# Section 2.7

## Definitions

### Stretch or Shrink

:::{prf:definition} Vertically Stretch or Shrink
:label: vertStretchShrink
Suppose that $a>0$. If a point $(x,y)$ lies on the graph of $y=f(x)$, then the point $(x,ay)$ lies on the graph of $y=af(x)$.

* If $a>1$, then the graph of $y=af(x)$ is a **vertical stretching** of the graph of $y=f(x)$.
* If $0<a<1$, then the graph of $y=af(x)$ is a **vertical shrinking** of the grapph of $y=f(x)$.
:::

:::{prf:definition} Horizontal Stretch or Shrink
:label: horStretchShrink
Suppose that $a>0$. If a point $(x,y)$ lines on teh graph $y=f(x)$, then the point $(\frac{x}{a},y)$ lies on the graph of $y=f(ax)$.

* If $0<a<1$, then the graph of $y=f(ax)$ is a **horizontal stretching** of the graph of $y=f(x)$.
* If $a>1$, then the graph of $y=f(ax)$ is a **horizontal shrinking** of the graph of $y=f(x)$.
:::

### Reflecting

:::{prf:definition} Reflecting across an Axis
:label: reflectXYAxis
* The graph of $y=-f(x)$ is the same as the graph of $y=f(x)$ reflected across the $x$-axis.
* The graph of $y=f(-x)$ is the same as the graph of $y=f(x)$ reflected across the $y$-axis.
* The graph of $y=-f(-x)$ is the same as the graph of $y=f(x)$ reflected across the line $y=x$.
:::

:::{prf:definition} Symmetry
:label: XYsymmetry
* The graph of an equation is symmetric with respect to the $y$-axis if the replacement of $x$ with $-x$ results in an equivalent equation.
* The graph of an equation is symmetric with respect to the $x$-axis if the replacement of $y$ with $-y$ results in an equivalent equation.
* The graph of an equation is symmetric with respect to the origin if the replacement of both $x$ with $-x$ and $y$ with $-y$ at the same time results in an equivalent equation.
:::

:::{prf:definition} Odd or Even
:label: oddEvenFunctions
* A function $f$ is an **even function** if $f(-x)=f(x)$ for all $x$ in the domain of $f$.
* A function $f$ is an **odd function** if $f(-x)=-f(x)$ for all $x$ in the domain of $f$.
:::

### Vertical or Horizontal Translations

:::{prf:definition} Vertical Translations
:label: vertTrans
Given a function $g$ defined by $g(x)=f(x)+k$, where $k$ is any real number:
* For every point $(x,y)$ on $f$, there will be a point $(x,y+k)$ on the graph of $g$.
* The graph of $g$ will be the same as the graph of $f$, but translated up $k$ units if $k>0$ or down $|k|$ units if $k<0$.
:::

:::{prf:definition} Horizontal Translations
:label: horTrans
Given a function $g$ defined by $g(x)=f(x-h)$, where $h$ is any real number:
* For every point $(x,y)$ on $f$, there will be a point $(x+h,y)$ on the graph of $g$.
* The graph of $g$ will be the same as the graph of $f$, but translated right $h$ units if $h>0$ or right $|h|$ units if $h<0$.
:::

## Examples

::::{prf:example}
:label: sketchGraph1
Draw the graph for $f(x)=\frac{1}{2}(x-1)^2+2$.
:::{dropdown} Solution:
1. The graph we translate is $y=x^2$.
2. The graph will be shrink vertically by a factor of $\frac{1}{2}$.
3. The graph will be translated $1$ unit to the right.
4.  The graph will be translated $2$ units upward.

step-by-step:
1. ![Parent Function](images/sketchGraph1.1.png)
2. ![Vertical shrink](images/sketchGraph1.2.png)
3. ![Translate right](images/sketchGraph1.3.png)
4. ![Translate up](images/sketchGraph1.4.png)
:::
::::
::::{prf:example}
:label: sketchGraph2
Draw the graph of $f(x)=-2\sqrt{x+1}-1$.
:::{dropdown} Solution:
1. The graph we translate is $y=\sqrt{x}$.
2. The graph will be stretched vertically by a factor of $2$.
3. The graph will be reflected across the $x$-axis.
4. The graph will be translated $1$ unit to the left.
5. The graph will be translated $1$ units downward.

step-by-step:
1. ![Parent Function](images/sketchGraph2.1.png)
2. ![Vertical stretch](images/sketchGraph2.2.png)
3. ![Reflect across x](images/sketchGraph2.3.png)
4. ![Translate left](images/sketchGraph2.4.png)
5. ![Translate down](images/sketchGraph2.5.png)
:::
::::

::::{prf:example}
:label: sketchGraph3
Draw the graph of $f(x)=\sqrt{-2x}+1$.
:::{dropdown} Solution:
1. The graph we translate is $y=\sqrt{x}$.
2. The graph will shrink horizontally by a factor of $\frac{1}{2}$.
3. The graph will be reflected across the $y$-axis.
4. The graph will have no translation left or right.
5. The graph will translate 1 unit upward.

step-by-step
1. ![Parent Function](images/sketchGraph2.1.png)
2. ![Horizontal Shrink](images/sketchGraph3.2.png)
3. ![Reflect across y](images/sketchGraph3.3.png)
4. skip
5. ![Translate up](images/sketchGraph3.5.png)
:::
::::