Discover the answers to your questions at Westonci.ca, where experts share their knowledge and insights with you. Explore a wealth of knowledge from professionals across various disciplines on our comprehensive Q&A platform. Connect with a community of professionals ready to provide precise solutions to your questions quickly and accurately.
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.
We hope you found this helpful. Feel free to come back anytime for more accurate answers and updated information. Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. Thank you for visiting Westonci.ca. Stay informed by coming back for more detailed answers.
