## 1. Stretches

### 1.1. Definition :

### 1.2. Vertical Stretch : A vertical stretch is along the x axis by a factor of lal. When a is between 1 and -1, the function becomes shorter. When b is greater than 1 and -1, the function becomes longer. However, when a is a negative number, it will also cause a reflection along the x axis

1.2.1. Function: y = af(x)

1.2.2. Mapping Notation: (x,y)-->(x,ay)

### 1.3. Horizontal Stretch : A horizontal stretch is along the y axis by a factor of 1 / lbl. When b is between 1 and -1, the function becomes wider. When b is greater than 1 and -1, the function becomes narrower. However, when b is a negative number, it will also cause a reflection along the y axis

1.3.1. Function: y = f(bx)

1.3.2. Mapping Notation: (x,y)-->(x/b,y)

## 2. Inverses

### 2.1. Definition : When f is a function with domain A and range B, the inverse of the function has domain B and range A and is donated by f-1

### 2.2. Function : f=f-1 where -1 does not represent an exponent

### 2.3. Inverses of a Function : done by changing the x-coordinates by the y-coordinates. The graph of the inverse is the graph of the relation reflected in the line y=x. The horizontal line test can determine if the inverse will be a function. If the inverse is not a function, this can be done by restricting the domain

## 3. Combining Transformations

### 3.1. Definition : Combing transformations uses stretches, reflections, and translations and their functions to form a new function

### 3.2. Combined Transformations Functions: y - k = a f[b ( x - h )] or y = a f[b ( x - h )] + k

### 3.3. : Correct Sequence for Translations

3.3.1. 1. Horizontal stretch about the y-axis by a factor of 1/ lbl

3.3.2. 2. Reflection in y axis when b is a negative

3.3.3. 3. Vertical stretch about the x-axis by factor of lal

3.3.4. 4. Reflection in x-axis when a is a neagative

3.3.5. 5. Horizontal and vertical translations of h or k units

## 4. Translations

### 4.1. Definition : A translation can move a function up, down, left, or right but will not change the orientation or shape.

### 4.2. Vertical Translations : Translations up and down are a result of the changes in the k value. In a function, it can be expressed like this: y - k = f(x) or y = f(x) + k where f is the function and k is the vertical translation. For the function y - k = f(x) remember that the signs will be the opposite. When k is greater than 0, the translation is up and when k is a negative number, the translation is down. In mapping notation, it is written as (x,y)-->(x,y+k)

4.2.1. Function: y - k = f(x) or y = f(x) + k

4.2.2. Mapping Notation: (x,y)-->(x,y+k)

### 4.3. Horizontal Translations : Translation right and left are a result of the changes in the h value. In a function, it is expressed like this: y = f ( x - h ) where f is the function and h is the horizontal translation. However, remember that the h value sign will be opposite. When is h is greater than 0, the translation is to the right. When h is less than 0 (a negative number), the translation is to the left

4.3.1. Function: y = f ( x - h )

4.3.2. Mapping Notation: (x,y)-->(x+h,y)

## 5. Reflections

### 5.1. Definition : The reflection of a graph creates a mirror image in a line, also known as the line of reflection. They do not change the shape of the graph but rather the orientation

### 5.2. X-axis Reflection : Occurs when y is a negative number, causing the y values to switch from positive to negative and vice versa. When there is a reflection on the x-axis, it causes the y values to switch from positive to negative or from negative to positive. In the combined function, the x axis reflection happens when a is a negative number

5.2.1. Function: y = -f(x)

5.2.2. Mapping Notation: (x,y)-->(x,-y)

### 5.3. Y-axis Reflection : Occurs when x is a negative number, causing the x values to switch from positive to negative and vice versa. When reflected on the y-axis, it changes the positive x values to negative x values or negative x-values to positive x-values. In the combined function, the y axis reflection happens when b is a negative number

5.3.1. Function: y = f(-x)

5.3.2. Mapping Notation: (x,y)-->(-x,y)