>[!summary]
Functions that are shifted are moved from this original position, but are not deformed.
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Functions that are compressed, stretched or reflected are deformed from their original function.
>
>
> **Summarized Generalization:**
>The following is a summarized generalization of this note
>
>
| Function | Transformation | Mapping |
| -------------- | ------------------------------------------------------------------------------------ | ------------------------ |
| $y = f(x) + k$ | If $k > 0$ translation is up<br>If $k < 0$ translation is down | $(x,y) \mapsto (x,y+k)$ |
| $y = f(x-h)$ | If $h > 0$ the translation is to the right<br>If $h < 0$ the translation is the left | $(x,y) \mapsto (x+h, y)$ |
>
>
| Function | Transformation | Mapping |
| ----------- | ------------------------------------------------------------------ | ------------------------------- |
| $y = -f(x)$ | Reflection of x-axis | $(x,y) \mapsto (x,-y)$ |
| $y = f(-x)$ | Relfection of y-axis | $(x,y) \mapsto (-x,y)$ |
| $y = af(x)$ | Vertical stretch by a factor of \|a\| | $(x,y) \mapsto (x,ay)$ |
| $y = f(bx)$ | Horizontal Stretch by a factor of $\frac{1}{\left \| b \right \|}$ | $(x,y) \mapsto (\frac{x}{b},y)$ |
>[!info]+ Read Time
**⏱ 5 mins**
# Shifts
Function shifts are when a function is shifted up or down. When a function is shifted, the shifted form is not manipulated in a way that deforms it from the original function. Other than it being displaced differently from the original function, it is fundamentally the same.
## Proof
Suppose we have a function f(x) whose function plotted looks like this on a table and graph.
| x | f(x) |
| --- | ---- |
| 1 | 1 |
| 2 | 2 |
| 3 | 3 |
| 4 | 4 |
![[desmos-graph.png]]
>[!warning] Assumptions
Assume we use the same function f(x) (same table outputs)
Now let's affect the function by subtracting the **output** of **f(x)**. Let's subtract by 1 for this example.
| x | f(x) - 1 |
| --- | -------- |
| 1 | 0 |
| 2 | 1 |
| 3 | 2 |
| 4 | 3 |
>[!info] Notice
Notice that when we subtract the output of f(x), the graph is **vertically** affected (The output is affected)
In a graph, this looks like
![[desmos-graph (1).png]]
>[!warning] Assumptions
Assume we use the same function f(x) (same table outputs)
Now let's affect the function **before the output of f(x)**. Let's assume by subtracting 1 before the output.
| x | f(x-h) |
| --- | ------ |
| 1 | 0 |
| 2 | 1 |
| 3 | 2 |
| 4 | 3 |
>[!info] Notice
That affects the input of the function before the output affects the graph horizontally.
![[desmos-graph (2).png]]
## Generalization
| Function | Transformation | Mapping |
| -------------- | ------------------------------------------------------------------------------------ | ------------------------ |
| $y = f(x) + k$ | If $k > 0$ translation is up<br>If $k < 0$ translation is down | $(x,y) \mapsto (x,y+k)$ |
| $y = f(x-h)$ | If $h > 0$ the translation is to the right<br>If $h < 0$ the translation is the left | $(x,y) \mapsto (x+h, y)$ |
# Stretching, Compression & Reflections
When a function is stretched, compressed or reflected, the new function is deformed. It is **physically** different from the original function.
## Proof
Suppose we have a function with an output in the form such as $y = f(x^2)$. Assume this is true for the following parts. Below is a table and graph of this function.
| x | $f(x^2)$ |
| --- | -------- |
| 1 | 1 |
| 2 | 4 |
| 3 | 9 |
| 4 | 16 |
![[desmos-graph (3).png]]
Now, suppose we affect this function by multiplying by a variable or sign, we get the following result and its graph. For this proof, let's change the original function above to **multiplying** our **output by negative 1**.
>[!warning] Notice
Be sure to be aware how that multiplying the input vs output is different, just as changing the shift of the input vs output is different as well.
| x | $-f(x^2)$ |
| --- | --------- |
| 1 | -1 |
| 2 | -4 |
| 3 | -9 |
| 4 | -16 |
![[desmos-graph (4).png]]
Now let's look at our original function again and **multiply our input** **by 1**.
>[!warning] Notice
Be sure to be aware that multiplying the input vs output is different, just as changing the shift of the input vs output is different as well.
| x | $-x^2$ | $f(-x^2)$ |
| --- | ------ | --------- |
| 1 | -1 | 1 |
| 2 | -2 | 4 |
| 3 | -3 | 9 |
| 4 | -4 | 16 |
![[desmos-graph (5).png]]
>[!info] Assumed New original function
For this next proof will assume our original function is $y = f(|x|)$ with the table and graph being below. We do this to better visualize the next part of strenching and compressions.
| x | f(\|x\|) | y |
| --- | -------- | --- |
| -2 | 2 | 2 |
| -1 | 1 | 1 |
| 0 | 0 | 0 |
| 1 | 1 | 1 |
| 2 | 2 | 2 |
![[desmos-graph (6).png]]
Now using our new original function, let's multiply that input by -2.
>[!warning] Notice
An increase in our output compared to the original function, we note this as a compression, because the graphs look more condensed.
| x | f(\|x\|) | f(\|-2x\|) | y |
| --- | -------- | ---------- | --- |
| -2 | 2 | 4 | 4 |
| -1 | 1 | 2 | 2 |
| 0 | 0 | 0 | 0 |
| 1 | 1 | 2 | 2 |
| 2 | 2 | 4 | 4 |
![[desmos-graph (7).png]]
Now using our original function once more. Let's multiply our output by -2. Notice how the two graphs differ.
| x | f(\|x\|) | -2f(\|x\|) | y |
| --- | -------- | ---------- | --- |
| -2 | 2 | -4 | -4 |
| -1 | 1 | -2 | -2 |
| 0 | 0 | 0 | 0 |
| 1 | 1 | -2 | -2 |
| 2 | 2 | -4 | -4 |
![[desmos-graph (8).png]]
## Generalization
| Function | Transformation | Mapping |
| ----------- | ------------------------------------------------------------------ | ------------------------------- |
| $y = -f(x)$ | Reflection of x-axis | $(x,y) \mapsto (x,-y)$ |
| $y = f(-x)$ | Relfection of y-axis | $(x,y) \mapsto (-x,y)$ |
| $y = af(x)$ | Vertical stretch by a factor of \|a\| | $(x,y) \mapsto (x,ay)$ |
| $y = f(bx)$ | Horizontal Stretch by a factor of $\frac{1}{\left \| b \right \|}$ | $(x,y) \mapsto (\frac{x}{b},y)$ |
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