Pre Calculus 12 - Chapter 1

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)

6. Vocabulary

6.1. Transformations: change made to a figure or a relation such that the figure or the graph of the relation is shifted or changed in shape

6.2. Mapping: relating one set of points or another set of points so they correspond exactly. In mapping notation it is shown as (x,y)-->(x,y+4)

6.3. Translation:A slide transformation that results in a shift of the original figure without changing its shape. Vertical and horizontal translations are types of transformations with the equations of the forms y-k=f(x) and y=f(x-h) respectively. H representing horizontal translations, k representing vertical translations, and f representing the function

6.4. Image Point: the point that is the result of a transformation of a point on the original graph

6.5. Reflection: a transformation where each point of the original graph has an image point resulting from a reflection of a line

6.6. Invariant Point: a point on the graph that remains unchanged after a transformation

6.7. Stretch: a transformation in which the distance of each x-coordinate or y- coordinate from the line of reflection is multiplied by some scale factor. Scale factors between 0 and 1 result in the point moving closer to the line of reflection; scale factors greater than 1 result in the point moving farther away from the line of reflection

6.8. Inverse of a Function: If f is a function with domain A and range B, the inverse of a function, if it exists, is denoted by f-1 and has domain B and range S. f-1 maps y to x if and only if f maps x to y

6.9. Horizontal line test: A test used to determine if an inverse if a function. If it is possible for a horizontal line to intersect the graph of a relation more than once, the the inverse of the relation is not a function