [tex]\(y\)[/tex] values is calculated using the formula:
[tex]\[ \text{Percent Change} = \frac{\Delta y}{|y_{\text{initial}}|} \times 100\% \][/tex]
Let's calculate these step-by-step:
1. [tex]\( \text{From } x = -3 \text{ to } x = -2 \)[/tex]:
[tex]\[ \frac{26 - 80}{80} \times 100 = \frac{-54}{80} \times 100 = -67.5\% \][/tex]
2. [tex]\( \text{From } x = -2 \text{ to } x = -1 \)[/tex]:
[tex]\[ \frac{8 - 26}{26} \times 100 = \frac{-18}{26} \times 100 \approx -69.23\% \][/tex]
3. [tex]\( \text{From } x = -1 \text{ to } x = 0 \)[/tex]:
[tex]\[ \frac{2 - 8}{8} \times 100 = \frac{-6}{8} \times 100 = -75\% \][/tex]
4. [tex]\( \text{From } x = 0 \text{ to } x = 1 \)[/tex]:
[tex]\[ \frac{0 - 2}{2} \times 100 = \frac{-2}{2} \times 100 = -100\% \][/tex]
5. [tex]\( \text{From } x = 1 \text{ to } x = 2 \)[/tex]:
[tex]\[ \frac{-\frac{2}{3} - 0}{|0|} \text{ is undefined as the initial value is 0}\][/tex]
6. [tex]\( \text{From } x = 2 \text{ to } x = 3 \)[/tex]:
[tex]\[ \frac{-\frac{8}{9} - (-\frac{2}{3})}{|-\frac{2}{3}|} \times 100 = \frac{-\frac{8}{9} + \frac{2}{3}}{\frac{2}{3}} \times 100 = \frac{-\frac{8}{9} + \frac{6}{9}}{\frac{2}{3}} \times 100 = \frac{-\frac{2}{9}}{\frac{2}{3}} \times 100 = \frac{-\frac{2}{9}}{\frac{2}{9}} \times 100 = -22.2\% \][/tex]
Average Percent Change (where applicable):
[tex]\[ \text{Average} = \frac{-67.5 + (-69.23) + (-75) + (-100) + (-22.2)}{5} \approx -68.386\%\][/tex]
### Part D: Domain and Range
Domain:
- The domain is all [tex]\( x \)[/tex]-values for which the function is defined.
- Given the x-values in the table: [tex]\( x = -3, -2, -1, 0, 1, 2, 3 \)[/tex]
- The domain is: [tex]\([-3, 3]\)[/tex]
Range:
- The range consists of all [tex]\( y \)[/tex]-values the function takes.
- Given the y-values in the table: [tex]\( y = 80, 26, 8, 2, 0, -\frac{2}{3}, -\frac{8}{9} \)[/tex]
- The range is: [tex]\([-8/9, 80]\)[/tex]
Thus, summarizing:
- Part A:
- As [tex]\( x \to \infty \)[/tex], [tex]\( y \to -1 \)[/tex]
- As [tex]\( x \to -\infty \)[/tex], [tex]\( y \to \infty \)[/tex]
- Part B:
- Positive intervals: [tex]\((-3, -2)\)[/tex], [tex]\((-2, -1)\)[/tex], [tex]\((-1, 0)\)[/tex]
- Negative intervals: [tex]\((1, 2)\)[/tex], [tex]\((2, 3)\)[/tex]
- Part C:
- Percent changes: -67.5%, -69.23%, -75%, -100%, -22.2%
- Average percent change (where applicable): approximately -68.386%
- Part D:
- Domain: [tex]\([-3, 3]\)[/tex]
- Range: [tex]\([-8/9, 80]\)[/tex]
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