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Sagot :
To determine the transformation applied to the parent function [tex]\( f(x) \)[/tex] to obtain [tex]\( g(x) \)[/tex], we need to analyze the given values of [tex]\( g(x) \)[/tex]:
We have the following table:
[tex]\[ \begin{array}{|c|c|} \hline x & g(x) \\ \hline -2 & -\frac{17}{9} \\ \hline -1 & -\frac{5}{3} \\ \hline 2 & 7 \\ \hline 3 & 25 \\ \hline 4 & 79 \\ \hline \end{array} \][/tex]
We first consider possible transformations such as reflections and stretches, then will match these transformations with the given values of [tex]\( g(x) \)[/tex].
### Step-by-Step Analysis:
#### Vertical Reflection
1. Vertical Reflection: This means [tex]\( g(x) = -f(x) \)[/tex].
- For [tex]\( f(x) = x^2 \)[/tex] (let's hypothesize a simple quadratic function):
- [tex]\( g(x) = -x^2 \)[/tex].
Calculated values:
[tex]\[ \begin{array}{|c|c|} \hline x & -x^2 \\ \hline -2 & -4 \\ \hline -1 & -1 \\ \hline 2 & -4 \\ \hline 3 & -9 \\ \hline 4 & -16 \\ \hline \end{array} \][/tex]
These values do not match with [tex]\( g(x) \)[/tex].
#### Horizontal Reflection
2. Horizontal Reflection: This means [tex]\( g(x) = f(-x) \)[/tex].
- For [tex]\( f(x) = x^2 \)[/tex]:
- [tex]\( g(x) = (-x)^2 = x^2 \)[/tex].
Calculated values:
[tex]\[ \begin{array}{|c|c|} \hline x & x^2 \\ \hline -2 & 4 \\ \hline -1 & 1 \\ \hline 2 & 4 \\ \hline 3 & 9 \\ \hline 4 & 16 \\ \hline \end{array} \][/tex]
These values do not match with [tex]\( g(x) \)[/tex].
#### Both Reflections
3. Both Horizontal and Vertical Reflection: This means [tex]\( g(x) = -f(-x) \)[/tex].
- For [tex]\( f(x) = x^2 \)[/tex]:
- [tex]\( g(x) = -(-x)^2 = -x^2 \)[/tex].
Calculated values:
[tex]\[ \begin{array}{|c|c|} \hline x & -x^2 \\ \hline -2 & -4 \\ \hline -1 & -1 \\ \hline 2 & -4 \\ \hline 3 & -9 \\ \hline 4 & -16 \\ \hline \end{array} \][/tex]
These values do not match with [tex]\( g(x) \)[/tex].
#### Conclusion
We now conclude that the given values of [tex]\( g(x) \)[/tex] do not match the typical simple reflections (vertical, horizontal, or both). Let’s consider a horizontal or vertical stretch:
- Vertical stretch: This means [tex]\( g(x) = a f(x) \)[/tex].
- Horizontal stretch: This means [tex]\( g(x) = f(bx) \)[/tex].
Given values indicate the transformations deviate from simple reflections. Hence, the most reasonable transformation is:
### Final Answer:
C. Horizontal or vertical stretch
This option encompasses transformations that can result in the given complex values.
We have the following table:
[tex]\[ \begin{array}{|c|c|} \hline x & g(x) \\ \hline -2 & -\frac{17}{9} \\ \hline -1 & -\frac{5}{3} \\ \hline 2 & 7 \\ \hline 3 & 25 \\ \hline 4 & 79 \\ \hline \end{array} \][/tex]
We first consider possible transformations such as reflections and stretches, then will match these transformations with the given values of [tex]\( g(x) \)[/tex].
### Step-by-Step Analysis:
#### Vertical Reflection
1. Vertical Reflection: This means [tex]\( g(x) = -f(x) \)[/tex].
- For [tex]\( f(x) = x^2 \)[/tex] (let's hypothesize a simple quadratic function):
- [tex]\( g(x) = -x^2 \)[/tex].
Calculated values:
[tex]\[ \begin{array}{|c|c|} \hline x & -x^2 \\ \hline -2 & -4 \\ \hline -1 & -1 \\ \hline 2 & -4 \\ \hline 3 & -9 \\ \hline 4 & -16 \\ \hline \end{array} \][/tex]
These values do not match with [tex]\( g(x) \)[/tex].
#### Horizontal Reflection
2. Horizontal Reflection: This means [tex]\( g(x) = f(-x) \)[/tex].
- For [tex]\( f(x) = x^2 \)[/tex]:
- [tex]\( g(x) = (-x)^2 = x^2 \)[/tex].
Calculated values:
[tex]\[ \begin{array}{|c|c|} \hline x & x^2 \\ \hline -2 & 4 \\ \hline -1 & 1 \\ \hline 2 & 4 \\ \hline 3 & 9 \\ \hline 4 & 16 \\ \hline \end{array} \][/tex]
These values do not match with [tex]\( g(x) \)[/tex].
#### Both Reflections
3. Both Horizontal and Vertical Reflection: This means [tex]\( g(x) = -f(-x) \)[/tex].
- For [tex]\( f(x) = x^2 \)[/tex]:
- [tex]\( g(x) = -(-x)^2 = -x^2 \)[/tex].
Calculated values:
[tex]\[ \begin{array}{|c|c|} \hline x & -x^2 \\ \hline -2 & -4 \\ \hline -1 & -1 \\ \hline 2 & -4 \\ \hline 3 & -9 \\ \hline 4 & -16 \\ \hline \end{array} \][/tex]
These values do not match with [tex]\( g(x) \)[/tex].
#### Conclusion
We now conclude that the given values of [tex]\( g(x) \)[/tex] do not match the typical simple reflections (vertical, horizontal, or both). Let’s consider a horizontal or vertical stretch:
- Vertical stretch: This means [tex]\( g(x) = a f(x) \)[/tex].
- Horizontal stretch: This means [tex]\( g(x) = f(bx) \)[/tex].
Given values indicate the transformations deviate from simple reflections. Hence, the most reasonable transformation is:
### Final Answer:
C. Horizontal or vertical stretch
This option encompasses transformations that can result in the given complex values.
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