Answer :
Sure, let's walk through the transformation of the function [tex]\( f(x) = 2 + x^2 \)[/tex] to [tex]\( j(x) = 2 + \left(\frac{1}{2} x\right)^2 \)[/tex].
### Step-by-Step Solution
1. Identify the original function:
The original function is given by:
[tex]\[ f(x) = 2 + x^2 \][/tex]
2. Describe the transformation:
The transformation is applied inside the squared term of the function. In [tex]\( j(x) = 2 + \left(\frac{1}{2} x\right)^2 \)[/tex], the term [tex]\( x \)[/tex] inside the square is replaced by [tex]\( \frac{1}{2} x \)[/tex].
3. Apply the transformation:
To transform [tex]\( f(x) \)[/tex] to [tex]\( j(x) \)[/tex], replace [tex]\( x \)[/tex] inside the square with [tex]\( \frac{1}{2} x \)[/tex]:
[tex]\[ j(x) = 2 + \left(\frac{1}{2} x\right)^2 \][/tex]
4. Simplify the transformed function:
Simplify the square of the term inside the function:
[tex]\[ j(x) = 2 + \left(\frac{1}{2} x\right)^2 = 2 + \left(\frac{x}{2}\right)^2 = 2 + \frac{x^2}{4} \][/tex]
So, the transformed function [tex]\( j(x) \)[/tex] is:
[tex]\[ j(x) = 2 + \frac{x^2}{4} \][/tex]
5. Evaluate at example values:
To understand how the values of [tex]\( f(x) \)[/tex] compare to [tex]\( j(x) \)[/tex], let's evaluate them at specific [tex]\( x \)[/tex]-values.
Consider the interval [tex]\( x \)[/tex] ranging from -10 to 10. Here's how they look:
- [tex]\( x \)[/tex] values:
```
[-10.0, -9.7979798, -9.5959596, -9.39393939, -9.19191919, ... , 9.39393939, 9.5959596, 9.7979798, 10.0]
```
- For each [tex]\( x \)[/tex] value, calculate [tex]\( f(x) \)[/tex] and [tex]\( j(x) \)[/tex]:
- [tex]\( f(x) \)[/tex] values:
```
[102.0, 98.00040812162024, 94.08244056728904, 90.24609733700643, 86.49137843077237, ... , 94.08244056728904, 98.00040812162024, 102.0]
```
- [tex]\( j(x) \)[/tex] values:
```
[27.0, 26.00010203040506, 25.02061014182226, 24.061524334251608, 23.122844607693093, ... , 25.02061014182226, 26.00010203040506, 27.0]
```
### Graphical Representation
To better visualize how [tex]\( f(x) \)[/tex] and [tex]\( j(x) \)[/tex] compare:
- [tex]\( f(x) = 2 + x^2 \)[/tex] is a parabola opening upwards with a vertex at (0,2).
- [tex]\( j(x) = 2 + \left( \frac{1}{2} x \right)^2 = 2 + \frac{x^2}{4} \)[/tex] is also a parabola opening upwards, but it is wider and grows more slowly than [tex]\( f(x) \)[/tex] because the [tex]\( \frac{x^2}{4} \)[/tex] term increases more slowly than [tex]\( x^2 \)[/tex].
### Summary
- We applied a horizontal scaling transformation to the function [tex]\( f(x) \)[/tex] to obtain [tex]\( j(x) \)[/tex].
- [tex]\( j(x) \)[/tex] is a wider version of [tex]\( f(x) \)[/tex], which means for the same [tex]\( x \)[/tex]-values, [tex]\( j(x) \)[/tex] will have lower values compared to [tex]\( f(x) \)[/tex].
- This is visually supported by evaluating the functions over a range of [tex]\( x \)[/tex]-values and comparing [tex]\( f(x) \)[/tex] and [tex]\( j(x) \)[/tex].
By comparing these values, we can see how the shape of the graph changes with the transformation applied inside the function.
### Step-by-Step Solution
1. Identify the original function:
The original function is given by:
[tex]\[ f(x) = 2 + x^2 \][/tex]
2. Describe the transformation:
The transformation is applied inside the squared term of the function. In [tex]\( j(x) = 2 + \left(\frac{1}{2} x\right)^2 \)[/tex], the term [tex]\( x \)[/tex] inside the square is replaced by [tex]\( \frac{1}{2} x \)[/tex].
3. Apply the transformation:
To transform [tex]\( f(x) \)[/tex] to [tex]\( j(x) \)[/tex], replace [tex]\( x \)[/tex] inside the square with [tex]\( \frac{1}{2} x \)[/tex]:
[tex]\[ j(x) = 2 + \left(\frac{1}{2} x\right)^2 \][/tex]
4. Simplify the transformed function:
Simplify the square of the term inside the function:
[tex]\[ j(x) = 2 + \left(\frac{1}{2} x\right)^2 = 2 + \left(\frac{x}{2}\right)^2 = 2 + \frac{x^2}{4} \][/tex]
So, the transformed function [tex]\( j(x) \)[/tex] is:
[tex]\[ j(x) = 2 + \frac{x^2}{4} \][/tex]
5. Evaluate at example values:
To understand how the values of [tex]\( f(x) \)[/tex] compare to [tex]\( j(x) \)[/tex], let's evaluate them at specific [tex]\( x \)[/tex]-values.
Consider the interval [tex]\( x \)[/tex] ranging from -10 to 10. Here's how they look:
- [tex]\( x \)[/tex] values:
```
[-10.0, -9.7979798, -9.5959596, -9.39393939, -9.19191919, ... , 9.39393939, 9.5959596, 9.7979798, 10.0]
```
- For each [tex]\( x \)[/tex] value, calculate [tex]\( f(x) \)[/tex] and [tex]\( j(x) \)[/tex]:
- [tex]\( f(x) \)[/tex] values:
```
[102.0, 98.00040812162024, 94.08244056728904, 90.24609733700643, 86.49137843077237, ... , 94.08244056728904, 98.00040812162024, 102.0]
```
- [tex]\( j(x) \)[/tex] values:
```
[27.0, 26.00010203040506, 25.02061014182226, 24.061524334251608, 23.122844607693093, ... , 25.02061014182226, 26.00010203040506, 27.0]
```
### Graphical Representation
To better visualize how [tex]\( f(x) \)[/tex] and [tex]\( j(x) \)[/tex] compare:
- [tex]\( f(x) = 2 + x^2 \)[/tex] is a parabola opening upwards with a vertex at (0,2).
- [tex]\( j(x) = 2 + \left( \frac{1}{2} x \right)^2 = 2 + \frac{x^2}{4} \)[/tex] is also a parabola opening upwards, but it is wider and grows more slowly than [tex]\( f(x) \)[/tex] because the [tex]\( \frac{x^2}{4} \)[/tex] term increases more slowly than [tex]\( x^2 \)[/tex].
### Summary
- We applied a horizontal scaling transformation to the function [tex]\( f(x) \)[/tex] to obtain [tex]\( j(x) \)[/tex].
- [tex]\( j(x) \)[/tex] is a wider version of [tex]\( f(x) \)[/tex], which means for the same [tex]\( x \)[/tex]-values, [tex]\( j(x) \)[/tex] will have lower values compared to [tex]\( f(x) \)[/tex].
- This is visually supported by evaluating the functions over a range of [tex]\( x \)[/tex]-values and comparing [tex]\( f(x) \)[/tex] and [tex]\( j(x) \)[/tex].
By comparing these values, we can see how the shape of the graph changes with the transformation applied inside the function.
