Answer :
Certainly! Let's tackle each of these transformations one by one and analyze their effects on the polynomial function [tex]\( f(x) \)[/tex].
### 1. When [tex]\( f(x) \)[/tex] becomes [tex]\( f(x) - 1 \)[/tex]
#### Y-intercept:
The [tex]\( y \)[/tex]-intercept of a function is the value of the function at [tex]\( x = 0 \)[/tex]. For the original function [tex]\( f(x) \)[/tex], the [tex]\( y \)[/tex]-intercept is [tex]\( f(0) \)[/tex].
When [tex]\( f(x) \)[/tex] becomes [tex]\( f(x) - 1 \)[/tex], the new [tex]\( y \)[/tex]-intercept will be:
[tex]\[ f(0) - 1 \][/tex]
This means the [tex]\( y \)[/tex]-intercept decreases by 1 unit.
#### Regions where the graph is increasing or decreasing:
Shifting the graph of [tex]\( f(x) \)[/tex] down by 1 unit does not change the regions where the graph is increasing or decreasing. The slopes of the tangents at any point on the graph remain the same because the transformation is purely vertical.
#### End behavior:
The end behavior of the polynomial function remains unchanged with a vertical shift. As [tex]\( x \)[/tex] approaches infinity or negative infinity, the polynomial function continues to dominate, and the overall shape of the graph doesn't change except for being shifted 1 unit downward.
### Summary for [tex]\( f(x) - 1 \)[/tex]:
- Y-intercept: Decreases by 1 unit.
- Increasing/Decreasing behavior: Remains unchanged.
- End behavior: Remains unchanged.
### 2. When [tex]\( f(x) \)[/tex] becomes [tex]\( -f(x) + 1 \)[/tex]
#### Y-intercept:
For the original function [tex]\( f(x) \)[/tex], the [tex]\( y \)[/tex]-intercept is [tex]\( f(0) \)[/tex].
When [tex]\( f(x) \)[/tex] becomes [tex]\( -f(x) + 1 \)[/tex], the new [tex]\( y \)[/tex]-intercept will be:
[tex]\[ -(f(0)) + 1 \][/tex]
This means the new [tex]\( y \)[/tex]-intercept is the negation of the original intercept plus 1.
#### Regions where the graph is increasing or decreasing:
Negating the function [tex]\( f(x) \)[/tex] will flip the graph over the [tex]\( x \)[/tex]-axis. Therefore:
- Regions where the graph was increasing will now be decreasing.
- Regions where the graph was decreasing will now be increasing.
Adding the constant 1 will shift the entire graph up by 1 unit, but this does not affect the increasing or decreasing nature of the function, only their respective positions.
#### End behavior:
- For even functions: Even functions are symmetric about the [tex]\( y \)[/tex]-axis. When you reflect an even function about the [tex]\( x \)[/tex]-axis, the end behavior at both infinity and negative infinity will be the same but reflected about the [tex]\( x \)[/tex]-axis. Adding 1 will just shift the entire graph up by 1 unit.
- For odd functions: Odd functions have rotational symmetry about the origin. Reflecting an odd function about the [tex]\( x \)[/tex]-axis inverts its end behavior, and adding 1 will again just shift the entire graph up by 1 unit.
### Summary for [tex]\( -f(x) + 1 \)[/tex]:
- Y-intercept: Becomes the negation of the original intercept plus 1.
- Increasing/Decreasing behavior: Increasing regions become decreasing, and vice versa.
- End behavior:
- For even functions: Reflected about the [tex]\( x \)[/tex]-axis but remain the same in terms of shape.
- For odd functions: Reflected about the [tex]\( x \)[/tex]-axis.
By understanding these transformations, you can predict how the graph of a polynomial function will be altered with respect to its y-intercept, regions of increasing and decreasing, and its end behavior.
### 1. When [tex]\( f(x) \)[/tex] becomes [tex]\( f(x) - 1 \)[/tex]
#### Y-intercept:
The [tex]\( y \)[/tex]-intercept of a function is the value of the function at [tex]\( x = 0 \)[/tex]. For the original function [tex]\( f(x) \)[/tex], the [tex]\( y \)[/tex]-intercept is [tex]\( f(0) \)[/tex].
When [tex]\( f(x) \)[/tex] becomes [tex]\( f(x) - 1 \)[/tex], the new [tex]\( y \)[/tex]-intercept will be:
[tex]\[ f(0) - 1 \][/tex]
This means the [tex]\( y \)[/tex]-intercept decreases by 1 unit.
#### Regions where the graph is increasing or decreasing:
Shifting the graph of [tex]\( f(x) \)[/tex] down by 1 unit does not change the regions where the graph is increasing or decreasing. The slopes of the tangents at any point on the graph remain the same because the transformation is purely vertical.
#### End behavior:
The end behavior of the polynomial function remains unchanged with a vertical shift. As [tex]\( x \)[/tex] approaches infinity or negative infinity, the polynomial function continues to dominate, and the overall shape of the graph doesn't change except for being shifted 1 unit downward.
### Summary for [tex]\( f(x) - 1 \)[/tex]:
- Y-intercept: Decreases by 1 unit.
- Increasing/Decreasing behavior: Remains unchanged.
- End behavior: Remains unchanged.
### 2. When [tex]\( f(x) \)[/tex] becomes [tex]\( -f(x) + 1 \)[/tex]
#### Y-intercept:
For the original function [tex]\( f(x) \)[/tex], the [tex]\( y \)[/tex]-intercept is [tex]\( f(0) \)[/tex].
When [tex]\( f(x) \)[/tex] becomes [tex]\( -f(x) + 1 \)[/tex], the new [tex]\( y \)[/tex]-intercept will be:
[tex]\[ -(f(0)) + 1 \][/tex]
This means the new [tex]\( y \)[/tex]-intercept is the negation of the original intercept plus 1.
#### Regions where the graph is increasing or decreasing:
Negating the function [tex]\( f(x) \)[/tex] will flip the graph over the [tex]\( x \)[/tex]-axis. Therefore:
- Regions where the graph was increasing will now be decreasing.
- Regions where the graph was decreasing will now be increasing.
Adding the constant 1 will shift the entire graph up by 1 unit, but this does not affect the increasing or decreasing nature of the function, only their respective positions.
#### End behavior:
- For even functions: Even functions are symmetric about the [tex]\( y \)[/tex]-axis. When you reflect an even function about the [tex]\( x \)[/tex]-axis, the end behavior at both infinity and negative infinity will be the same but reflected about the [tex]\( x \)[/tex]-axis. Adding 1 will just shift the entire graph up by 1 unit.
- For odd functions: Odd functions have rotational symmetry about the origin. Reflecting an odd function about the [tex]\( x \)[/tex]-axis inverts its end behavior, and adding 1 will again just shift the entire graph up by 1 unit.
### Summary for [tex]\( -f(x) + 1 \)[/tex]:
- Y-intercept: Becomes the negation of the original intercept plus 1.
- Increasing/Decreasing behavior: Increasing regions become decreasing, and vice versa.
- End behavior:
- For even functions: Reflected about the [tex]\( x \)[/tex]-axis but remain the same in terms of shape.
- For odd functions: Reflected about the [tex]\( x \)[/tex]-axis.
By understanding these transformations, you can predict how the graph of a polynomial function will be altered with respect to its y-intercept, regions of increasing and decreasing, and its end behavior.
